Milnor conjecture on slice genus of torus knots

Remarks

For any knot, $g_4(K) \le u(K)$: each crossing change can be realized by adding a handle to a surface in $B^4$. An explicit unknotting shows $u(T(p, q)) \le (p-1)(q-1)/2$. The hard part is the lower bound $g_4(T(p, q)) \ge (p-1)(q-1)/2$. Kronheimer and Mrowka proved it in 1993 using gauge theory. Rasmussen later gave a combinatorial proof with his invariant from Khovanov homology. Example: for the trefoil knot $T(2, 3)$, the formula gives $g_4 = u = 1$. Milnor asked the question in his study of singular points of complex plane curves, where torus knots appear as links of singularities.

Going deeper

For a student

The upper bounds are elementary, but lower bounds on the slice genus need deep tools. Kronheimer and Mrowka obtained the bound from gauge theory, through lower bounds on the genus of embedded surfaces in $4$-manifolds. Rasmussen's $s$-invariant, the Ozsváth–Szabó $\tau$-invariant from knot Floer homology, and the signature give bounds of the form $g_4(K) \ge \abs{s(K)}/2$ and $g_4(K) \ge \abs{\tau(K)}$. For torus knots, the $s$- and $\tau$-invariants are large enough to give the exact value.