# The Milnor Conjecture on the Unknotting Number of a Torus Knot

Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. (This is the same geometric genus in the Yau-Durfee Theorem: $\mu \geq n! \ p_{g}$ for $n > 2$, See Y93 or Y06.)

Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.

Consider the Brieskorn-Pham polynomial $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. In particular, for $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of the local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.

In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.

(Edited) Question(s): Is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$ for $n > 2$? Is there a geometric genus-like object for $n = 2$? If so, how might it relate to invariants of the torus knots?

(Also posted on MO.)

Thanks!

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@user02138: I think this question would be fine for MathOverflow. –  Qiaochu Yuan Nov 8 '10 at 12:27
Typo: I think in your 3rd paragraph you mean to say the homology of the Milnor fibre is free of rank $(p-1)(q-1)$, rather than saying the homology is $(p-1)(q-1)$. More broadly speaking, this Milnor conjecture isn't an algebraic-geometric conjecture as at some point you have to talk about unknotting numbers and that's not an idea that fits with algebraic geometry. So I can't see a positive answer to your question. –  Ryan Budney Nov 8 '10 at 16:06
@Ryan: fixed. thanks. –  user02138 Nov 8 '10 at 16:10
@Ryan: I don't understand your comment. Kronheimer and Mrowka proved the Milnor conjecture by showing something about genera of projective curves. The relation between torus knots and singularity theory is well-established and questions of knot invariants falls nicely within the realm of algebraic geometry. For instance, see "Singular Points of Complex Hypersurfaces" by Milnor. –  user02138 Nov 8 '10 at 16:20
@Ryan: Thanks for being so helpful! I really appreciate it. –  user02138 Nov 8 '10 at 18:42