A note on cobordisms of algebraic knots

Abstract

In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant ν + : we study its behaviour with respect to connected sums, providing an explicit formula in the case of L-space knots and proving subadditivity in general.

A full-twist inequality for the ν+-invariant

“…Here we note that Theorem 1.1 gives an inequality for J#(−K) rather than J and K. However, by subadditivity of ν + [1], we also have the following result for J and K. Theorem 1.2. Suppose that K is deformed into J by a positive full-twist with n-linking.…”

Truncated Heegaard Floer homology and knot concordance invariants

“…Similarly, consider the chain maps v ′ s : C{i = 0} → C{min(i, j − s) = 0}, v + ′ s : C{i ≥ 0} → C{min(i, j − s) ≥ 0}, consisting of quotienting by C{i = 0, j ≤ s} followed by the inclusion. Ozsváth and Szabó [11] show that these maps induce the maps from (1) and (2).…”

“…defined by taking the quotient by C{i < 0, j = s}, respectively C{i < 0, j ≥ s}, followed by the inclusions. The large integer surgery formula of Ozsváth-Szabó [11] asserts that the maps v s and v + s induce the maps from (1) and (2). Similarly, consider the chain maps v ′ s : C{i = 0} → C{min(i, j − s) = 0}, v + ′ s : C{i ≥ 0} → C{min(i, j − s) ≥ 0}, consisting of quotienting by C{i = 0, j ≤ s} followed by the inclusion.…”

“…Another question is how the invariants ν n (K) behave under connected sum. It is known that ν + (K) is subadditive by [2]. That is, ν + (K#L) ≤ ν + (K) + ν + (L)…”

Genus One Cobordisms Between Torus Knots

“…Trotter's classical knot signature [Tro62] and the Tristram-Levine signatures [Tri69,Lev69] allow to determine the cobordism distance of most torus knots of two fixed braid indices up to a constant [Baa12]. The modern Heegaard Floer concordance invariants ν + [HW16] and Υ [OSS17] lead to better bounds on cobordism distance depending on the braid indices [BCG17,FK17]. And for small braid indices, these invariants allow to compute the cobordism distance completely [Fel16,BFLZ16].…”