Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Alexander polynomials of ribbon knots and virtual knots

Sheng Bai

TL;DR

The paper develops a novel framework tying the Alexander polynomial of ribbon knots in $\mathbb{Z}HS^3$ to intrinsic singularities via the half Alexander polynomial $A_R(t)$, with the key identity $\Delta(K)(t)=A_R(t)A_R(t^{-1})$. It introduces ribbon diagrams/graphs and a ribbon-matrix $R(t)$ whose determinant $A_R(t)=|R(t)|$ encodes vast information beyond $\Delta(t)$, including simplified computational formulas (contracted and path-type) and a 3D-topological interpretation. The approach generalizes to general and virtual knots through Gauss-diagram techniques, yielding new determinant-based formulas and contractions that bridge classical and virtual knot theory, and it connects to the Blanchfield pairing and Seifert-matrix data. The work also explores limitations (half Alexander polynomials are not ribbon-knot invariants) and extends to band-sum generalizations, with implications for symmetric unions and fusion-number questions, providing both new computational tools and deeper structural insights into knot invariants.

Abstract

We find that Alexander polynomial of a ribbon knot in $ \mathbb{Z}HS^3 $ is determined by the intrinsic singularity information of its ribbon, and give a formula to calculate Alexander polynomial of a ribbon knot by that. We define half Alexander polynomial $ A_R (t) $, an invariant of oriented ribbons, and in fact the Alexander polynomial of the ribbon knot is $ A_R (t) A_R (t^{-1}) $. We give two useful simplified formulas for half Alexander polynomial. We characterize completely the polynomials arising as half Alexander polynomials of ribbons. The above study unexpectedly leads us to discover new formulas for Alexander polynomial of general knots and virtual knots in terms of Gauss diagrams.

Alexander polynomials of ribbon knots and virtual knots

TL;DR

The paper develops a novel framework tying the Alexander polynomial of ribbon knots in to intrinsic singularities via the half Alexander polynomial , with the key identity . It introduces ribbon diagrams/graphs and a ribbon-matrix whose determinant encodes vast information beyond , including simplified computational formulas (contracted and path-type) and a 3D-topological interpretation. The approach generalizes to general and virtual knots through Gauss-diagram techniques, yielding new determinant-based formulas and contractions that bridge classical and virtual knot theory, and it connects to the Blanchfield pairing and Seifert-matrix data. The work also explores limitations (half Alexander polynomials are not ribbon-knot invariants) and extends to band-sum generalizations, with implications for symmetric unions and fusion-number questions, providing both new computational tools and deeper structural insights into knot invariants.

Abstract

We find that Alexander polynomial of a ribbon knot in is determined by the intrinsic singularity information of its ribbon, and give a formula to calculate Alexander polynomial of a ribbon knot by that. We define half Alexander polynomial , an invariant of oriented ribbons, and in fact the Alexander polynomial of the ribbon knot is . We give two useful simplified formulas for half Alexander polynomial. We characterize completely the polynomials arising as half Alexander polynomials of ribbons. The above study unexpectedly leads us to discover new formulas for Alexander polynomial of general knots and virtual knots in terms of Gauss diagrams.

Paper Structure

This paper contains 42 sections, 19 theorems, 65 equations, 35 figures.

Table of Contents

  1. Introduction
  2. Alexander polynomial of ribbon knots in $\mathbb{Z}HS^3$
  3. Alexander polynomial of general knots and virtual knots
  4. Background
  5. Alexander polynomials in $\mathbb{Z}HS^3$
  6. Alexander polynomial of virtual knots
  7. Alexander module of ribbon knots in $\mathbb{Z}HS^3$
  8. Ribbon diagram and ribbon graph
  9. Half Alexander polynomial and the main theorem
  10. Proof of Theorem 3.2.2
  11. From ribbon to Seifert surface
  12. Homological basis on $F_g$
  13. Seifert matrix
  14. Read from ribbon graph
  15. Alexander Invariants of ribbon knots
  16. ...and 27 more sections

Key Result

Theorem 1.1.1

Given a ribbon $R$ for a ribbon knot $K$ in $\mathbb{Z}HS^3$, let $A_R (t)$ be the half Alexander polynomial determined by the ribbon graph. Then the Conway-normalized Alexander polynomial of $K$ is $\Delta (t) = A_R (t) A_R (t^{-1})$. Especially, Alexander polynomial of a ribbon knot is determine

Figures (35)

  • Figure 1: Fox differential of Wirtinger presentation.
  • Figure 2: A ribbon and its ribbon diagram.
  • Figure 3: A ribbon diagram and its ribbon graph.
  • Figure 4: Ribbon graph drawn in a 3 dimensional manner.
  • Figure 5: Surgery and isotopy.
  • ...and 30 more figures

Theorems & Definitions (50)

  • Definition 1.0.1
  • Theorem 1.1.1
  • Corollary 1.1.2
  • Proposition 2.2.1
  • proof
  • Definition 3.1.1
  • Definition 3.1.2
  • Lemma 3.2.1
  • proof
  • Theorem 3.2.2
  • ...and 40 more