text: in Ni content and overlap the range of diogenites where mdark is obtained through Eq. 6. and polymict eucrites. According to Warren et i 2.5. Fe and siderophile elements al. (2009), Ni content in howardites varies in the − − − As the Gamma Ray and Neutron Detector range 10 5000 µg/g, i.e. 10 3 0.5 wt%. If we (GRaND) on-board the Dawn spacecraft mea- assume, based on the values reported by Warren sured the global Fe content of minerals on the sur- et al. (2009), an average Ni content of 1 µg/g for face of Vesta (Prettyman et al., 2012; Yamashita eucrites and 40 µg/g for diogenites, th∼e resulting et al., 2013), we used our model to assess what a 2:1 mixture will have a Ni content of 14 µg/g. continuous ﬂux of impactors would imply from the This is also the order of magnitude of the Ni con- point of view of the contamination of the vestan tent of the more Ni-poor samples of howardites regolith by Fe and siderophile elements. Concern- analyzed by Warren et al. (2009) and will be used ing the latter, we focused our analysis on the role in the following as our native Ni content of the of Ni, which has been suggested by Warren et al. vestan crust. (2009) to play an important role as tracer of the Contrary to HED meteorites, the order of regolith’s maturity and enrichment. magnitude of the Fe and Ni contents of chon- In contrast to the cases of the dark material and drites (both ordinary and carbonaceous) is much the water-equivalent material discussed in Sect. less variable. Based on the values reported by g, the howarditic material composing most of the Jarosewich (1990), the average Fe and Ni contents vestan regolith (De Sanctis et al., 2012b; Pretty- of all (both dark and non-dark) impactors in our man et al., 2012) would contain amounts of Fe model can be assumed to be respectively 20 wt% and Ni, even if uncontaminated. The mean Fe (i.e. FFe = 0.2) and 1 wt% (i.e. FNi = 0.01). The contents for diogenites and basaltic eucrites are amount of Fe delivered on Vesta by impactors of respectively 13 wt% and 14.7 wt% (Jarosewich a given size Di over a temporal interval ∆T can 1990; Prettyman et al. 2012, Supplementary Ma- then be expressed as terial). A 2:1 eucrite-diogenite mixture, sug- mdark × gested to be the average compositional mixture mFe = i F (8) i F Fe of howardites (Warren et al., 2009), would have d an average Fe content of 14.1 wt%, which is con- where mdark is obtained through Eq. 6 and is sistent with the mean Fe content of 13.8 wt% i divided by the value F to obtain the total mass of howardites (Jarosewich 1990; Prettyman et al. d delivered by dark and non-dark impactors. A sim- 2012, Supplementary Material). In order to be ilar equation holds for the case of Ni. able to compare our results with the measure- 2.6. Crater saturation and ejecta blanketing ments of GRaND, we will follow Yamashita et al. (2013) and assume that the native Fe content of The eﬀects of crater saturation and of the blan- the vestan regolith is 13.8 wt%. keting due to crater ejecta are two important fac- The Ni content of the diﬀerent materials com- tors that must be considered in discussing the con- posing the vestan crust spans a greater range of tamination of Vesta. The estimates of the extent values than Fe. The Ni content of diogenites of the vestan surface aﬀected by craters and of − × − − varies between 5 200 µg/g, i.e. 5 10 4 0.02 the level of crater saturation reached over a given wt% (Warren et al., 2009). Monomict and cu- timespan allow to assess how long the dark ma- mulate eucrites have a lower Ni content, ranging terial can survive on the surface of Vesta before − between 0.1 10 µg/g (Warren et al., 2009), i.e. being removed by the impacts of non-dark aster- − − − 10 5 10 3 wt%. Polymict eucrites have a Ni oids and whether we should expect to be able content similar to that of diogenites, with one ex- to link most dark material to speciﬁc cratering ceptional polylmict eucrite sample as rich in Ni as events or not. The assessment of the degree to ∼ 1000 µg/g, i.e. 0.1 wt% (Warren et al., 2009). which crater ejecta cover the surface of the aster- Howardites are characterized by a large variability oid, on the other hand, would supply information 9 ##LTLine##

text: in

text: Ni

text: content

text: and

text: overlap

text: the

text: range

text: of

text: diogenites

text: where

text: mdark

text: is

text: obtained

text: through

text: Eq.

text: and

text: polymict

text: eucrites.

text: According

text: to

text: Warren

text: et

text: i

text: Fe

text: and

text: siderophile

text: elements

text: al.

text: Ni

text: content

text: in

text: howardites

text: varies

text: in

text: the

text: As

text: the

text: Gamma

text: Ray

text: and

text: Neutron

text: Detector

text: range

text: µg/g,

text: i.e.

text: wt%.

text: If

text: we

text: (GRaND)

text: on-board

text: the

text: Dawn

text: spacecraft

text: mea-

text: assume,

text: based

text: on

text: the

text: values

text: reported

text: by

text: Warren

text: sured

text: the

text: global

text: Fe

text: content

text: of

text: minerals

text: on

text: the

text: sur-

text: et

text: al.

text: an

text: average

text: Ni

text: content

text: of

text: µg/g

text: for

text: face

text: of

text: Vesta

text: (Prettyman

text: et

text: al.,

text: Yamashita

text: eucrites

text: and

text: µg/g

text: for

text: diogenites,

text: th∼e

text: resulting

text: et

text: al.,

text: we

text: used

text: our

text: model

text: to

text: assess

text: what

text: a

text: mixture

text: will

text: have

text: a

text: Ni

text: content

text: of

text: µg/g.

text: continuous

text: ﬂux

text: of

text: impactors

text: would

text: imply

text: from

text: the

text: This

text: is

text: also

text: the

text: order

text: of

text: magnitude

text: of

text: the

text: Ni

text: con-

text: point

text: of

text: view

text: of

text: the

text: contamination

text: of

text: the

text: vestan

text: tent

text: of

text: the

text: more

text: Ni-poor

text: samples

text: of

text: howardites

text: regolith

text: by

text: Fe

text: and

text: siderophile

text: elements.

text: Concern-

text: analyzed

text: by

text: Warren

text: et

text: al.

text: and

text: will

text: be

text: used

text: ing

text: the

text: latter,

text: we

text: focused

text: our

text: analysis

text: on

text: the

text: role

text: in

text: the

text: following

text: as

text: our

text: native

text: Ni

text: content

text: of

text: the

text: of

text: Ni,

text: which

text: has

text: been

text: suggested

text: by

text: Warren

text: et

text: al.

text: vestan

text: crust.

text: to

text: play

text: an

text: important

text: role

text: as

text: tracer

text: of

text: the

text: Contrary

text: to

text: HED

text: meteorites,

text: the

text: order

text: of

text: regolith’s

text: maturity

text: and

text: enrichment.

text: magnitude

text: of

text: the

text: Fe

text: and

text: Ni

text: contents

text: of

text: chon-

text: In

text: contrast

text: to

text: the

text: cases

text: of

text: the

text: dark

text: material

text: and

text: drites

text: (both

text: ordinary

text: and

text: carbonaceous)

text: is

text: much

text: the

text: water-equivalent

text: material

text: discussed

text: in

text: Sect.

text: less

text: variable.

text: Based

text: on

text: the

text: values

text: reported

text: by

text: g,

text: the

text: howarditic

text: material

text: composing

text: most

text: of

text: the

text: Jarosewich

text: the

text: average

text: Fe

text: and

text: Ni

text: contents

text: vestan

text: regolith

text: (De

text: Sanctis

text: et

text: al.,

text: Pretty-

text: of

text: all

text: (both

text: dark

text: and

text: non-dark)

text: impactors

text: in

text: our

text: man

text: et

text: al.,

text: would

text: contain

text: amounts

text: of

text: Fe

text: model

text: can

text: be

text: assumed

text: to

text: be

text: respectively

text: wt%

text: and

text: Ni,

text: even

text: if

text: uncontaminated.

text: The

text: mean

text: Fe

text: (i.e.

text: FFe

text: and

text: wt%

text: (i.e.

text: FNi

text: The

text: contents

text: for

text: diogenites

text: and

text: basaltic

text: eucrites

text: are

text: amount

text: of

text: Fe

text: delivered

text: on

text: Vesta

text: by

text: impactors

text: of

text: respectively

text: wt%

text: and

text: wt%

text: (Jarosewich

text: a

text: given

text: size

text: Di

text: over

text: a

text: temporal

text: interval

text: ∆T

text: can

text: Prettyman

text: et

text: al.

text: Supplementary

text: Ma-

text: then

text: be

text: expressed

text: as

text: terial).

text: A

text: eucrite-diogenite

text: mixture,

text: sug-

text: mdark

text: gested

text: to

text: be

text: the

text: average

text: compositional

text: mixture

text: mFe

text: i

text: F

text: i

text: F

text: Fe

text: of

text: howardites

text: (Warren

text: et

text: al.,

text: would

text: have

text: d

text: an

text: average

text: Fe

text: content

text: of

text: wt%,

text: which

text: is

text: con-

text: where

text: mdark

text: is

text: obtained

text: through

text: Eq.

text: and

text: is

text: sistent

text: with

text: the

text: mean

text: Fe

text: content

text: of

text: wt%

text: i

text: divided

text: by

text: the

text: value

text: F

text: to

text: obtain

text: the

text: total

text: mass

text: of

text: howardites

text: (Jarosewich

text: Prettyman

text: et

text: al.

text: d

text: delivered

text: by

text: dark

text: and

text: non-dark

text: impactors.

text: A

text: sim-

text: Supplementary

text: Material).

text: In

text: order

text: to

text: be

text: ilar

text: equation

text: holds

text: for

text: the

text: case

text: of

text: Ni.

text: able

text: to

text: compare

text: our

text: results

text: with

text: the

text: measure-

text: Crater

text: saturation

text: and

text: ejecta

text: blanketing

text: ments

text: of

text: GRaND,

text: we

text: will

text: follow

text: Yamashita

text: et

text: al.

text: and

text: assume

text: that

text: the

text: native

text: Fe

text: content

text: of

text: The

text: eﬀects

text: of

text: crater

text: saturation

text: and

text: of

text: the

text: blan-

text: the

text: vestan

text: regolith

text: is

text: wt%.

text: keting

text: due

text: to

text: crater

text: ejecta

text: are

text: two

text: important

text: fac-

text: The

text: Ni

text: content

text: of

text: the

text: diﬀerent

text: materials

text: com-

text: tors

text: that

text: must

text: be

text: considered

text: in

text: discussing

text: the

text: con-

text: posing

text: the

text: vestan

text: crust

text: spans

text: a

text: greater

text: range

text: of

text: tamination

text: of

text: Vesta.

text: The

text: estimates

text: of

text: the

text: extent

text: values

text: than

text: Fe.

text: The

text: Ni

text: content

text: of

text: diogenites

text: of

text: the

text: vestan

text: surface

text: aﬀected

text: by

text: craters

text: and

text: of

text: varies

text: between

text: µg/g,

text: i.e.

text: the

text: level

text: of

text: crater

text: saturation

text: reached

text: over

text: a

text: given

text: wt%

text: (Warren

text: et

text: al.,

text: Monomict

text: and

text: cu-

text: timespan

text: allow

text: to

text: assess

text: how

text: long

text: the

text: dark

text: ma-

text: mulate

text: eucrites

text: have

text: a

text: lower

text: Ni

text: content,

text: ranging

text: terial

text: can

text: survive

text: on

text: the

text: surface

text: of

text: Vesta

text: before

text: between

text: µg/g

text: (Warren

text: et

text: al.,

text: i.e.

text: being

text: removed

text: by

text: the

text: impacts

text: of

text: non-dark

text: aster-

text: wt%.

text: Polymict

text: eucrites

text: have

text: a

text: Ni

text: oids

text: and

text: whether

text: we

text: should

text: expect

text: to

text: be

text: able

text: content

text: similar

text: to

text: that

text: of

text: diogenites,

text: with

text: one

text: ex-

text: to

text: link

text: most

text: dark

text: material

text: to

text: speciﬁc

text: cratering

text: ceptional

text: polylmict

text: eucrite

text: sample

text: as

text: rich

text: in

text: Ni

text: as

text: events

text: or

text: not.

text: The

text: assessment

text: of

text: the

text: degree

text: to

text: µg/g,

text: i.e.

text: wt%

text: (Warren

text: et

text: al.,

text: which

text: crater

text: ejecta

text: cover

text: the

text: surface

text: of

text: the

text: aster-

text: Howardites

text: are

text: characterized

text: by

text: a

text: large

text: variability

text: oid,

text: on

text: the

text: other

text: hand,

text: would

text: supply

text: information

text: ##LTLine##


text: 6 e e µ V µ+ V gA µ+ γ g − g gA µ− µ− V A γ W V p p ν FIG. 4. Typical representatives of muon pair production by → electron-proton collision due to a new force. The parity vio- FIG. 5. Diagram that leads to the decay W µνV . lation will come about due to the presence of the g coupling A in the interference with the pure QED diagrams. parameter space, one should expect a very strong con- straint on the model. Carrying out explicit calculation diagrams, leading to an asymmetry in the muon pair- in the limit of g (cid:29) g , as implied by (g − 2) , and to production cross section by the longitudinally polarized V A µ leading order in m /m and m /m , we arrive at electrons, V W µ W σσLL −+ σσRR ∝ κ(gA/e). (19) Γ (W → µνV ) = 512g√V22π3 G(cid:16) Fmm2V5W(cid:17) (cid:18) (cid:19) (20) g 2 10 MeV 2 = 1.74 GeV V− . While the rate for such a process is rather low, new 10 2 m V high intensity polarized electron beam facilities can con- ceivably be used to search for such an eﬀect. Because of the prompt decay of V to an electron- 5. HIGH-ENERGY CONSTRAINTS positron pair, and the small value of m , this decay will V be similar to W → µνγ. In any case, the additional So far we have dealt with the low-energy observables channel leads to the increase of the total W width. The that are only mildly sensitive to the fact that the eﬀec- contribution in Eq. (20) should be compared against the tive Lagrangian (4) at a generic point in {gV , gA} pa- current experimental value for the W width, dominated rameter space does not respect the full gauge invariance by measurements at the Tevatron [34], of the SM. Speciﬁcally, we have insisted that the muon neutrinos are uncharged under the new force, due to the fact that their interactions are well-known and do not Γ = 2.085 ± 0.042 GeV. (21) W have any room for O(G ) new physics eﬀects, let alone F stronger-than-G eﬀects as suggested by the r discrep- F p ancy. We also do not assume any direct coupling of V Given the agreement of this with SM expectations for to W-bosons other than via the kinetic mixing κ. It is W → (cid:96)ν and W →hadrons, we limit the contribution of then clear that the SM charged current processes accom- the µνV mode to the W width to twice its error, leading panied by the emission of the light vector boson V from to a branching the muon line will be drastically diﬀerent from a similar process with an emission of a photon. In particular, the B (W → µνV ) < 4.0% (22) interaction of the longitudinal part of the V boson will be enhanced with energy due to the absence of the con- servation of the corresponding current. As pointed out at 2σ. This translates to a limit on the coupling of V in Refs. [30–32], direct production of V from muons in to muons of (cid:16) (cid:17) K → µνV decays can be enhanced by a factor of m2/m2 µ V for the V + A current, and even more for the V − A cur- − m rent. In the latter case it is advantageous to study very g < 2.2 × 10 3 V . (23) high-energy processes (see e.g. Ref. [33]), where the en- V 10 MeV hancement can scale as (Energy)2/m2 . V One of the best known charged current processes is It is clear that a large correction to W decay is an the leptonic decay of W boson. When g (cid:54)= g (in other example of strong high-energy constraints resulting from V A words, when the coupling of V boson to the left-handed the lack of the consistent SM embedding of the starting muon is not zero) the decay W → µνV will be enhanced point in Eq. (4). There are other processes that can be by m2 /m2 , with the onset of an eﬀectively strong cou- equally problematic for such models. For example, inser- W V pling when (gV − gA)mW /mV >∼ 1. Since this parame- tion of the virtual V line into the µν loop in the W self- ter is indeed larger than one for the interesting part of energy diagram will result in the shift of m and will W ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: e

font: default

text: e

text: µ

text: V

text: µ+

text: V

text: gA

text: µ+

text: γ

text: g

text: g

text: gA

text: µ−

text: µ−

text: V

text: A

text: γ

text: W

text: V

text: p

text: p

text: ν

text: FIG.

text: Typical

label: caption

text: representatives

text: of

text: muon

text: pair

text: production

text: by

text: electron-proton

text: collision

text: due

text: to

text: a

text: new

text: force.

text: The

text: parity

text: vio-

text: FIG.

text: Diagram

text: that

text: leads

text: to

text: the

text: decay

text: W

text: µνV

text: lation

text: will

text: come

text: about

text: due

text: to

text: the

text: presence

text: of

text: the

text: g

text: coupling

text: A

text: in

text: the

text: interference

text: with

text: the

text: pure

text: QED

text: diagrams.

text: parameter

text: space,

text: one

text: should

text: expect

text: a

text: very

text: strong

text: con-

text: straint

text: on

text: the

text: model.

text: Carrying

text: out

text: explicit

text: calculation

text: diagrams,

text: leading

text: to

text: an

text: asymmetry

text: in

text: the

text: muon

text: pair-

text: in

text: the

text: limit

text: of

text: g

text: g

text: as

text: implied

text: by

text: (g

text: and

text: to

text: production

text: cross

text: section

text: by

text: the

text: longitudinally

text: polarized

text: V

text: A

text: µ

text: leading

text: order

text: in

text: m

text: /m

text: and

text: m

text: /m

text: we

text: arrive

text: at

text: electrons,

text: V

text: W

text: µ

text: W

text: σσLL

text: σσRR

text: κ(gA/e).

text: Γ

text: (W

text: µνV

text: g

text: MeV

text: GeV

text: V−

text: While

text: the

text: rate

text: for

text: such

text: a

text: process

text: is

text: rather

text: low,

text: new

text: m

text: V

text: high

text: intensity

text: polarized

text: electron

text: beam

text: facilities

text: can

text: con-

text: ceivably

text: be

text: used

text: to

text: search

text: for

text: such

text: an

text: eﬀect.

text: Because

text: of

text: the

text: prompt

text: decay

text: of

text: V

text: to

text: an

text: electron-

text: HIGH-ENERGY

text: CONSTRAINTS

text: positron

text: pair,

text: and

text: the

text: small

text: value

text: of

text: m

text: this

text: decay

text: will

text: V

text: be

text: similar

text: to

text: W

text: µνγ.

text: In

text: any

text: case,

text: the

text: additional

text: So

text: far

text: we

text: have

text: dealt

text: with

text: the

text: low-energy

text: observables

text: channel

text: leads

text: to

text: the

text: increase

text: of

text: the

text: total

text: W

text: width.

text: The

text: that

text: are

text: only

text: mildly

text: sensitive

text: to

text: the

text: fact

text: that

text: the

text: eﬀec-

text: contribution

text: in

text: Eq.

text: should

text: be

text: compared

text: against

text: the

text: tive

text: Lagrangian

text: at

text: a

text: generic

text: point

text: in

text: {gV

text: gA}

text: pa-

text: current

text: experimental

text: value

text: for

text: the

text: W

text: width,

text: dominated

text: rameter

text: space

text: does

text: not

text: respect

text: the

text: full

text: gauge

text: invariance

text: by

text: measurements

text: at

text: the

text: Tevatron

text: of

text: the

text: SM.

text: Speciﬁcally,

text: we

text: have

text: insisted

text: that

text: the

text: muon

text: neutrinos

text: are

text: uncharged

text: under

text: the

text: new

text: force,

text: due

text: to

text: the

text: fact

text: that

text: their

text: interactions

text: are

text: well-known

text: and

text: do

text: not

text: Γ

text: GeV.

text: W

text: have

text: any

text: room

text: for

text: O(G

text: new

text: physics

text: eﬀects,

text: let

text: alone

text: F

text: stronger-than-G

text: eﬀects

text: as

text: suggested

text: by

text: the

text: r

text: discrep-

text: F

text: p

text: ancy.

text: We

text: also

text: do

text: not

text: assume

text: any

text: direct

text: coupling

text: of

text: V

text: Given

text: the

text: agreement

text: of

text: this

text: with

text: SM

text: expectations

text: for

text: to

text: W-bosons

text: other

text: than

text: via

text: the

text: kinetic

text: mixing

text: κ.

text: It

text: is

text: W

text: and

text: W

text: →hadrons,

text: we

text: limit

text: the

text: contribution

text: of

text: then

text: clear

text: that

text: the

text: SM

text: charged

text: current

text: processes

text: accom-

text: the

text: µνV

text: mode

text: to

text: the

text: W

text: width

text: to

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text: Electronic chemical potentials of porous metal-organic frameworks ∗ Keith T. Butler, Christopher H. Hendon, and Aron Walsh Department of Chemistry, University of Bath, Claverton Down, Bath, BA2 7AY, UK \ { }{ } Received May 26, 2020; E-mail: markcomm footercolor a.walsh@bath.ac.uk a b c Abstract: The binding energy of an electron in a material is a fundamental characteristic, which determines a wealth of important chemical and physical properties. For metal-organic frameworks this quantity is hitherto unknown. We present a general approach for determining the vacuum level of porous r = 6 r = 5, 7, 8 r = 9 metal-organic frameworks and apply it to obtain the ﬁrst ioni- d e f sation energy for six prototype materials including zeolitic, co- valent and ionic frameworks. This approach for valence band alignment can explain observations relating to the electrochem- ical, optical and electrical properties of porous frameworks. r = 10 r = 7.5 r = 7.5 Metal-organic frameworks (MOFs) are hybrid materials Figure 1. Structures of archetype porous frameworks. a: MOF- that combine both organic and inorganic functional motifs. 5, b: HKUST-1, c: ZIF-8, d: COF-1M, e: CPO-27-Mg and f: Owing to the porous structure and large surface-area of some MIL-125. The largest pores of each framework are emphasised by MOFs, they have been the subject of a concerted research burgundy spheres, with the pore radii (r) described in ˚Angstr¨om. eﬀort in ﬁelds such as gas storage and catalysis.1–3 Recently HKUST-1 has three notably diﬀerent pore sizes, emphasised with grey spheres.16 their unique combination of optical and electronic proper- ties has led to interest in incorporating them into photocat- alytic, photovoltaic and electrochemical devices;4–12 how- tallographic unit cells consisting of hundreds of atoms (see ever, the rational design of MOFs for these applications is 1), direct computational treatment of the surface electronic hampered by the lack of a reference scale for the electronic structure using quantitative methods is intractable. An al- levels that control these functionalities. We demonstrate a ternative approach is required. procedure, using computational chemistry, which allows us A common feature of ‘designer’ MOFs is porosity, with to establish values for the binding of electrons in porous pore sizes ranging from 2 ˚A to 50 ˚A in radius.17 We demon- frameworks, by accessing a vacuum potential level at the strate that the electrostatic potential at the centre of the centre of the pores. The resulting valence band alignment pore provides a reference that can be used to place the elec- for six archetype porous materials explains observations re- tronic energy levels of MOFs on a common energy scale. lating to the electrochemical, optical and electrical proper- ties of these materials,13 and highlights a novel avenue for Following the validation of this approach, we report the va- lence band energy of six familiar frameworks (1), including tuning the performance of photo- and electro-active hybrid a key metal-organic framework (MOF-5),18 two of the high- frameworks. est performing gas-storage coordination frameworks (CPO- The ionisation energy of an atom is well deﬁned, i.e. 27-Mg and HKUST-1),19,20 a covalent organic framework the energy required to remove an electron in the gaseous state, for example H → H+ + e− (φ = 1312 kJ/mol). (COF-1M),21,22 a zeolitic imidazolate framework (ZIF-8)23 (g) (g) and a material of Institut Lavoisier (MIL-125)24 that dif- For molecules, the same process occurs, but there are dis- fer in both local and extended connectivity. This approach tinct vertical and adiabatic ionisation energies depending deﬁnes the reference potentials necessary for rational design on whether atomic relaxation takes place. The ionisation of MOFs for electronic devices and photocatalytic applica- energy is more diﬃcult to deﬁne for a solid owing to the tions. The generality and low computational overhead make anisotropy of an electron parting the lattice. The termina- it suitable for incorporation into materials screening proce- tion of the crystal, and the associated structural and elec- dures.25 trostatic variations, result in a large spread of measured and Chemical interactions are predominately “near- computed values. The bulk binding energy of an electron in sighted”.26 The valence electron density taken through a solid can be computed, for example, based on electrostatic grounds,14 or through the application of quantum chemistry a plane of MIL-125, which is composed of TiO2 octahedra with appropriate boundary conditions.15 and 1,4-benzenedicarboxylate (bdc), is shown in 2. The electrons are conﬁned to the hybrid framework, with strong Density functional theory (DFT) is one of the most widely localisation at anionic (oxide) centres. The associated elec- used electronic structure techniques in computational mate- trostatic potential resulting from the nuclear and electronic rials chemistry. Indeed, the application of DFT to MOFs distributions is also shown. These interactions extend fur- has resulted in hundreds of reports to date. However, the ther from the atomic centres, e.g. the potential energy of surface science of MOFs is still in its infancy; there are few − interacting quadrupole moments varies with distance as r 5; models describing the atomic or electronic changes that oc- however, they rapidly decay towards the centre of the pore cur at a crystal boundary. Furthermore, due to large crys- 1 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: Electronic

text: chemical

text: potentials

text: of

text: porous

text: metal-organic

text: frameworks

text: Keith

label: author

text: T.

text: Butler,

text: Christopher

text: H.

text: Hendon,

text: and

text: Aron

text: Walsh

text: Department

text: of

text: Chemistry,

text: University

text: of

text: Bath,

text: Claverton

text: Down,

text: Bath,

text: UK

text: Received

text: May

text: E-mail:

text: markcomm

text: footercolor

text: a.walsh@bath.ac.uk

text: a

text: b

text: c

text: Abstract:

text: The

label: abstract

text: binding

text: energy

text: of

text: an

text: electron

text: in

text: a

text: material

text: is

text: a

text: fundamental

text: characteristic,

text: which

text: determines

text: a

text: wealth

text: of

text: important

text: chemical

text: and

text: physical

text: properties.

text: For

text: metal-organic

text: frameworks

text: this

text: quantity

text: is

text: hitherto

text: unknown.

text: We

text: present

text: a

text: general

text: approach

text: for

text: determining

text: the

text: vacuum

text: level

text: of

text: porous

text: r

text: r

text: r

text: metal-organic

text: frameworks

text: and

text: apply

text: it

text: to

text: obtain

text: the

text: ﬁrst

text: ioni-

text: d

text: e

text: f

text: sation

text: energy

text: for

text: six

text: prototype

text: materials

text: including

text: zeolitic,

text: co-

text: valent

text: and

text: ionic

text: frameworks.

text: This

text: approach

text: for

text: valence

text: band

text: alignment

text: can

text: explain

text: observations

text: relating

text: to

text: the

text: electrochem-

text: ical,

text: optical

text: and

text: electrical

text: properties

text: of

text: porous

text: frameworks.

text: r

text: r

text: r

text: Metal-organic

text: frameworks

text: (MOFs)

text: are

text: hybrid

text: materials

text: Figure

text: Structures

text: of

text: archetype

text: porous

text: frameworks.

text: a:

text: MOF-

text: that

text: combine

text: both

text: organic

text: and

text: inorganic

text: functional

text: motifs.

text: b:

text: c:

text: d:

text: e:

text: and

text: f:

text: Owing

text: to

text: the

text: porous

text: structure

text: and

text: large

text: surface-area

text: of

text: some

text: The

text: largest

text: pores

text: of

text: each

text: framework

text: are

text: emphasised

text: by

text: MOFs,

text: they

text: have

text: been

text: the

text: subject

text: of

text: a

text: concerted

text: research

text: burgundy

text: spheres,

text: with

text: the

text: pore

text: radii

text: (r)

text: described

text: in

text: ˚Angstr¨om.

text: eﬀort

text: in

text: ﬁelds

text: such

text: as

text: gas

text: storage

text: and

text: Recently

text: has

text: three

text: notably

text: diﬀerent

text: pore

text: sizes,

text: emphasised

text: with

text: grey

text: their

text: unique

text: combination

text: of

text: optical

text: and

text: electronic

text: proper-

text: ties

text: has

text: led

text: to

text: interest

text: in

text: incorporating

text: them

text: into

text: photocat-

text: alytic,

text: photovoltaic

text: and

text: electrochemical

text: how-

text: tallographic

text: unit

text: cells

text: consisting

text: of

text: hundreds

text: of

text: atoms

text: (see

text: ever,

text: the

text: rational

text: design

text: of

text: MOFs

text: for

text: these

text: applications

text: is

text: direct

text: computational

text: treatment

text: of

text: the

text: surface

text: electronic

text: hampered

text: by

text: the

text: lack

text: of

text: a

text: reference

text: scale

text: for

text: the

text: electronic

text: structure

text: using

text: quantitative

text: methods

text: is

text: intractable.

text: An

text: al-

text: levels

text: that

text: control

text: these

text: functionalities.

text: We

text: demonstrate

text: a

text: ternative

text: approach

text: is

text: required.

text: procedure,

text: using

text: computational

text: chemistry,

text: which

text: allows

text: us

text: A

text: common

text: feature

text: of

text: ‘designer’

text: MOFs

text: is

text: porosity,

text: with

text: to

text: establish

text: values

text: for

text: the

text: binding

text: of

text: electrons

text: in

text: porous

text: pore

text: sizes

text: ranging

text: from

text: ˚A

text: to

text: ˚A

text: in

text: We

text: demon-

text: frameworks,

text: by

text: accessing

text: a

text: vacuum

text: potential

text: level

text: at

text: the

text: strate

text: that

text: the

text: electrostatic

text: potential

text: at

text: the

text: centre

text: of

text: the

text: centre

text: of

text: the

text: pores.

text: The

text: resulting

text: valence

text: band

text: alignment

text: pore

text: provides

text: a

text: reference

text: that

text: can

text: be

text: used

text: to

text: place

text: the

text: elec-

text: for

text: six

text: archetype

text: porous

text: materials

text: explains

text: observations

text: re-

text: tronic

text: energy

text: levels

text: of

text: MOFs

text: on

text: a

text: common

text: energy

text: scale.

text: lating

text: to

text: the

text: electrochemical,

text: optical

text: and

text: electrical

text: proper-

text: ties

text: of

text: these

text: and

text: highlights

text: a

text: novel

text: avenue

text: for

text: Following

text: the

text: validation

text: of

text: this

text: approach,

text: we

text: report

text: the

text: va-

text: lence

text: band

text: energy

text: of

text: six

text: familiar

text: frameworks

text: including

text: tuning

text: the

text: performance

text: of

text: photo-

text: and

text: electro-active

text: hybrid

text: a

text: key

text: metal-organic

text: framework

text: two

text: of

text: the

text: high-

text: frameworks.

text: est

text: performing

text: gas-storage

text: coordination

text: frameworks

text: (CPO-

text: The

text: ionisation

text: energy

text: of

text: an

text: atom

text: is

text: well

text: deﬁned,

text: i.e.

text: and

text: a

text: covalent

text: organic

text: framework

text: the

text: energy

text: required

text: to

text: remove

text: an

text: electron

text: in

text: the

text: gaseous

text: state,

text: for

text: example

text: H

text: H+

text: e−

text: (φ

text: kJ/mol).

text: a

text: zeolitic

text: imidazolate

text: framework

text: (g)

text: (g)

text: and

text: a

text: material

text: of

text: Institut

text: Lavoisier

text: that

text: dif-

text: For

text: molecules,

text: the

text: same

text: process

text: occurs,

text: but

text: there

text: are

text: dis-

text: fer

text: in

text: both

text: local

text: and

text: extended

text: connectivity.

text: This

text: approach

text: tinct

text: vertical

text: and

text: adiabatic

text: ionisation

text: energies

text: depending

text: deﬁnes

text: the

text: reference

text: potentials

text: necessary

text: for

text: rational

text: design

text: on

text: whether

text: atomic

text: relaxation

text: takes

text: place.

text: The

text: ionisation

text: of

text: MOFs

text: for

text: electronic

text: devices

text: and

text: photocatalytic

text: applica-

text: energy

text: is

text: more

text: diﬃcult

text: to

text: deﬁne

text: for

text: a

text: solid

text: owing

text: to

text: the

text: tions.

text: The

text: generality

text: and

text: low

text: computational

text: overhead

text: make

text: anisotropy

text: of

text: an

text: electron

text: parting

text: the

text: lattice.

text: The

text: termina-

text: it

text: suitable

text: for

text: incorporation

text: into

text: materials

text: screening

text: proce-

text: tion

text: of

text: the

text: crystal,

text: and

text: the

text: associated

text: structural

text: and

text: elec-

text: trostatic

text: variations,

text: result

text: in

text: a

text: large

text: spread

text: of

text: measured

text: and

text: Chemical

text: interactions

text: are

text: predominately

text: “near-

text: computed

text: values.

text: The

text: bulk

text: binding

text: energy

text: of

text: an

text: electron

text: in

text: The

text: valence

text: electron

text: density

text: taken

text: through

text: a

text: solid

text: can

text: be

text: computed,

text: for

text: example,

text: based

text: on

text: electrostatic

text: or

text: through

text: the

text: application

text: of

text: quantum

text: chemistry

text: a

text: plane

text: of

text: which

text: is

text: composed

text: of

text: octahedra

text: with

text: appropriate

text: boundary

text: and

text: (bdc),

text: is

text: shown

text: in

text: The

text: electrons

text: are

text: conﬁned

text: to

text: the

text: hybrid

text: framework,

text: with

text: strong

text: Density

text: functional

text: theory

text: (DFT)

text: is

text: one

text: of

text: the

text: most

text: widely

text: localisation

text: at

text: anionic

text: (oxide)

text: centres.

text: The

text: associated

text: elec-

text: used

text: electronic

text: structure

text: techniques

text: in

text: computational

text: mate-

text: trostatic

text: potential

text: resulting

text: from

text: the

text: nuclear

text: and

text: electronic

text: rials

text: chemistry.

text: Indeed,

text: the

text: application

text: of

text: DFT

text: to

text: MOFs

text: distributions

text: is

text: also

text: shown.

text: These

text: interactions

text: extend

text: fur-

text: has

text: resulted

text: in

text: hundreds

text: of

text: reports

text: to

text: date.

text: However,

text: the

text: ther

text: from

text: the

text: atomic

text: centres,

text: e.g.

text: the

text: potential

text: energy

text: of

text: surface

text: science

text: of

text: MOFs

text: is

text: still

text: in

text: its

text: infancy;

text: there

text: are

text: few

text: interacting

text: quadrupole

text: moments

text: varies

text: with

text: distance

text: as

text: r

text: models

text: describing

text: the

text: atomic

text: or

text: electronic

text: changes

text: that

text: oc-

text: however,

text: they

text: rapidly

text: decay

text: towards

text: the

text: centre

text: of

text: the

text: pore

text: cur

text: at

text: a

text: crystal

text: boundary.

text: Furthermore,

text: due

text: to

text: large

text: crys-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: RIBBON GRAPHS AND BIALGEBRA OF LAGRANGIAN SUBSPACES VICTOR KLEPTSYN AND EVGENY SMIRNOV Dedicated to the memory of Sergei Vassilievich Duzhin Abstract. To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection ma- trix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of L-spaces, becoming changes of bases in this vector space. Finally, we deﬁne a bialgebra structure on the span of L-spaces, which is anal- ogous to the 4-bialgebra structure on chord diagrams. Contents 1. Introduction 1 2. Ribbon graphs 3 3. L-spaces of ribbon graphs 6 4. Bialgebra structures 13 5. Miscellaneous 18 Acknowledgements 21 References 21 1. Introduction Chord diagrams appear as an essential tool in the study of Vassiliev ﬁnite-type knot invariants. They are constructed as follows: to each singular knot, i.e., a knot with a ﬁnite number of simple self-intersection points, we assign a circle, corre- sponding to the preimage of the knot, with the preimages of the self-intersection points joined by chords. Chord diagrams span a graded vector space (over an arbitrary ﬁeld) with grad- ing corresponding to the number of chords. Since the ﬁnite-type knot invariants produce functions invariant under 4-term relations, they can be considered as func- tions on the quotient of this vector space by the subspace generated by the 4-term relations. The latter vector space can be equipped with operations of multiplication and comultiplication, turning it into a bialgebra. The combinatorial data encoded in chord diagrams are quite involved. In some situations one can pass from chord diagrams to their intersection matrices: these are symmetric 0-1 matrices with rows and columns corresponding to the chords, such that each element is 1 if the corresponding two chords intersect and 0 otherwise Date: May 26, 2020. 2010 Mathematics Subject Classiﬁcation. 57M20, 57M27 (primary), 57M15, 05C25 (secondary). Key words and phrases. Ribbon graph, chord diagram, Vassiliev knot invariant. 1 ##LTLine##

text: RIBBON

text: GRAPHS

text: AND

text: BIALGEBRA

text: OF

text: LAGRANGIAN

text: SUBSPACES

text: VICTOR

text: KLEPTSYN

text: AND

text: EVGENY

text: SMIRNOV

text: Dedicated

text: to

text: the

text: memory

text: of

text: Sergei

text: Vassilievich

text: Duzhin

text: Abstract.

text: To

text: each

text: ribbon

text: graph

text: we

text: assign

text: a

text: so-called

text: L-space,

text: which

text: is

text: a

text: Lagrangian

text: subspace

text: in

text: an

text: even-dimensional

text: vector

text: space

text: with

text: the

text: standard

text: symplectic

text: form.

text: This

text: invariant

text: generalizes

text: the

text: notion

text: of

text: the

text: intersection

text: ma-

text: trix

text: of

text: a

text: chord

text: diagram.

text: Moreover,

text: the

text: actions

text: of

text: Morse

text: perestroikas

text: (or

text: taking

text: a

text: partial

text: dual)

text: and

text: Vassiliev

text: moves

text: on

text: ribbon

text: graphs

text: are

text: reinterpreted

text: nicely

text: in

text: the

text: language

text: of

text: L-spaces,

text: becoming

text: changes

text: of

text: bases

text: in

text: this

text: vector

text: space.

text: Finally,

text: we

text: deﬁne

text: a

text: bialgebra

text: structure

text: on

text: the

text: span

text: of

text: L-spaces,

text: which

text: is

text: anal-

text: ogous

text: to

text: the

text: structure

text: on

text: chord

text: diagrams.

text: Contents

text: Introduction

text: Ribbon

text: graphs

text: L-spaces

text: of

text: ribbon

text: graphs

text: Bialgebra

text: structures

text: Miscellaneous

text: Acknowledgements

text: References

text: Introduction

text: Chord

text: diagrams

text: appear

text: as

text: an

text: essential

text: tool

text: in

text: the

text: study

text: of

text: Vassiliev

text: ﬁnite-type

text: knot

text: invariants.

text: They

text: are

text: constructed

text: as

text: follows:

text: to

text: each

text: singular

text: knot,

text: i.e.,

text: a

text: knot

text: with

text: a

text: ﬁnite

text: number

text: of

text: simple

text: self-intersection

text: points,

text: we

text: assign

text: a

text: circle,

text: corre-

text: sponding

text: to

text: the

text: preimage

text: of

text: the

text: knot,

text: with

text: the

text: preimages

text: of

text: the

text: self-intersection

text: points

text: joined

text: by

text: chords.

text: Chord

text: diagrams

text: span

text: a

text: graded

text: vector

text: space

text: (over

text: an

text: arbitrary

text: ﬁeld)

text: with

text: grad-

text: ing

text: corresponding

text: to

text: the

text: number

text: of

text: chords.

text: Since

text: the

text: ﬁnite-type

text: knot

text: invariants

text: produce

text: functions

text: invariant

text: under

text: relations,

text: they

text: can

text: be

text: considered

text: as

text: func-

text: tions

text: on

text: the

text: quotient

text: of

text: this

text: vector

text: space

text: by

text: the

text: subspace

text: generated

text: by

text: the

text: relations.

text: The

text: latter

text: vector

text: space

text: can

text: be

text: equipped

text: with

text: operations

text: of

text: multiplication

text: and

text: comultiplication,

text: turning

text: it

text: into

text: a

text: bialgebra.

text: The

text: combinatorial

text: data

text: encoded

text: in

text: chord

text: diagrams

text: are

text: quite

text: involved.

text: In

text: some

text: situations

text: one

text: can

text: pass

text: from

text: chord

text: diagrams

text: to

text: their

text: intersection

text: matrices:

text: these

text: are

text: symmetric

text: matrices

text: with

text: rows

text: and

text: columns

text: corresponding

text: to

text: the

text: chords,

text: such

text: that

text: each

text: element

text: is

text: if

text: the

text: corresponding

text: two

text: chords

text: intersect

text: and

text: otherwise

text: Date:

text: May

text: Mathematics

text: Subject

text: Classiﬁcation.

text: (primary),

text: (secondary).

text: Key

text: words

text: and

text: phrases.

text: Ribbon

text: graph,

text: chord

text: diagram,

text: Vassiliev

text: knot

text: invariant.

text: ##LTLine##


text: 2 VICTOR KLEPTSYN AND EVGENY SMIRNOV (all the diagonal entries are 0). Alternatively, instead of matrices one can consider their intersection graphs. As in the case of chord diagrams, we can introduce the four-term relation and bialgebra structure on the algebra spanned by intersection matrices. The correspondence between chord diagrams and their intersection ma- trices is neither injective nor surjective; however, weight systems on intersection matrices can be lifted to a weight system on chord diagrams, so intersection matri- ces can serve a valuable source of new weight systems. The notion of chord diagram admits several natural generalizations. First, we can study (singular) links instead of knots; they give rise to multicomponent chord diagrams. Further, following V. Arnold [1], instead of knots we can consider smooth curves in the plane with possible transversal self-intersections. In this case chord diagrams are replaced by framed chord diagrams, with chords of two diﬀerent types. These chords can be thought of as ribbons joining pairs of segments on a circle; these ribbons can be either untwisted or twisted. If all ribbons are untwisted, the diagram is unframed, i.e., ordinary. For framed chord diagrams one can also introduce the four-term relation; this was done by S. Lando in [12], cf. also [14]. However, the space of framed chord diagrams modulo the four-term relation does not form a bialgebra anymore. It admits a coalgebra structure, but the multiplication of framed chord diagrams is not well-deﬁned. Recently M. Karev [11] showed that this space is a module over the algebra of unframed chord diagrams. However, for a framed diagram we still can deﬁne the notion of its intersection matrix (or, equivalently, of its intersection framed graph, i.e., a graph with colored vertices of two diﬀerent colors, corresponding to twisted and untwisted ribbons). The four-term relation descends to the space of intersection graphs. In [13] it was shown that the space of framed graphs modulo the four-term relation still forms a bialgebra (while the space of chord diagrams does not). Both unframed and framed chord diagrams can be considered in a much more general context: they are ribbon graphs, or fat graphs, with one vertex. Informally speaking, ribbon graphs are surfaces glued from disks representing vertices and ribbons representing edges (formal deﬁnition will be given in Sec. ). Thus, one can think of ribbon graphs with more than one vertex as of framed multicomponent chord diagrams. In this paper we introduce a new object, called the L-space, for each ribbon graph. For a ribbon graph with n edges, this is a Lagrangian subspace in a 2n- F dimensional space over with a standard symplectic form. It generalizes the 2 notion of the intersection matrix. A vector space spanned by L-spaces has a bialge- bra structure, which extends the bialgebra structure on the space spanned by the set of (framed) intersection matrices. The notions of Vassiliev moves and four-term relations for chord diagrams can be generalized to ribbon graphs. We show that these notions are reinterpreted nicely in terms of L-spaces: they correspond to certain base changes in the ambient space. Moreover, for ribbon graphs there is an operation of partial dual, or a Morse perestroika with respect to a subset of chords, deﬁned by Chmutov [8]; analogues of this operation also appeared before in the study of graph invariants (see [2] and [3]). We show that this operation also descends to L-spaces as a particularly nice base change. Finally, we deﬁne the bialgebra of L-spaces purely combinatorially. This notion generalizes the notion of the bialgebra of graphs, introduced by S. Lando in [13]. More speciﬁcally, we deﬁne a bialgebra structure on the vector space freely spanned by all possible Lagrangian spaces. This bialgebra contains the bialgebra of framed graphs. The Vassiliev moves, introduced earlier, allow us to extend the four-term

text: VICTOR

text: KLEPTSYN

text: AND

text: EVGENY

text: SMIRNOV

text: (all

text: the

text: diagonal

text: entries

text: are

text: Alternatively,

text: instead

text: of

text: matrices

text: one

text: can

text: consider

text: their

text: intersection

text: graphs.

text: As

text: in

text: the

text: case

text: of

text: chord

text: diagrams,

text: we

text: can

text: introduce

text: the

text: four-term

text: relation

text: and

text: bialgebra

text: structure

text: on

text: the

text: algebra

text: spanned

text: by

text: intersection

text: matrices.

text: The

text: correspondence

text: between

text: chord

text: diagrams

text: and

text: their

text: intersection

text: ma-

text: trices

text: is

text: neither

text: injective

text: nor

text: surjective;

text: however,

text: weight

text: systems

text: on

text: intersection

text: matrices

text: can

text: be

text: lifted

text: to

text: a

text: weight

text: system

text: on

text: chord

text: diagrams,

text: so

text: intersection

text: matri-

text: ces

text: can

text: serve

text: a

text: valuable

text: source

text: of

text: new

text: weight

text: systems.

text: The

text: notion

text: of

text: chord

text: diagram

text: admits

text: several

text: natural

text: generalizations.

text: First,

text: we

text: can

text: study

text: (singular)

text: links

text: instead

text: of

text: knots;

text: they

text: give

text: rise

text: to

text: multicomponent

text: chord

text: diagrams.

text: Further,

text: following

text: V.

text: Arnold

text: instead

text: of

text: knots

text: we

text: can

text: consider

text: smooth

text: curves

text: in

text: the

text: plane

text: with

text: possible

text: transversal

text: self-intersections.

text: In

text: this

text: case

text: chord

text: diagrams

text: are

text: replaced

text: by

text: framed

text: chord

text: diagrams,

text: with

text: chords

text: of

text: two

text: diﬀerent

text: types.

text: These

text: chords

text: can

text: be

text: thought

text: of

text: as

text: ribbons

text: joining

text: pairs

text: of

text: segments

text: on

text: a

text: circle;

text: these

text: ribbons

text: can

text: be

text: either

text: untwisted

text: or

text: twisted.

text: If

text: all

text: ribbons

text: are

text: untwisted,

text: the

text: diagram

text: is

text: unframed,

text: i.e.,

text: ordinary.

text: For

text: framed

text: chord

text: diagrams

text: one

text: can

text: also

text: introduce

text: the

text: four-term

text: relation;

text: this

text: was

text: done

text: by

text: S.

text: Lando

text: in

text: cf.

text: also

text: However,

text: the

text: space

text: of

text: framed

text: chord

text: diagrams

text: modulo

text: the

text: four-term

text: relation

text: does

text: not

text: form

text: a

text: bialgebra

text: anymore.

text: It

text: admits

text: a

text: coalgebra

text: structure,

text: but

text: the

text: multiplication

text: of

text: framed

text: chord

text: diagrams

text: is

text: not

text: well-deﬁned.

text: Recently

text: M.

text: Karev

text: showed

text: that

text: this

text: space

text: is

text: a

text: module

text: over

text: the

text: algebra

text: of

text: unframed

text: chord

text: diagrams.

text: However,

text: for

text: a

text: framed

text: diagram

text: we

text: still

text: can

text: deﬁne

text: the

text: notion

text: of

text: its

text: intersection

text: matrix

text: (or,

text: equivalently,

text: of

text: its

text: intersection

text: framed

text: graph,

text: i.e.,

text: a

text: graph

text: with

text: colored

text: vertices

text: of

text: two

text: diﬀerent

text: colors,

text: corresponding

text: to

text: twisted

text: and

text: untwisted

text: ribbons).

text: The

text: four-term

text: relation

text: descends

text: to

text: the

text: space

text: of

text: intersection

text: graphs.

text: In

text: it

text: was

text: shown

text: that

text: the

text: space

text: of

text: framed

text: graphs

text: modulo

text: the

text: four-term

text: relation

text: still

text: forms

text: a

text: bialgebra

text: (while

text: the

text: space

text: of

text: chord

text: diagrams

text: does

text: not).

text: Both

text: unframed

text: and

text: framed

text: chord

text: diagrams

text: can

text: be

text: considered

text: in

text: a

text: much

text: more

text: general

text: context:

text: they

text: are

text: ribbon

text: graphs,

text: or

text: fat

text: graphs,

text: with

text: one

text: vertex.

text: Informally

text: speaking,

text: ribbon

text: graphs

text: are

text: surfaces

text: glued

text: from

text: disks

text: representing

text: vertices

text: and

text: ribbons

text: representing

text: edges

text: (formal

text: deﬁnition

text: will

text: be

text: given

text: in

text: Sec.

text: Thus,

text: one

text: can

text: think

text: of

text: ribbon

text: graphs

text: with

text: more

text: than

text: one

text: vertex

text: as

text: of

text: framed

text: multicomponent

text: chord

text: diagrams.

text: In

text: this

text: paper

text: we

text: introduce

text: a

text: new

text: object,

text: called

text: the

text: L-space,

text: for

text: each

text: ribbon

text: graph.

text: For

text: a

text: ribbon

text: graph

text: with

text: n

text: edges,

text: this

text: is

text: a

text: Lagrangian

text: subspace

text: in

text: a

text: F

text: dimensional

text: space

text: over

text: with

text: a

text: standard

text: symplectic

text: form.

text: It

text: generalizes

text: the

text: notion

text: of

text: the

text: intersection

text: matrix.

text: A

text: vector

text: space

text: spanned

text: by

text: L-spaces

text: has

text: a

text: bialge-

text: bra

text: structure,

text: which

text: extends

text: the

text: bialgebra

text: structure

text: on

text: the

text: space

text: spanned

text: by

text: the

text: set

text: of

text: (framed)

text: intersection

text: matrices.

text: The

text: notions

text: of

text: Vassiliev

text: moves

text: and

text: four-term

text: relations

text: for

text: chord

text: diagrams

text: can

text: be

text: generalized

text: to

text: ribbon

text: graphs.

text: We

text: show

text: that

text: these

text: notions

text: are

text: reinterpreted

text: nicely

text: in

text: terms

text: of

text: L-spaces:

text: they

text: correspond

text: to

text: certain

text: base

text: changes

text: in

text: the

text: ambient

text: space.

text: Moreover,

text: for

text: ribbon

text: graphs

text: there

text: is

text: an

text: operation

text: of

text: partial

text: dual,

text: or

text: a

text: Morse

text: perestroika

text: with

text: respect

text: to

text: a

text: subset

text: of

text: chords,

text: deﬁned

text: by

text: Chmutov

text: analogues

text: of

text: this

text: operation

text: also

text: appeared

text: before

text: in

text: the

text: study

text: of

text: graph

text: invariants

text: (see

text: and

text: We

text: show

text: that

text: this

text: operation

text: also

text: descends

text: to

text: L-spaces

text: as

text: a

text: particularly

text: nice

text: base

text: change.

text: Finally,

text: we

text: deﬁne

text: the

text: bialgebra

text: of

text: L-spaces

text: purely

text: combinatorially.

text: This

text: notion

text: generalizes

text: the

text: notion

text: of

text: the

text: bialgebra

text: of

text: graphs,

text: introduced

text: by

text: S.

text: Lando

text: in

text: More

text: speciﬁcally,

text: we

text: deﬁne

text: a

text: bialgebra

text: structure

text: on

text: the

text: vector

text: space

text: freely

text: spanned

text: by

text: all

text: possible

text: Lagrangian

text: spaces.

text: This

text: bialgebra

text: contains

text: the

text: bialgebra

text: of

text: framed

text: graphs.

text: The

text: Vassiliev

text: moves,

text: introduced

text: earlier,

text: allow

text: us

text: to

text: extend

text: the

text: four-term


text: RIBBON GRAPHS AND BIALGEBRA OF LAGRANGIAN SUBSPACES 13 4. Bialgebra structures A well-known feature of chord diagrams is that they generate a bialgebra. In this section we ﬁrst recall this construction and then provide its generalizations: the graph bialgebra, due to Lando [13], and the bialgebra of L-spaces, which is the main construction of this paper. K (cid:76) Let be a ﬁeld (or, even more generally, a commutative and associative ring). A K For n > 0, let be a -vector space formally spanned by all chord diagrams with nA K A A n chords; we set = and = ≥ . 0 n 0 n For any chord diagram c and any pair of neighboring chords deﬁning Vassiliev moves, the alternating sum − − c v (c) v (c) + v v (c) 1 2 1 2 is called a four-term element. A A Let (4) be the subspace of generated by all four-term elements. Denote by M M A A the quotient space: = / (4). This graded vector space can be turned into a graded bialgebra by introducing the operations of multiplication and comultipli- cation. M A To deﬁne the comultiplication on , let us ﬁrst deﬁne it on . Take a chord ⊂ diagram c with n chords. Let V (c) be the set of its chords; for a subset I V (c) denote by c the diagram formed(cid:88)by all chords from the set I. Then the I A → A ⊗ A comultiplication µ: takes c into ⊗ µ(c) = c c (cid:48). I I (cid:116) (cid:48) I I =V (c) A ⊂ A ⊗ A A ⊗ A A routine check shows that µ( (4)) (4) + (4), so this operation M → M ⊗ M determines a comultiplication µ: on the quotient space. Multiplication of two chord diagrams is deﬁned as their “connected sum”. To multiply two chord diagrams c and c , we make a puncture in each of the two 1 2 circles and attach these two circles one to another along this puncture, obtaining a new diagram c #c . Clearly, this operation is not well-deﬁned, since the result 1 2 depends on the positions of the punctures. However, it is not hard to show that all such chord diagrams are congruent modulo the four-term relations. This allows M · A us to deﬁne the product on as follows: c c = c #c mod (4). Clearly, it is 1 2 1 2 commutative. One can also show that these two operations satisfy the axioms of a bialgebra, thus obtaining the following theorem (cf. [14, Theorem 6.1.12]). M Theorem 4.1. The vector space of chord diagrams is a graded commutative and K cocommutative bialgebra over with respect to the operations introduced above. In this subsection we deﬁne the bialgebra of graphs. This is a purely combina- torial anagolue of the bialgebra of chord diagrams. It was introduced by Lando in (cid:76) [13]. We mostly follow this paper and [14, Sec. 6.4]. G K Let be a graded -vector space freely spanned by all graphs (not necessarily n G G connected) on n vertices, and let = ≥ . This vector space admits a struc- n 0 n G ⊗ G → G ture of a bialgebra in the following way. The multiplication m: k n k+n brings a pair of graphs into their disjoint union. The unit of this multiplication is represented by the empty graph. ⊂ For a graph Γ and a subset J V (Γ) of its vertices, denote by Γ the restriction J of Γ to the set J: a subgraph of Γ formed by the vertices from J and the edges of ##LTLine## ##LTLine## ##LTLine##

text: RIBBON

text: GRAPHS

text: AND

text: BIALGEBRA

text: OF

text: LAGRANGIAN

text: SUBSPACES

label: section

text: Bialgebra

text: structures

text: A

text: well-known

text: feature

text: of

text: chord

text: diagrams

text: is

text: that

text: they

text: generate

text: a

text: bialgebra.

text: In

text: this

text: section

text: we

text: ﬁrst

text: recall

text: this

text: construction

text: and

text: then

text: provide

text: its

text: generalizations:

text: the

text: graph

text: bialgebra,

text: due

text: to

text: Lando

text: and

text: the

text: bialgebra

text: of

text: L-spaces,

text: which

text: is

text: the

text: main

text: construction

text: of

text: this

text: paper.

text: K

text: Let

text: be

text: a

text: ﬁeld

text: (or,

text: even

text: more

text: generally,

text: a

text: commutative

text: and

text: associative

text: ring).

text: A

text: K

text: For

text: n

text: let

text: be

text: a

text: -vector

text: space

text: formally

text: spanned

text: by

text: all

text: chord

text: diagrams

text: with

text: K

text: A

text: A

text: n

text: chords;

text: we

text: set

text: and

text: n

text: n

text: For

text: any

text: chord

text: diagram

text: c

text: and

text: any

text: pair

text: of

text: neighboring

text: chords

text: deﬁning

text: Vassiliev

text: moves,

text: the

text: alternating

text: sum

text: c

text: v

text: (c)

text: v

text: (c)

text: v

text: v

text: (c)

text: is

text: called

text: a

text: four-term

text: element.

text: A

text: A

text: Let

text: be

text: the

text: subspace

text: of

text: generated

text: by

text: all

text: four-term

text: elements.

text: Denote

text: by

text: M

text: M

text: A

text: A

text: the

text: quotient

text: space:

text: This

text: graded

text: vector

text: space

text: can

text: be

text: turned

text: into

text: a

text: graded

text: bialgebra

text: by

text: introducing

text: the

text: operations

text: of

text: multiplication

text: and

text: comultipli-

text: cation.

text: M

text: A

text: To

text: deﬁne

text: the

text: comultiplication

text: on

text: let

text: us

text: ﬁrst

text: deﬁne

text: it

text: on

text: Take

text: a

text: chord

text: diagram

text: c

text: with

text: n

text: chords.

text: Let

text: V

text: (c)

text: be

text: the

text: set

text: of

text: its

text: chords;

text: for

text: a

text: subset

text: I

text: V

text: (c)

text: denote

text: by

text: c

text: the

text: diagram

text: all

text: chords

text: from

text: the

text: set

text: I.

text: Then

text: the

text: I

text: A

text: A

text: A

text: comultiplication

text: µ:

text: takes

text: c

text: into

text: µ(c)

text: c

text: c

text: I

text: I

text: I

text: I

text: =V

text: (c)

text: A

text: A

text: A

text: A

text: A

text: A

text: routine

text: check

text: shows

text: that

text: µ(

text: so

text: this

text: operation

text: M

text: M

text: M

text: determines

text: a

text: comultiplication

text: µ:

text: on

text: the

text: quotient

text: space.

text: Multiplication

text: of

text: two

text: chord

text: diagrams

text: is

text: deﬁned

text: as

text: their

text: “connected

text: sum”.

text: To

text: multiply

text: two

text: chord

text: diagrams

text: c

text: and

text: c

text: we

text: make

text: a

text: puncture

text: in

text: each

text: of

text: the

text: two

text: circles

text: and

text: attach

text: these

text: two

text: circles

text: one

text: to

text: another

text: along

text: this

text: puncture,

text: obtaining

text: a

text: new

text: diagram

text: c

text: #c

text: Clearly,

text: this

text: operation

text: is

text: not

text: well-deﬁned,

text: since

text: the

text: result

text: depends

text: on

text: the

text: positions

text: of

text: the

text: punctures.

text: However,

text: it

text: is

text: not

text: hard

text: to

text: show

text: that

text: all

text: such

text: chord

text: diagrams

text: are

text: congruent

text: modulo

text: the

text: four-term

text: relations.

text: This

text: allows

text: M

text: A

text: us

text: to

text: deﬁne

text: the

text: product

text: on

text: as

text: follows:

text: c

text: c

text: c

text: #c

text: mod

text: Clearly,

text: it

text: is

text: commutative.

text: One

text: can

text: also

text: show

text: that

text: these

text: two

text: operations

text: satisfy

text: the

text: axioms

text: of

text: a

text: bialgebra,

text: thus

text: obtaining

text: the

text: following

text: theorem

text: (cf.

text: Theorem

text: M

text: Theorem

text: The

text: vector

text: space

text: of

text: chord

text: diagrams

text: is

text: a

text: graded

text: commutative

text: and

text: K

text: cocommutative

text: bialgebra

text: over

text: with

text: respect

text: to

text: the

text: operations

text: introduced

text: above.

text: In

text: this

text: subsection

text: we

text: deﬁne

text: the

text: bialgebra

text: of

text: graphs.

text: This

text: is

text: a

text: purely

text: combina-

text: torial

text: anagolue

text: of

text: the

text: bialgebra

text: of

text: chord

text: diagrams.

text: It

text: was

text: introduced

text: by

text: Lando

text: in

text: We

text: mostly

text: follow

text: this

text: paper

text: and

text: Sec.

text: G

text: K

text: Let

text: be

text: a

text: graded

text: -vector

text: space

text: freely

text: spanned

text: by

text: all

text: graphs

text: (not

text: necessarily

text: n

text: G

text: G

text: connected)

text: on

text: n

text: vertices,

text: and

text: let

text: This

text: vector

text: space

text: admits

text: a

text: struc-

text: n

text: n

text: G

text: G

text: G

text: ture

text: of

text: a

text: bialgebra

text: in

text: the

text: following

text: way.

text: The

text: multiplication

text: m:

text: k

text: n

text: k+n

text: brings

text: a

text: pair

text: of

text: graphs

text: into

text: their

text: disjoint

text: union.

text: The

text: unit

text: of

text: this

text: multiplication

text: is

text: represented

text: by

text: the

text: empty

text: graph.

text: For

text: a

text: graph

text: Γ

text: and

text: a

text: subset

text: J

text: V

text: (Γ)

text: of

text: its

text: vertices,

text: denote

text: by

text: Γ

text: the

text: restriction

text: J

text: of

text: Γ

text: to

text: the

text: set

text: J:

text: a

text: subgraph

text: of

text: Γ

text: formed

text: by

text: the

text: vertices

text: from

text: J

text: and

text: the

text: edges

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: RIBBON GRAPHS AND BIALGEBRA OF LAGRANGIAN SUBSPACES 15 A natural idea would be to extend the bialgebra structure on the space spanned by chord diagrams (or, equivalently, orientable one-vertex ribbon graphs) to the space of all ribbon graphs. Unfortunately, this fails even on the stage of framed chord diagrams (one-vertex ribbon graphs, not necessarily orientable), modulo the four-term relation. As it was discussed in [12], it is unclear whether the multiplica- tion of framed chord diagrams deﬁned in the same way as in the previous subsection is well-deﬁned. (Recently M. Karev [11] showed that framed chord diagrams form a module over the bialgebra of chord diagrams.) However, the L-spaces of ribbon graphs do have a bialgebra structure. We have seen earlier that L-spaces can be viewed as generalizations of intersection matrices/graphs, so this structure is naturally extended from the graph bialgebra. Just as in the case of graphs, the multiplication and comultiplication of L-spaces is well-deﬁned even without the four-term relation. We have seen in Example 3.2 that L-spaces of ribbon graphs generalize the notion of intersection matrices for chord diagrams. Intersection matrices are deﬁned by their adjacency graphs; the only diﬀerence is that they depend on a speciﬁc ordering of chords, while the vertices in graphs are not ordered. To deﬁne a proper analogue of an intersection graph, we will consider not just L-spaces, but rather their orbits under the action of a symmetric group. F (cid:104) (cid:105) F Let 2n = e , f , . . . , e , f be a 2n-dimensional vector space over with a 2 1 1 n n 2 standard skew-symmetric form deﬁned by (e , f ) = δ . Consider a Lagrangian i j ij Grassmannian LGr(n): this is the set of all maximal (i.e., n-dimensional) isotropic F subspaces in 2n. We will consider it just as a ﬁnite set, without using any addi- 2 tional structure on it. The symmetric group S acts on V by simultaneous per- n mutations of e ’s and f ’s. This action preserves the symplectic form and yields an i i K action of S on LGr(n). Consider the (cid:77)-vector space spanned by the set of orbits n L LGr(n)/S of the latter action; denote it by . Let n n L L = , n ≥ n 0 L K where = . 0∈ G Let Γ be a graph. Pick an arbitrary numbering of its vertices and take the n × | intersection matrix M(Γ). Consider an n 2n-matrix (M(Γ) Id ). Its rows span ⊂ F n a Lagrangian subspace L(Γ) 2n. This gives us an embedding 2 G → L . L L of the bialgebra of graphs into . Our next goal is to show that has a bialgebra G structure compatible with the multiplication and comultiplication on . To do this, L let us deﬁne the operations on . ⊂ F ∈ F The multiplication on is deﬁned as follows. Let L 2n and L 2m be two Lagrangian subspaces; then their direct sum L ⊕ L1 ⊂ F22(m+n) =2 F2n2⊕ F2m is 1 2 2 2 2 also Lagrangian. Then the multiplication map L ⊗ L → L n m m+n is deﬁned by · (cid:55)→ ⊕ [L ] [L ] [L L ], 1 2 1 2 where the square brackets stand for orbits of the symmetric groups on the corre- sponding Lagrangian Grassmannians. Clearly, this product is well-deﬁned, com- { } ⊂ F G mutative, associative, has a unit 0 0, and its restriction to gives the multi- G 2 plication on obtained by taking the disjoint union of graphs. L To deﬁne comultiplication in , we need to generalize the operation of restricting a graph to a subset of its vertices. This is done by symplectic reduction.

text: RIBBON

text: GRAPHS

text: AND

text: BIALGEBRA

text: OF

text: LAGRANGIAN

text: SUBSPACES

text: A

text: natural

text: idea

text: would

text: be

text: to

text: extend

text: the

text: bialgebra

text: structure

text: on

text: the

text: space

text: spanned

text: by

text: chord

text: diagrams

text: (or,

text: equivalently,

text: orientable

text: one-vertex

text: ribbon

text: graphs)

text: to

text: the

text: space

text: of

text: all

text: ribbon

text: graphs.

text: Unfortunately,

text: this

text: fails

text: even

text: on

text: the

text: stage

text: of

text: framed

text: chord

text: diagrams

text: (one-vertex

text: ribbon

text: graphs,

text: not

text: necessarily

text: orientable),

text: modulo

text: the

text: four-term

text: relation.

text: As

text: it

text: was

text: discussed

text: in

text: it

text: is

text: unclear

text: whether

text: the

text: multiplica-

text: tion

text: of

text: framed

text: chord

text: diagrams

text: deﬁned

text: in

text: the

text: same

text: way

text: as

text: in

text: the

text: previous

text: subsection

text: is

text: well-deﬁned.

text: (Recently

text: M.

text: Karev

text: showed

text: that

text: framed

text: chord

text: diagrams

text: form

text: a

text: module

text: over

text: the

text: bialgebra

text: of

text: chord

text: diagrams.)

text: However,

text: the

text: L-spaces

text: of

text: ribbon

text: graphs

text: do

text: have

text: a

text: bialgebra

text: structure.

text: We

text: have

text: seen

text: earlier

text: that

text: L-spaces

text: can

text: be

text: viewed

text: as

text: generalizations

text: of

text: intersection

text: matrices/graphs,

text: so

text: this

text: structure

text: is

text: naturally

text: extended

text: from

text: the

text: graph

text: bialgebra.

text: Just

text: as

text: in

text: the

text: case

text: of

text: graphs,

text: the

text: multiplication

text: and

text: comultiplication

text: of

text: L-spaces

text: is

text: well-deﬁned

text: even

text: without

text: the

text: four-term

text: relation.

text: We

text: have

text: seen

text: in

text: Example

text: that

text: L-spaces

text: of

text: ribbon

text: graphs

text: generalize

text: the

text: notion

text: of

text: intersection

text: matrices

text: for

text: chord

text: diagrams.

text: Intersection

text: matrices

text: are

text: deﬁned

text: by

text: their

text: adjacency

text: graphs;

text: the

text: only

text: diﬀerence

text: is

text: that

text: they

text: depend

text: on

text: a

text: speciﬁc

text: ordering

text: of

text: chords,

text: while

text: the

text: vertices

text: in

text: graphs

text: are

text: not

text: ordered.

text: To

text: deﬁne

text: a

text: proper

text: analogue

text: of

text: an

text: intersection

text: graph,

text: we

text: will

text: consider

text: not

text: just

text: L-spaces,

text: but

text: rather

text: their

text: orbits

text: under

text: the

text: action

text: of

text: a

text: symmetric

text: group.

text: F

text: F

text: Let

text: e

text: f

text: e

text: f

text: be

text: a

text: vector

text: space

text: over

text: with

text: a

text: n

text: n

text: standard

text: skew-symmetric

text: form

text: deﬁned

text: by

text: (e

text: f

text: δ

text: Consider

text: a

text: Lagrangian

text: i

text: j

text: ij

text: Grassmannian

text: LGr(n):

text: this

text: is

text: the

text: set

text: of

text: all

text: maximal

text: (i.e.,

text: n-dimensional)

text: isotropic

text: F

text: subspaces

text: in

text: We

text: will

text: consider

text: it

text: just

text: as

text: a

text: ﬁnite

text: set,

text: without

text: using

text: any

text: addi-

text: tional

text: structure

text: on

text: it.

text: The

text: symmetric

text: group

text: S

text: acts

text: on

text: V

text: by

text: simultaneous

text: per-

text: n

text: mutations

text: of

text: e

text: ’s

text: and

text: f

text: ’s.

text: This

text: action

text: preserves

text: the

text: symplectic

text: form

text: and

text: yields

text: an

text: i

text: i

text: K

text: action

text: of

text: S

text: on

text: LGr(n).

text: Consider

text: the

text: space

text: spanned

text: by

text: the

text: set

text: of

text: orbits

text: n

text: L

text: LGr(n)/S

text: of

text: the

text: latter

text: action;

text: denote

text: it

text: by

text: Let

text: n

text: n

text: L

text: L

text: n

text: n

text: L

text: K

text: where

text: 0∈

text: G

text: Let

text: Γ

text: be

text: a

text: graph.

text: Pick

text: an

text: arbitrary

text: numbering

text: of

text: its

text: vertices

text: and

text: take

text: the

text: n

text: intersection

text: matrix

text: M(Γ).

text: Consider

text: an

text: n

text: (M(Γ)

text: Id

text: Its

text: rows

text: span

text: F

text: n

text: a

text: Lagrangian

text: subspace

text: L(Γ)

text: This

text: gives

text: us

text: an

text: embedding

text: G

text: L

text: L

text: L

text: of

text: the

text: bialgebra

text: of

text: graphs

text: into

text: Our

text: next

text: goal

text: is

text: to

text: show

text: that

text: has

text: a

text: bialgebra

text: G

text: structure

text: compatible

text: with

text: the

text: multiplication

text: and

text: comultiplication

text: on

text: To

text: do

text: this,

text: L

text: let

text: us

text: deﬁne

text: the

text: operations

text: on

text: F

text: F

text: The

text: multiplication

text: on

text: is

text: deﬁned

text: as

text: follows.

text: Let

text: L

text: and

text: L

text: be

text: two

text: Lagrangian

text: subspaces;

text: then

text: their

text: direct

text: sum

text: L

text: is

text: also

text: Lagrangian.

text: Then

text: the

text: multiplication

text: map

text: L

text: L

text: L

text: n

text: m

text: m+n

text: is

text: deﬁned

text: by

text: [L

text: [L

text: [L

text: L

text: where

text: the

text: square

text: brackets

text: stand

text: for

text: orbits

text: of

text: the

text: symmetric

text: groups

text: on

text: the

text: corre-

text: sponding

text: Lagrangian

text: Grassmannians.

text: Clearly,

text: this

text: product

text: is

text: well-deﬁned,

text: com-

text: F

text: G

text: mutative,

text: associative,

text: has

text: a

text: unit

text: and

text: its

text: restriction

text: to

text: gives

text: the

text: multi-

text: G

text: plication

text: on

text: obtained

text: by

text: taking

text: the

text: disjoint

text: union

text: of

text: graphs.

text: L

text: To

text: deﬁne

text: comultiplication

text: in

text: we

text: need

text: to

text: generalize

text: the

text: operation

text: of

text: restricting

text: a

text: graph

text: to

text: a

text: subset

text: of

text: its

text: vertices.

text: This

text: is

text: done

text: by

text: symplectic

text: reduction.


text: RIBBON GRAPHS AND BIALGEBRA OF LAGRANGIAN SUBSPACES 23

text: RIBBON

text: GRAPHS

text: AND

text: BIALGEBRA

text: OF

text: LAGRANGIAN

text: SUBSPACES


text: 8 VICTOR KLEPTSYN AND EVGENY SMIRNOV ribbon graphs (or, what is the same, of multicomponent framed chord diagrams). Our next goal is to present a (what we think is a) reasonable analogue of this notion. Let G be a ribbon graph with n chords (for the moment, we will suppose for simplicity that these chords are numbered). Consider a 2n-dimensional symplectic F space V = 2n with the base e , . . . , e , f , . . . , f , and the intersection form deﬁned 2 1 n 1 n by (e , f ) = δ (and vanishing on any oth(cid:98)er couple of base vectors). Our goal is to i j ij ⊂ associate to the graph G a Lagrangian (i.e. maximal isotropic) subspace L(G) V . To do this, ﬁrst consider the surface G obtained from G by removing an open disk from each of its vertices. We will call this surface punctured ribbon graph; this is a compact surface with boundary. Next, for each edge E of G consider two paths: i let α connect the opposite sides of the edge E and let β connect along E the i i i i boundaries of the punctures in the vertices adjacent to E , as shown on Figure 8. i β i E i α i (cid:98) Figure 8. Two cycles corresponding to an edge; here ∂G is shown \ by bold lines, the boundary of punctures ∂G ∂G by thin ones, (punctured) vertices are ﬁlled with grey color, the edge E is un- i (cid:98) ﬁlled. (cid:98) (cid:98) The paths α and β deﬁne classes in the ﬁrst relative homology group of G i i ∈ F modulo its boundary: [α ], [β ] H (G, ∂G; ) (we suppose that all homology i F i 1 2 groups have coeﬃcients in(cid:98) )(cid:98). (cid:98) 2 F Now, the intersection index deﬁnes a natural (cid:98) -valued pairing between the rela- 2 F F tive homology group H (G, ∂G; ) and the ﬁrst homology group H (G; ). Con- 1 2 ∈ F 1 2 sider the intersection indices of a cycle γ H (G; ) with [α ] and [β ]. Roughly 1 2 i i speaking, the intersection index with [α ] tells us whether γ “crosses” E from one i i shore to the other, while the intersection index with [β ] tells us whether γ goes i along the boundary of the vertex from the left to the right near one of the endpoints of E . i (cid:88) (cid:88) Let us group a(cid:98)ll these intersection indices together, deﬁning a map F → ∩ · ∩ · (1) ϕ : H (G; ) V, ϕ (γ) = (γ [α ]) e + (γ [β ]) f . G 1 2 G i i i i i i This provides us with the following Deﬁnition 3.1. L-space, associated to the ribbon graph G, is deﬁned as the image of the map ϕ : G ⊂ (cid:98) L(G) := Im ϕ V. (cid:98) G F One can immediately note that L(G) is at most n-dimensional. Indeed, dim H (G; ) = 1 2 n+1, as G is homotopy equivalent to the wedge of n+1 circle. However, the “outer” ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: VICTOR

text: KLEPTSYN

text: AND

text: EVGENY

text: SMIRNOV

text: ribbon

text: graphs

text: (or,

text: what

text: is

text: the

text: same,

text: of

text: multicomponent

text: framed

text: chord

text: diagrams).

text: Our

text: next

text: goal

text: is

text: to

text: present

text: a

text: (what

text: we

text: think

text: is

text: a)

text: reasonable

text: analogue

text: of

text: this

text: notion.

text: Let

text: G

text: be

text: a

text: ribbon

text: graph

text: with

text: n

text: chords

text: (for

text: the

text: moment,

text: we

text: will

text: suppose

text: for

text: simplicity

text: that

text: these

text: chords

text: are

text: numbered).

text: Consider

text: a

text: symplectic

text: F

text: space

text: V

text: with

text: the

text: base

text: e

text: e

text: f

text: f

text: and

text: the

text: intersection

text: form

text: deﬁned

text: n

text: n

text: by

text: (e

text: f

text: δ

text: (and

text: vanishing

text: on

text: any

text: couple

text: of

text: base

text: vectors).

text: Our

text: goal

text: is

text: to

text: i

text: j

text: ij

text: associate

text: to

text: the

text: graph

text: G

text: a

text: Lagrangian

text: (i.e.

text: maximal

text: isotropic)

text: subspace

text: L(G)

text: V

text: To

text: do

text: this,

text: ﬁrst

text: consider

text: the

text: surface

text: G

text: obtained

text: from

text: G

text: by

text: removing

text: an

text: open

text: disk

text: from

text: each

text: of

text: its

text: vertices.

text: We

text: will

text: call

text: this

text: surface

text: punctured

text: ribbon

text: graph;

text: this

text: is

text: a

text: compact

text: surface

text: with

text: boundary.

text: Next,

text: for

text: each

text: edge

text: E

text: of

text: G

text: consider

text: two

text: paths:

text: i

text: let

text: α

text: connect

text: the

text: opposite

text: sides

text: of

text: the

text: edge

text: E

text: and

text: let

text: β

text: connect

text: along

text: E

text: the

text: i

text: i

text: i

text: i

text: boundaries

text: of

text: the

text: punctures

text: in

text: the

text: vertices

text: adjacent

text: to

text: E

text: as

text: shown

text: on

text: Figure

text: i

text: β

text: i

text: E

text: i

text: α

text: i

text: Figure

text: Two

text: cycles

text: corresponding

text: to

text: an

text: edge;

text: here

text: ∂G

text: is

text: shown

text: by

text: bold

text: lines,

text: the

text: boundary

text: of

text: punctures

text: ∂G

text: ∂G

text: by

text: thin

text: ones,

text: (punctured)

text: vertices

text: are

text: ﬁlled

text: with

text: grey

text: color,

text: the

text: edge

text: E

text: is

text: un-

text: i

text: ﬁlled.

text: The

text: paths

text: α

text: and

text: β

text: deﬁne

text: classes

text: in

text: the

text: ﬁrst

text: relative

text: homology

text: group

text: of

text: G

text: i

text: i

text: F

text: modulo

text: its

text: boundary:

text: [α

text: [β

text: H

text: (G,

text: ∂G;

text: (we

text: suppose

text: that

text: all

text: homology

text: i

text: F

text: i

text: groups

text: have

text: coeﬃcients

text: F

text: Now,

text: the

text: intersection

text: index

text: deﬁnes

text: a

text: natural

text: -valued

text: pairing

text: between

text: the

text: rela-

text: F

text: F

text: tive

text: homology

text: group

text: H

text: (G,

text: ∂G;

text: and

text: the

text: ﬁrst

text: homology

text: group

text: H

text: (G;

text: Con-

text: F

text: sider

text: the

text: intersection

text: indices

text: of

text: a

text: cycle

text: γ

text: H

text: (G;

text: with

text: [α

text: and

text: [β

text: Roughly

text: i

text: i

text: speaking,

text: the

text: intersection

text: index

text: with

text: [α

text: tells

text: us

text: whether

text: γ

text: “crosses”

text: E

text: from

text: one

text: i

text: i

text: shore

text: to

text: the

text: other,

text: while

text: the

text: intersection

text: index

text: with

text: [β

text: tells

text: us

text: whether

text: γ

text: goes

text: i

text: along

text: the

text: boundary

text: of

text: the

text: vertex

text: from

text: the

text: left

text: to

text: the

text: right

text: near

text: one

text: of

text: the

text: endpoints

text: of

text: E

text: i

text: Let

text: us

text: group

text: these

text: intersection

text: indices

text: together,

text: deﬁning

text: a

text: map

text: F

text: ϕ

text: H

text: (G;

text: V,

text: ϕ

text: (γ)

text: (γ

text: [α

text: e

text: (γ

text: [β

text: f

text: G

text: G

text: i

text: i

text: i

text: i

text: i

text: i

text: This

text: provides

text: us

text: with

text: the

text: following

text: Deﬁnition

text: L-space,

text: associated

text: to

text: the

text: ribbon

text: graph

text: G,

text: is

text: deﬁned

text: as

text: the

text: image

text: of

text: the

text: map

text: ϕ

text: G

text: L(G)

text: Im

text: ϕ

text: V.

text: G

text: F

text: One

text: can

text: immediately

text: note

text: that

text: L(G)

text: is

text: at

text: most

text: n-dimensional.

text: Indeed,

text: dim

text: H

text: (G;

text: as

text: G

text: is

text: homotopy

text: equivalent

text: to

text: the

text: wedge

text: of

text: circle.

text: However,

text: the

text: “outer”

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 VICTOR KLEPTSYN AND EVGENY SMIRNOV where a are the elements of the intersection matrix A(c), and the sum a + a ij F ij ji vanishes in , as this matrix is symmetric. 2 (cid:98) Proof of Theorem 3.4. Let us start with an alternative description of the space L(G). Namely, the surface G is homotopy equivalent to a 4-valent graph Γ on n vertices, obtained from G by contracting each ribbon E to the corresponding i vertex v (and the rest to its edges); see Fig. 10. i β i β i E i α α i i P i Figure 10. Contracting a ribbon graph to a 4-graph Then the four half-edges, starting from a vertex v , correspond to the four vertices i of the corresponding ribbon E (considered as a rectangle). Finally, these half-edges i are joined if the corresponding vertices of the ribbons are joined by a p(cid:98)art of the boundary of a vertex in the ribbon graph. F The homotopy equivalence gives us an ide(cid:88)ntiﬁcation between H (G; ) and F 1 2 H (Γ; ). Now, for each ribbon E the corresponding summand 1 2 i ∩ · ∩ · ϕ (γ) := (γ [α ]) e + (γ [β ]) f G,i i i i i i in (1) under this identiﬁcation corresponds to a map F → ϕ : H (Γ; ) V, Γ,i 1 2 ∈ F that counts the “crossing” by a cycle γ H (Γ; ) of the associated α and β (see 1 2 i i Fig. 10, right). For any i, denote by X the set of half-edges of Γ, adjacent to the vertex P . F → i i Let Ri : H1(Γ; 2) 2Xi be the restriction map, sending each cycle γ to the set of half-edges, adjacent to P , that it includes (in particular, R takes values only in i i subsets of X of even cardinality). It is then straightforward to check the following. i Lemma 3.6. ∈ F (i) For any γ H (Γ; ), if R (γ) consists of zero or four half-edges, one 1 2 i ∈ { } has ϕ (γ) = 0. Otherwise, ϕ (γ) e , f , e + f . Γ,i (cid:48) ∈ F Γ,i i i i i (cid:48) (cid:54) (ii) For any γ, γ H (Γ; ), one has ϕ (γ) = ϕ (γ ) = 0 if and only if (cid:48) 1 2 Γ,i Γ,i R (γ) and R (γ ) consist of two half-edges each, and either co(cid:80)incide, or i i are complementary in X . i (cid:98) Applying Lemma 3.6, we get that dim Ker ϕ = 1, where ϕ = ϕ is Γ Γ Fi Γ,i the map that corresponds to ϕ under the identiﬁcation between H (G; ) and F ∈ G F 1 2 H (Γ; ). Indeed, if for γ H (Γ; ) one has ϕ (γ) = 0, then for any vertex P 1 2 1 2 Γ i one has ϕ (γ) = 0, and hence γ visits any vertex P either by zero, or by all the Γ,i i four adjacent half-edges. An induction argument (together with the connectedness ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: VICTOR

text: KLEPTSYN

text: AND

text: EVGENY

text: SMIRNOV

text: where

text: a

text: are

text: the

text: elements

text: of

text: the

text: intersection

text: matrix

text: A(c),

text: and

text: the

text: sum

text: a

text: a

text: ij

text: F

text: ij

text: ji

text: vanishes

text: in

text: as

text: this

text: matrix

text: is

text: symmetric.

text: Proof

text: of

text: Theorem

text: Let

text: us

text: start

text: with

text: an

text: alternative

text: description

text: of

text: the

text: space

text: L(G).

text: Namely,

text: the

text: surface

text: G

text: is

text: homotopy

text: equivalent

text: to

text: a

text: graph

text: Γ

text: on

text: n

text: vertices,

text: obtained

text: from

text: G

text: by

text: contracting

text: each

text: ribbon

text: E

text: to

text: the

text: corresponding

text: i

text: vertex

text: v

text: (and

text: the

text: rest

text: to

text: its

text: edges);

text: see

text: Fig.

text: i

text: β

text: i

text: β

text: i

text: E

text: i

text: α

text: α

text: i

text: i

text: P

text: i

text: Figure

text: Contracting

text: a

text: ribbon

text: graph

text: to

text: a

text: Then

text: the

text: four

text: half-edges,

text: starting

text: from

text: a

text: vertex

text: v

text: correspond

text: to

text: the

text: four

text: vertices

text: i

text: of

text: the

text: corresponding

text: ribbon

text: E

text: (considered

text: as

text: a

text: rectangle).

text: Finally,

text: these

text: half-edges

text: i

text: are

text: joined

text: if

text: the

text: corresponding

text: vertices

text: of

text: the

text: ribbons

text: are

text: joined

text: by

text: a

text: of

text: the

text: boundary

text: of

text: a

text: vertex

text: in

text: the

text: ribbon

text: graph.

text: F

text: The

text: homotopy

text: equivalence

text: gives

text: us

text: an

text: between

text: H

text: (G;

text: and

text: F

text: H

text: (Γ;

text: Now,

text: for

text: each

text: ribbon

text: E

text: the

text: corresponding

text: summand

text: i

text: ϕ

text: (γ)

text: (γ

text: [α

text: e

text: (γ

text: [β

text: f

text: G,i

text: i

text: i

text: i

text: i

text: i

text: in

text: under

text: this

text: identiﬁcation

text: corresponds

text: to

text: a

text: map

text: F

text: ϕ

text: H

text: (Γ;

text: V,

text: Γ,i

text: F

text: that

text: counts

text: the

text: “crossing”

text: by

text: a

text: cycle

text: γ

text: H

text: (Γ;

text: of

text: the

text: associated

text: α

text: and

text: β

text: (see

text: i

text: i

text: Fig.

text: right).

text: For

text: any

text: i,

text: denote

text: by

text: X

text: the

text: set

text: of

text: half-edges

text: of

text: Γ,

text: adjacent

text: to

text: the

text: vertex

text: P

text: F

text: i

text: i

text: Let

text: Ri

text: be

text: the

text: restriction

text: map,

text: sending

text: each

text: cycle

text: γ

text: to

text: the

text: set

text: of

text: half-edges,

text: adjacent

text: to

text: P

text: that

text: it

text: includes

text: (in

text: particular,

text: R

text: takes

text: values

text: only

text: in

text: i

text: i

text: subsets

text: of

text: X

text: of

text: even

text: cardinality).

text: It

text: is

text: then

text: straightforward

text: to

text: check

text: the

text: following.

text: i

text: Lemma

text: F

text: (i)

text: For

text: any

text: γ

text: H

text: (Γ;

text: if

text: R

text: (γ)

text: consists

text: of

text: zero

text: or

text: four

text: half-edges,

text: one

text: i

text: has

text: ϕ

text: (γ)

text: Otherwise,

text: ϕ

text: (γ)

text: e

text: f

text: e

text: f

text: Γ,i

text: F

text: Γ,i

text: i

text: i

text: i

text: i

text: (ii)

text: For

text: any

text: γ,

text: γ

text: H

text: (Γ;

text: one

text: has

text: ϕ

text: (γ)

text: ϕ

text: (γ

text: if

text: and

text: only

text: if

text: Γ,i

text: Γ,i

text: R

text: (γ)

text: and

text: R

text: (γ

text: consist

text: of

text: two

text: half-edges

text: each,

text: and

text: either

text: or

text: i

text: i

text: are

text: complementary

text: in

text: X

text: i

text: Applying

text: Lemma

text: we

text: get

text: that

text: dim

text: Ker

text: ϕ

text: where

text: ϕ

text: ϕ

text: is

text: Γ

text: Γ

text: Fi

text: Γ,i

text: the

text: map

text: that

text: corresponds

text: to

text: ϕ

text: under

text: the

text: identiﬁcation

text: between

text: H

text: (G;

text: and

text: F

text: G

text: F

text: H

text: (Γ;

text: Indeed,

text: if

text: for

text: γ

text: H

text: (Γ;

text: one

text: has

text: ϕ

text: (γ)

text: then

text: for

text: any

text: vertex

text: P

text: Γ

text: i

text: one

text: has

text: ϕ

text: (γ)

text: and

text: hence

text: γ

text: visits

text: any

text: vertex

text: P

text: either

text: by

text: zero,

text: or

text: by

text: all

text: the

text: Γ,i

text: i

text: four

text: adjacent

text: half-edges.

text: An

text: induction

text: argument

text: (together

text: with

text: the

text: connectedness

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##