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The AJ conjecture and connected sums of torus knots

Xingru Zhang

Abstract

The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and $a$ have the same sign. Notably, in cases where $pq=ab$ but $p\ne a$, the recurrence polynomial $α(t,M,L)$ of $T(p,q)\#T(a,b)$ has repeated factors involving the variable $L$ after evaluation at $t=-1$. These appear to be the first examples of knots exhibiting this phenomenon. Therefore, the AJ conjecture requires a slight modification to accommodate this possibility.

The AJ conjecture and connected sums of torus knots

Abstract

The set of isotopy classes of nontrivial torus knots in is in bijection with the set of coprime integer pairs satisfying . We verify the AJ conjecture for the connected sums when and have the same sign. Notably, in cases where but , the recurrence polynomial of has repeated factors involving the variable after evaluation at . These appear to be the first examples of knots exhibiting this phenomenon. Therefore, the AJ conjecture requires a slight modification to accommodate this possibility.
Paper Structure (9 sections, 13 theorems, 136 equations)

This paper contains 9 sections, 13 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Some formulas
  3. Case $|p|>q>2$, $|a|>b>2$, $pq\ne ab$, $p$ and $a$ have the same sign
  4. Case $|p|>q>2$, $(a,b)=(p,q)$
  5. Case $|p|>q>2$, $|a|>b>2$, $pq=ab$, $p\ne a$
  6. Case $|p|>q>2$, $|a|>b=2$, $pq\ne 2a$, $p, a$ have the same sign
  7. Case $|p|>q>2$, $|a|>b=2$, $pq=2a$
  8. Case $|p|>q=2$, $|a|>b=2$, $p\ne a$ and $p,a$ have the same sign
  9. Case $|p|>q=2$, $|a|>b=2$, $p= a$

Key Result

Theorem 1.1

The $AJ$ conjecture holds for the connected sums $T(p,q)\#T(a,b)$ when $p$ and $a$ have the same sign.

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 4.1
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Lemma 6.1
  • ...and 3 more