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An example of semiquandle and u-polynomials for flat virtual knots via this semiquandle coloring

Nozomu Sekino

Abstract

Flat virtual links are some variant of links, and semiquandles are counterparts of quandles or biquandles, which axiomize the Reidemeister-like moves. In this paper, we give some example of semiquandle and introduce an invariant for flat virtual knot using it. We also explain that it relates to the u-polynomials for flat virtual knots.

An example of semiquandle and u-polynomials for flat virtual knots via this semiquandle coloring

Abstract

Flat virtual links are some variant of links, and semiquandles are counterparts of quandles or biquandles, which axiomize the Reidemeister-like moves. In this paper, we give some example of semiquandle and introduce an invariant for flat virtual knot using it. We also explain that it relates to the u-polynomials for flat virtual knots.

Paper Structure

This paper contains 14 sections, 4 theorems, 2 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The ring $\mathcal{S}_{n}$
  3. A basis of ${\mathcal{S}}_{n}$ as a $\mathbb{Z}$-module
  4. Unit elements of $\mathcal{S}_{n}$
  5. Semiquandle structure on $\mathcal{S}_{n}$
  6. Flat virtual links
  7. Definition of flat virtual links
  8. Gauss diagrams and u-polynomials
  9. Gauss diagrams
  10. u-polynomials
  11. Quasideterminant
  12. $\mathcal{S}_{n}$-coloring invariant of flat virtual knots
  13. Construction
  14. Some properties

Key Result

Lemma 2.1

The element $x\in \mathcal{S}_n$ is a unit element if and only if $\pi_{0,n}(x)$ is a unit element of $\mathcal{S}_{0}=\mathbb{Z}[s^{\pm1}]$.

Figures (12)

  • Figure 1: Left: flat crossing Right: virtual crossing
  • Figure 2: Examples of flat virtual knot diagrams.
  • Figure 3: Flat Reidemeister moves, where orientations are arbitrary.
  • Figure 4: Five moves which generate moves FR1, FR2 and FR3.
  • Figure 5: Coloring relation
  • ...and 7 more figures

Theorems & Definitions (37)

  • Remark 2.1
  • proof
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.1
  • proof
  • Definition 3.1
  • proof
  • Remark 3.1
  • Definition 4.1
  • ...and 27 more