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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 :

  1. 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?
  2. In general, how to detect whether a fibered knot $K$ is algebraic or not?
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What do you mean by "a fibred knot $K′$ in the isotopy class of $K$"? I'm asking this because, usually, knots are studied up to isotopy (and the question has the "knot-theory" tag). Anyway, regarding question 1: the figure-eight not is fibred but not algebraic. –  Marco Golla Sep 7 '11 at 9:45
On the other hand, if you're asking for given embeddings $S^1\to S^3 \subset \mathbb{C}^2$, you have that "most" deformations of an algebraic embedding are not algebraic. For example, any algebraic knot in this sense has to be transverse to the standard contact structure on $S^3$. –  Marco Golla Sep 7 '11 at 9:48

2 Answers 2

up vote 4 down vote accepted

Let me elaborate a bit on my comments.

First of all, algebraic knots (up to isotopy) have been classified. People refer to an unpublished paper by Bonahon and Siebenmann, The classification of algebraic links, and there appears to be a discussion of this fact in the book Three-dimensional link theory and invariants of plane curve singularities by Eisenbud and Neumann (ed. Princeton University Press): they're some iterated cables of the unknot (some positivity condition is involved, as -for example- negative torus knots are not algebraic).

1. Your question, on the other hand, seems to be more focused on a given embedding. Let's assume that your definition of algebraic is that you identify $S^3_\varepsilon$ with $S^3_1$ using the rescaling in $\mathbb{C}^2$, and define $K$ to be the embedding of $K\simeq S^1$ in $S^3$ using these identifications*. My guess would be that, using this definition, most smooth perturbations of $K$ are not algebraic (mostly because real functions are many more than complex ones).

On the other hand, it's easy to produce a non-algebraic knot (under the same definition) in any isotopy class of knots; call $J$ the canonical complex structure on $\mathbb{C}^2$: let's observe that if $K$ is algebraic, parametrised by $\gamma$, then $J(\dot\gamma)$ lies in $T(f^{-1}(0))$ and is linearly independent of $\dot\gamma$, therefore it can't lie in $TS^3$. Therefore, $\dot\gamma$ is never a complex tangency to $S^3$ (i.e. $\dot\gamma \not\in TS^3\cap JTS^3$, that is, $K$ is a transverse knot with respect to the standard contact structure on $S^3$). In particular, take a given algebraic $K$, then you can just bend it so that somewhere it has a complex tangency, and that's an obstruction to being algebraic. (As it happens, you can also ask for a Legendrian approximation, but that's more than we need)

2. Let me now assume that you're talking about algebraic knots as isotopy classes of algebraic knots (as in the previous definition). As I said, algebraic knots have been classified, so the question is -in principle- solved. Still, there are some nice obstructions to being algebraic. As you wrote, an algebraic knot gives an open book for $S^3$, and an open book supports a contact structure: an algebraic knot has to support the standard contact structure (the one defined above, i.e. the complex tangencies to $S^3$). Now, $S^3$ admits (infinitely many) overtwisted contact structures (Bennequin) and every contact structure is supported by an open book, which is unique up to stabilisations (Giroux). In particular, every contact structure admits an open book with connected binding. Let's take any overtwisted $\xi$ on $S^3$, take an open book with connected binding $K$ supporting $\xi$: $K$ is fibred (by definition), but not algebraic.

The question of whether a fibred knot (hence, an open book) supports the standard contact structure has in turn been completely solved (Etnyre and Van Horn-Morris): a fibred knot $K$ supports $\xi_{\rm st}$ if and only if it's quasi-positive, and in turn this holds if and only if $g(K) = \tau(K)$ (where $\tau$ is the concordance invariant of $K$ coming from knot Floer homology), and this holds if and only if the maximal self-linking number of $K$ (that is, the maximal self-linking number of a transverse representative of $K$ with respect to $\xi_{\rm st}$) is $2g(K)-1$.

In particular, the figure-eight knot has maximal self-linking number -3 (and $\tau = 0$), so it can't be algebraic. Also, the mirror of any nontrivial algebraic knot is not algebraic (since $\tau$ changes sign when taking the mirror, but probably also using "classical" inequalities for the self-linking number coming from knot polynomials).

* This embedding is defined up to self-diffeomorfisms of $S^1$.

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Thanks a lot for elaborate and nice answer and references too! –  Dheeraj Kulkarni Sep 7 '11 at 17:41
The term "algebraic knots" referred to in Bonahon-Siebenmann I believe refers to a different notion than that in Eisenbud-Neumann. The Bonahon-Siebenmann algebraic knots/links have a 2-fold branched cover which is a graph manifold (made from Seifert-fibered spaces). The complements of the Eisenbud-Neumann ones are graph manifolds, so there is some relation. –  Ian Agol Sep 8 '11 at 15:54
I couldn't check any of the references I've given for the classification, and I couldn't even find a proper statement (although I remember reading it, once). I'll be glad to find the statement somewhere. –  Marco Golla Sep 8 '11 at 19:32

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.

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By "detect" I meant obstruction for $K$ to be an algebraic knot. Thank you for illuminating answer. –  Dheeraj Kulkarni Sep 8 '11 at 8:48

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