TITLE: Computation of \tau invariant
QUESTION [5 upvotes]: I am trying to understand the following inequality, $$0 \leq \tau (K_{+}) - \tau(K_{-}) \leq 1$$ from the following paper by Livingston. \ https://arxiv.org/pdf/math/0311036.pdf . At page 737 , he argues, a negative crossing change converts  $-T_{2,3}$ into the unknot.So $(K_{+} \# -T_{2,3} \#-K_{-})$ bounds a disks with 2 double point. Resolving the double point we get a genus 1 surface.  I am trying to construct an explicit movie from $K_{+}$ to $T_{2,3} \# K_{-}$ with a genus one cobordism between them. 
   I end up  having a genus 2 cobordism instead of 1. 
   Please help me with the movie move.

REPLY [3 votes]: See Livingston's "Computations of the Ozsvath-Szabo knot concordance invariant" (https://arxiv.org/abs/math/0311036), Corollary 3.
Or see my thesis for a picture of Livingston's cobordism (p. 19, http://lewark.de/lukas/PhDthesis-Lukas-Lewark.pdf).