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On the skein polynomial for links

Boju Jiang, Jiajun Wang, Hao Zheng

Abstract

We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual "smoothing of a crossing" move. As by-products we have characterizations of these polynomials for knots, and for links with any given number of components.

On the skein polynomial for links

Abstract

We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual "smoothing of a crossing" move. As by-products we have characterizations of these polynomials for knots, and for links with any given number of components.

Paper Structure

This paper contains 6 sections, 16 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Braids and skein relators
  3. An algebraic reduction lemma
  4. Stabilizations
  5. Proof of Theorems \ref{['thm:HOMFLY']}--\ref{['thm:Alexander-Conway']}
  6. Proof of Theorems \ref{['thm:HOMFLY_knot']}--\ref{['thm:Alexander-Conway_knot']}

Key Result

Theorem 1.1

The skein polynomial $P_L \in \mathbb Z[a^{\pm1},z^{\pm1}]$ is the invariant of oriented links determined uniquely by the following four axioms.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • ...and 25 more