Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum enhancement polynomials associated with the canonical two-element tricbracket

Yuya Koda, Yuya Nishimura, Yuka Sakamoto

Abstract

Quantum enhancement polynomials are invariants for oriented links, defined in association with an algebraic structure called a tribracket. In this paper, we focus on the particular case of the canonical two-element tribracket. We prove that, in that case, the quantum enhancement polynomials can be recovered by five specific polynomials, which we refer to as the universal quantum enhancement polynomials. After presenting several notable properties of these polynomials, we show that they are strictly stronger than the Jones polynomial. Furthermore, we provide computational results for links with up to 10 crossings.

Quantum enhancement polynomials associated with the canonical two-element tricbracket

Abstract

Quantum enhancement polynomials are invariants for oriented links, defined in association with an algebraic structure called a tribracket. In this paper, we focus on the particular case of the canonical two-element tribracket. We prove that, in that case, the quantum enhancement polynomials can be recovered by five specific polynomials, which we refer to as the universal quantum enhancement polynomials. After presenting several notable properties of these polynomials, we show that they are strictly stronger than the Jones polynomial. Furthermore, we provide computational results for links with up to 10 crossings.
Paper Structure (15 sections, 30 theorems, 61 equations, 17 figures, 1 table)

This paper contains 15 sections, 30 theorems, 61 equations, 17 figures, 1 table.

Table of Contents

  1. Tribracket colorings of link diagrams and their quantum enhancement polynomials
  2. Examples
  3. Basic properties of quantum enhancement polynomials
  4. $X_2$-colorings of link diagrams
  5. Complementary colorings
  6. Mirror images
  7. The universal tribracket brackets for the canonical two-element tribraket
  8. Quantum enhancement invariants associated with universal tribracket brackets
  9. The universal quantum enhancement polynomials $\Phi^{(A^{(i)}, B^{(i)})}$ and linking numbers
  10. The universal quantum enhancement polynomials $\Phi^{(A^{(i)}, B^{(i)})}$ and Jones polynomials for links.
  11. The quantum enhancement polynomials $\Phi^{(A^{(i)}, B^{(i)})}$ and Jones polynomials for knots.
  12. Computer Experiments
  13. The quantum enhancement polynomials for torus links $T(2, q)$.
  14. The quantum enhancement polynomials for twist knots $\mathit{TW}(q)$
  15. The quantum enhancement polynomials for Thistlethwaite's link

Key Result

Theorem 1.5

Let $X$ be a tribracket and $(A, B)$ a tribracket bracket with respect to $X$ and a commutative ring. Then, for a diagram $D_L$ of $L$ and an $X$-coloring $C$ of $D_L$, $\beta_X^{(A, B)}(D_L, C)$ is invariant under $X$-colored Reidemeister moves.

Figures (17)

  • Figure 1: The rule for a region coloring around each vertex.
  • Figure 2: $k$ half-twists.
  • Figure 3: The standard diagram of the $(2,q)$-torus link $T(2, q)$.
  • Figure 4: The standard diagram of the $q$-twist knot $\mathit{TW}(q)$.
  • Figure 5: The coloring $C(0, \{D_1\}, \{D_2, D_3\})$.
  • ...and 12 more figures

Theorems & Definitions (69)

  • Definition 1.1
  • Remark
  • Example 1
  • Definition 1.2
  • Definition 1.3: Aggarwal--Nelson--Rivera ANR21ANR24
  • Definition 1.4: Aggarwal--Nelson--Rivera ANR21
  • Theorem 1.5: Aggarwal--Nelson--Rivera ANR21
  • Definition 1.6: Aggarwal--Nelson--Rivera ANR21
  • Example 2
  • Proposition 2.1
  • ...and 59 more