On the topological 4-genus of torus knots

Abstract

Abstract. We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.

The four-genus of a link, Levine–Tristram signatures and satellites

“…In Section 2 we construct examples of knots for which the upper and lower bounds provided by the two theorems above enable us to compute the 4-genus, and for which other methods for creating an upper bound do not seem to work. In particular we compare Theorem 1.7 with recent results on the topological 4-genus of [Fel15], [FM15] and [BFLL15].…”

“…(6) The infection curves (η 1 , η 2 ) intersect the obvious minimal genus Seifert surface for K, therefore only by stabilising this Seifert surface can we easily understand a Seifert surface for S(K, J). Lukas Lewark informs me that one can apply the techniques of [BFLL15] to reduce this stabilised Seifert surface, pushed into the 4-ball, to a genus 4 surface. It would be interesting to find some examples where the approach of [Fel15], [BFLL15] of excising a part of the Seifert surface with Alexander polynomial one, fails to find a sharp upper bound, but Theorem 1.7 does.…”

Heegaard Floer Homologies and Rational Cuspidal Curves. Lecture notes.

“…Observe that each of these sets is preserved by an action of Z q given by [1] · (z 1 , z 2 ) = (e 2πi/q · z 1 , e 2πip/q · z 2 ), and the orbits U 0 /Z q , U 1 /Z q are again solid tori. Finally, the quotient Σ/Z q is a torus.…”

“…The assumption that C is smooth is essential. For example, in [44] there are constructed locally flat embedded surfaces C in CP 2 such that χ(C) > −d(d − 3), where d is the degree of C. This problem is related to showing that the topological four-genus of some algebraic knots is strictly less than the smooth four-genus; see [103,1]. 2 In Heegaard Floer theory, the adjunction inequality is a key tool in proving many important theorems.…”

On classical upper bounds for slice genera

“…The algebraic genus of torus links satisfies[BFLL18] g alg (T p,q ) g(T p,q ) ≤ 3 4and for p ≥ q ≥ 3, and (p, q) ∈ {(3, 3), (4, 3), (5, 3), (6, 3), (4, 4)}:1 2 ≤ g alg (T p,q ) g(T p,q ) ≤ 6 7 = g alg (T 8,3 ) g(T 8,3 ).…”