I'm studying Milnor's Singularities of Complex Hypersurfaces, and a small, perhaps moot, point in Chapter 10 has me thinking in circles. (I asked a related but different question here).

Here is some relevant background material, most of which can be found in the book.

Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with an isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Milnor proves that the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$ is a fibration, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link which we denote by $K_{f}$. (It is known that all algebraic links are fibered links (by definition) as well as iterated torus links.)

In 1928, Brauner proved that for $f = z_{1}^{p} + z_{2}^{q}$, $K_{f}$ is a torus link $T_{p,q}$, which is a knot if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is equal to the delta invariant of the corresponding complex algebraic plane curve. By a theorem of Kronheimer and Mrowka in 1992, the conjecture is true.

In the same book, Milnor proves the relation $\mu = 2 \delta - r + 1$, where $\delta$ is the delta invariant, $r$ is the number analytically irreducible branches of $V_{f,0}$ passing through the origin and $\mu = \dim_{\mathbb{C}} \mathbb{C}\{ z_1, \dots, z_n \} / \langle \partial_1 f, \dots, \partial_n f \rangle$.

Questions: Is there an algebraic link that is not of the torus-type $T_{p,q}$; one whose crossing number, unknotting number, etc. is well-known? For example, what is the link of $f = z_1^{a} z_2^{b} + z_1^{c} z_{2}^{d}$, where not both $ac = 0$ and $bd = 0$ (provided that $a,b,c,d$ are non-negative integers such that $f$ has an isolated critical point at the origin)?

Can the delta invariant be defined for any non-degenerate weighted homogeneous polynomial of two variables? If so, how does one compute it? My understanding is that the delta invariant can be defined if and only if the polynomial is square-free, which leads me to yet another question: What is a square-free complex polynomial? If so, how does one reconcile the fact that the corresponding unknotting number exists but the delta invariant may not be defined, if these are suppose to be equal?

Edit: The question about square-free complex polynomials has been answered in the comments. Since the ring of convergent power series $\mathbb{C}\{ z_1, \dots, z_n \}$ is a UFD, then a series $f$ is square-free if and only if factored into irreducibles, $f = f_1^{r_1} \cdots f_{n}^{r_{n}}$, it is the case $r_{i} = 1$. Note, here $r = \sum_i r_i$. This is true (though not too simple to prove) if and only if $\mu$ is non-negative and finite.


  • $\begingroup$ I'm not sure I understand your question about square-free complex polynomials. $\mathbb{C}[x_1, ... x_n]$ is a UFD, so like all UFDs it has a natural notion of square-freeness (an element with no repeated irreducible factors). $\endgroup$ – Qiaochu Yuan Aug 8 '11 at 19:59
  • $\begingroup$ @Qiaochu Yuan: Thanks. $\endgroup$ – user02138 Aug 8 '11 at 20:19

Check out the Knot Atlas, which has a tabulation of known invariants for various knots. For example, the figure-8 knot $4_1$ has crossing number $4$ and unknotting number $1$ but is not a torus knot. A good way to rule out that a knot is a torus knot is to check if it has hyperbolic volume. All torus knots are of Seifert-fibered type, which means their complement can't be hyperbolic. In fact, any randomly chosen prime knot is highly likely to be hyperbolic, as the table shows.

By the way, unknotting number is often difficult to compute, so many of the entries in the table don't have this invariant, but maybe one of them which does will be what you're looking for.

Edit: It is known that any algebraic link, in this sense, is an iterated torus link. None of these examples is hyperbolic. An iterated torus link is an iterated satellite of a torus link by other torus links. I'm not sure what the status is on the crossing or unknotting numbers of such knots. Even computing the crossing number of a generic satellite is a difficult problem. This paper of Lackenby shows that how far we are from realistic results in that direction, but it may be that for the special case of iterated torus knots the answer is known.

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  • $\begingroup$ This seems more like a list of links up to crossing. Are algebraic links listed? $\endgroup$ – user02138 Aug 8 '11 at 21:40
  • $\begingroup$ Oops, sorry. I was thinking that most small links are algebraic, but that's using a different definition than this one. (I was thinking algebraic meant closure of an algebraic tangle, which is not the usage here.) According to one reference I just found, algebraic in your sense means "Iterated torus knot," but I don't know the precise definition of what that means. $\endgroup$ – Cheerful Parsnip Aug 8 '11 at 22:28

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