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A family of slice-torus invariants from the divisibility of Lee classes

Taketo Sano, Kouki Sato

TL;DR

The authors develop a family of slice-torus invariants \tilde{ss}_c from the c-divisibility of the reduced Lee class in a reduced Khovanov framework, parameterized by prime elements in a PID. They show that for (R,c)=(F[H],H) this invariant coincides with the Rasmussen invariant s^F, and prove equality with the unreduced ss_c in (F[H],H) and (\mathbb{Z},2) cases, while demonstrating that ss_3 is not slice-torus in general, implying independence from reduced and Rasmussen-type invariants. A detailed reduction of parameters, module structures, reduced homology, refined Lee classes, and cobordism behavior under connected sums and mirrors underlie the construction and its classification. They also provide computational evidence via a public program, revealing that s^Q can differ from s^{\mathbb{F}_2} and related invariants, and thus confirming nontrivial interactions between different coefficient systems. Overall, the work clarifies when the new reduced invariant recovers known Rasmussen-type invariants and when unreduced variants reveal new slice-torus information, with practical implications for distinguishing concordance classes.

Abstract

We give a family of slice-torus invariants $\tilde{ss}_c$, each defined from the $c$-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements $c$ in any principal ideal domain $R$. For the special case $(R, c) = (F[H], H)$ where $F$ is any field, we prove that $\tilde{ss}_c$ coincides with the Rasmussen invariant $s^F$ over $F$. Compared with the unreduced invariants $ss_c$ defined by the first author in a previous paper, we prove that $ss_c = \tilde{ss}_c$ for $(R, c) = (F[H], H)$ and $(\mathbb{Z}, 2)$. However for $(R, c) = (\mathbb{Z}, 3)$, computational results show that $ss_3$ is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants.

A family of slice-torus invariants from the divisibility of Lee classes

TL;DR

The authors develop a family of slice-torus invariants \tilde{ss}_c from the c-divisibility of the reduced Lee class in a reduced Khovanov framework, parameterized by prime elements in a PID. They show that for (R,c)=(F[H],H) this invariant coincides with the Rasmussen invariant s^F, and prove equality with the unreduced ss_c in (F[H],H) and (\mathbb{Z},2) cases, while demonstrating that ss_3 is not slice-torus in general, implying independence from reduced and Rasmussen-type invariants. A detailed reduction of parameters, module structures, reduced homology, refined Lee classes, and cobordism behavior under connected sums and mirrors underlie the construction and its classification. They also provide computational evidence via a public program, revealing that s^Q can differ from s^{\mathbb{F}_2} and related invariants, and thus confirming nontrivial interactions between different coefficient systems. Overall, the work clarifies when the new reduced invariant recovers known Rasmussen-type invariants and when unreduced variants reveal new slice-torus information, with practical implications for distinguishing concordance classes.

Abstract

We give a family of slice-torus invariants , each defined from the -divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements in any principal ideal domain . For the special case where is any field, we prove that coincides with the Rasmussen invariant over . Compared with the unreduced invariants defined by the first author in a previous paper, we prove that for and . However for , computational results show that is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants.
Paper Structure (25 sections, 82 theorems, 187 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 25 sections, 82 theorems, 187 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Khovanov homology
  3. Khovanov homology
  4. Lee classes
  5. Reduction of parameters
  6. Module structures and reduced homology
  7. Module structures
  8. Cobordism maps
  9. Reduced homology
  10. Reduced Lee classes
  11. Connected sums and mirrors
  12. Connected sums
  13. Mirrors
  14. The invariant $\widetilde{\mathit{ss}}_c$
  15. Setup
  16. ...and 10 more sections

Key Result

Theorem 1

For each prime element $c$ in a principal integral domain $R$, the value defined by is a slice-torus invariant. Here $K$ is a knot with diagram $D$, $d_c(D)$ the $c$-divisibility of the reduced Lee class $\widetilde{\alpha}_c(D)$, $w(D)$ the writhe and $r(D)$ the number of Seifert circles of $D$.

Figures (2)

  • Figure 1: Construction of the Lee cycle.
  • Figure 2: Visualization of the isomorphism class of $C_{h, t}$.

Theorems & Definitions (172)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: BarNatan:2004, Khovanov:2004
  • Definition 2.4
  • Definition 2.5
  • ...and 162 more