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Mock Alexander Polynomials

Neslihan Gügümcü, Louis H. Kauffman

Abstract

In this paper, we construct mock Alexander polynomials for starred links and linkoids in surfaces. These polynomials are defined as specific sums over states of link or linkoid diagrams that satisfy $f=n$, where $f$ denotes the number of regions and $n$ denotes the number of crossings of diagrams.

Mock Alexander Polynomials

Abstract

In this paper, we construct mock Alexander polynomials for starred links and linkoids in surfaces. These polynomials are defined as specific sums over states of link or linkoid diagrams that satisfy , where denotes the number of regions and denotes the number of crossings of diagrams.
Paper Structure (21 sections, 26 theorems, 37 equations, 46 figures)

This paper contains 21 sections, 26 theorems, 37 equations, 46 figures.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Alexander-Conway Polynomial
  4. Organization of the paper
  5. Linkoids and universes
  6. Admissible diagrams
  7. Results from Euler's formula
  8. Starred links and linkoids
  9. Mock Alexander polynomials for starred links and linkoids
  10. A state-sum polynomial for starred links and linkoids
  11. A matrix formulation for the potential
  12. An invariant of starred links and linkoids induced by the potential
  13. Reversal and mirror behavior of mock Alexander polynomials
  14. Skein relations
  15. More on starred link diagrams
  16. ...and 6 more sections

Key Result

Proposition 1

Let $U$ be the universe of a connected link or linkoid diagram with $n$ crossings that is tightly embedded in a closed, connected and orientable surface $\Sigma_g$. We have the following.

Figures (46)

  • Figure 1: Alexander Labeling.
  • Figure 2: The Alexander Polynomial.
  • Figure 3: States with Markers
  • Figure 4: A black hole.
  • Figure 5: State-sum calculation of Alexander polynomial
  • ...and 41 more figures

Theorems & Definitions (79)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 69 more