TITLE: Examples of Non-algebraic Fibered Knots?
QUESTION [7 upvotes]: I am currently reading a monograph by Jose Seade, " On the topology of isolated singularities in analytic spaces". 
 I have following questions but before asking questions I recall the definition of algebraic knots/ links
Definition : Let $ f : (\mathbb{C}^2,0) \rightarrow (\mathbb{C},0)$ be a holomorphic function with an isolated critical point at $0$. Then for sufficiently small $\epsilon >0$ we have $ K= \mathbb{S}^3_{\epsilon} \cap f^{-1}(0)$. The knot/link $K$ is called algebraic knot/link.  
By Milnor's fibration theorem for complex singularities, we get an open book decomposition $(\mathbb{S}^3,K) $ .

Questions : 

Here we think of knot as an embedded copy of $\mathbb{S}^1$ in $\mathbb{S}^3 $.  Suppose we have an algebraic knot $K$. Is there a fibered knot $K'$ in the isotopy class of $K$ which is not algebraic? 
In general, how to detect whether a fibered knot $K$ is algebraic or not?

REPLY [8 votes]: The algebraic knots are iterated cablings of the unknot, so have simplicial volume $=0$. 
Thus, "most" fibered knots will not be algebraic. 
There is an algorithm to find a minimal genus Seifert surface of a knot. 
Moreover, one may determine if it is a fiber of a fibration, and compute
the conjugacy class in the mapping class group of the monodromy of the fiber. 
There are various approaches, e.g. using sutured manifold theory. One may 
determine whether a mapping class is completely reducible (having no
pseudo-Anosov components), and therefore
whether the mapping torus has simplicial volume $=0$. In fact, you
may use this to determine which iterated cabling of the unknot you have
algorithmically. Then you can apply the criterion of Eisenbud and Neumann
to see if it is algebraic. 
This answer is interpreting "detect" as "does
there exist an algorithm", which I'm not sure if this is what you want. 
I think there may be ways of doing this now using Heegaard Floer homology,
although I'm not certain if one can detect simplicial volume $=0$ using
that invariant. 
I should remark also that the term "algebraic knot" can have other meanings
in topology.