Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Oriented Disingquandles and Invariants of Oriented Dichromatic Singular links

Mohd Ibrahim Sheikh, Mohamed Elhamdadi, Danish Ali

Abstract

We introduce and investigate oriented dichromatic singular links. We also introduce oriented disingquandles and use them to define counting invariants for oriented dichromatic singular links. We provide some examples to show that these invariants distinguish some oriented dichromatic singular links.

Oriented Disingquandles and Invariants of Oriented Dichromatic Singular links

Abstract

We introduce and investigate oriented dichromatic singular links. We also introduce oriented disingquandles and use them to define counting invariants for oriented dichromatic singular links. We provide some examples to show that these invariants distinguish some oriented dichromatic singular links.
Paper Structure (5 sections, 9 theorems, 16 equations, 8 figures, 3 tables)

This paper contains 5 sections, 9 theorems, 16 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Singular Links, Oriented Singquandles and Oriented Dichromatic Links
  3. Oriented Dichromatic Singular Links
  4. Oriented Disingquandles
  5. Invariants for Oriented Dichromatic Singular Links

Key Result

Lemma 2.4

CCE1 Let $(X, *_1, *_2, R_1, R_2)$ be an oriented singquandle. A non-empty subset $S$ of $X$ is an oriented sub-singquandle of $X$ if and only if $S$ is closed under the operations $*_1, *_2, R_1$ and $R_2$.

Figures (8)

  • Figure 1: Oriented Singular Links
  • Figure 2: Generating Set of Oriented Singular Reidemeister Moves
  • Figure 3: Oriented Dichromatic Links
  • Figure 4: Generalized Reidemeister Moves for Oriented Dichromatic Links
  • Figure 5: Oriented Dichromatic Singular Links
  • ...and 3 more figures

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 3.1
  • Proposition 3.2
  • Definition 4.1
  • Remark 4.2
  • Lemma 4.3
  • ...and 20 more