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On the slice-torus invariant $q_M$ from $\mathbb{Z}_2$-equivariant Seiberg--Witten Floer cohomology

Nobuo Iida, Taketo Sano, Kouki Sato, Masaki Taniguchi

TL;DR

The paper proves that the $ Z$-valued slice-torus invariant $q_M$ from $ Z_2$-equivariant Seiberg--Witten theory cannot be expressed as a linear combination of several established concordance invariants, by exhibiting $K=9_{42}$ with $q_M(9_{42})=-1$ while a wide class of other invariants vanish at this knot. The argument combines a key $L$-space result for the branched double cover $ Sigma_2(9_{42})$, vanishing surgery and instanton invariants, Bar-Natan/computational homology inputs for $ ilde{ss}_H$, and spectral-sequence analyses of $ rak{sl}_N$-concordance invariants $s_{ abla w, abla ext{α}}$ to rule out any linear combination. Additionally, the authors discuss the relationship of $q_M$ with the Bars and $ heta$-invariant, highlighting how $q_M$ can provide sharper 4-ball genus bounds in certain cases and offering a comprehensive survey of $q_M$ and $ heta$ values for prime knots with small crossing numbers. The work thus establishes the distinctness of $q_M$ within the landscape of slice-torus invariants and informs ongoing conjectures about their interrelations.

Abstract

We show that Iida--Taniguchi's $\mathbb{Z}$-valued slice-torus invariant $q_M$ cannot be realized as a linear combination of Rasmussen's $s$-invariant, Ozsváth--Szabó's $τ$-invariant, all of the $\mathfrak{sl}_N$-concordance invariants ($N \geq 2$), Baldwin--Sivek's instanton $τ$-invariant, Daemi--Imori--Sato--Scaduto--Taniguchi's instanton $\tilde{s}$-invariant and Sano--Sato's Rasmussen type invariants $\tilde{ss}_c$.

On the slice-torus invariant $q_M$ from $\mathbb{Z}_2$-equivariant Seiberg--Witten Floer cohomology

TL;DR

The paper proves that the -valued slice-torus invariant from -equivariant Seiberg--Witten theory cannot be expressed as a linear combination of several established concordance invariants, by exhibiting with while a wide class of other invariants vanish at this knot. The argument combines a key -space result for the branched double cover , vanishing surgery and instanton invariants, Bar-Natan/computational homology inputs for , and spectral-sequence analyses of -concordance invariants to rule out any linear combination. Additionally, the authors discuss the relationship of with the Bars and -invariant, highlighting how can provide sharper 4-ball genus bounds in certain cases and offering a comprehensive survey of and values for prime knots with small crossing numbers. The work thus establishes the distinctness of within the landscape of slice-torus invariants and informs ongoing conjectures about their interrelations.

Abstract

We show that Iida--Taniguchi's -valued slice-torus invariant cannot be realized as a linear combination of Rasmussen's -invariant, Ozsváth--Szabó's -invariant, all of the -concordance invariants (), Baldwin--Sivek's instanton -invariant, Daemi--Imori--Sato--Scaduto--Taniguchi's instanton -invariant and Sano--Sato's Rasmussen type invariants .
Paper Structure (4 sections, 7 theorems, 32 equations, 3 figures, 2 tables)

This paper contains 4 sections, 7 theorems, 32 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Proof of \ref{['main']}
  3. $q_M$ and $\theta$ for prime knots with small crossing number
  4. Tables

Key Result

Theorem 1.1

Let us denote by $s$, $\tau$, $s_{\partial \omega, \alpha}$, $\tau^\#$, $\tilde{s}$, and $\tilde{ss}_c$ the Rasmussen invariant Ra10, the Ozsváth--Szabó $\tau$-invariant OS03, the $\mathfrak{sl}_N$-concordance invariant ($N \geq 2$) Lobb09Wu09LL:2016 with any separable potential $\partial \omega$ eq where the convention of $9_{42}$ follows the knotinfo knotinfo. In particular, the invariant $q_M$

Figures (3)

  • Figure 1: Kirby moves which show $S^3_1(9_{42}) \cong S^3_{-1}(K)$ for some knot $K$ in $S^3$
  • Figure 2: Kirby moves which show that $S^3_1(-9_{42})$ bounds a smooth compact contractible 4-manifold
  • Figure 3: A diagram of $9_{42}$ from the KnotInfo knotinfo. We added names $A, B, C$ for three crossings and markings.

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • proof
  • Remark 1.4
  • Lemma 2.1
  • proof : Proof of \ref{['vanishing']}
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 9 more