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Singular twisted links and singular twisted virtual braids

Komal Negi, Madeti Prabhakar

Abstract

The concepts of twisted knot theory and singular knot theory inspire the introduction of singular twisted knot theory. This study showcases similar findings for singular twisted links, including the Alexander theorem and the Markov theorem derived from knot theory. Moreover, in this paper we define singular twisted virtual braids and their monoid structure. Additionally, we provide both a monoid and a reduced monoid presentation for singular twisted virtual braids.

Singular twisted links and singular twisted virtual braids

Abstract

The concepts of twisted knot theory and singular knot theory inspire the introduction of singular twisted knot theory. This study showcases similar findings for singular twisted links, including the Alexander theorem and the Markov theorem derived from knot theory. Moreover, in this paper we define singular twisted virtual braids and their monoid structure. Additionally, we provide both a monoid and a reduced monoid presentation for singular twisted virtual braids.
Paper Structure (9 sections, 16 theorems, 17 equations, 21 figures)

This paper contains 9 sections, 16 theorems, 17 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singular twisted link
  4. Singular twisted virtual braid
  5. $L_v$-moves and Markov-type theorems for singular twisted virtual braids
  6. Algebraic Markov type theorem for singular twisted virtual braids
  7. On threading moves of singular twisted virtual braids
  8. Reduced presentation for singular twisted virtual braid monoid
  9. Concluding remarks

Key Result

Theorem 2.9

The monoid $STVB_n$ is generated by standard generators, $\sigma_i^{\pm 1}$, $v_i$$(i=1, \dots, n-1)$, $\gamma_i$$(i=1, \dots, n)$, and $\tau_i$$(i=1, \dots, n-1)$ satisfy the following relations, where $e$ denotes the identity element:

Figures (21)

  • Figure 1: Reidemeister moves for singular twisted link diagrams
  • Figure 2: Singular twisted link diagrams
  • Figure 3: A singular twisted virtual braid on 3 strands
  • Figure 4: Classical braid moves
  • Figure 5: Virtual braid moves
  • ...and 16 more figures

Theorems & Definitions (35)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 25 more