Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the symmetric braid index of ribbon knots

Vitalijs Brejevs, Feride Ceren Kose

TL;DR

The paper defines the symmetric braid index $b_s(K)$ for ribbon knots via braided symmetric-union diagrams and proves a Khovanov homology–based classification for knots with $b_s(K) ext{ } s 3$, linking them to either $K_{p,q} ext{ ext{ ext{}} ext{#}} -K_{p,q}$ or certain Montesinos knots. It combines this with Lisca's 3-braid classification to show that knots with $b_s(K) ext{ } s 3$ lie in two families, while a third family yields knots with larger symmetric braid index; the results yield obstructions showing, for example, that some knots have $b_s(K)>b(K)$ and that chiral slice knots with determinant $1$ have $b_s(K) ext{ } s 4$. The work also provides explicit Kh formulas for Montesinos-type knots, develops obstructions via $q_{ ext{max}}$ and $q_{ ext{min}}$, and includes extensive data for ribbon knots up to $11$ crossings, highlighting both matches and disparities between symmetric and regular braid indices. Overall, the paper builds a bridge between knot homology, symmetric-union braidings, and the braiding complexity of ribbon knots, with concrete consequences for distinguishing knot types and guiding future classifications.

Abstract

We define the symmetric braid index $b_s(K)$ of a ribbon knot $K$ to be the smallest index of a braid whose closure yields a symmetric union diagram of $K$, and derive a Khovanov-homological characterisation of knots with $b_s(K)$ at most three. As applications, we show that there exist knots whose symmetric braid index is strictly greater than the braid index, and deduce that every chiral slice knot with determinant one has braid index at least four. We also calculate bounds for $b_s(K)$ for prime ribbon knots with at most 11 crossings.

On the symmetric braid index of ribbon knots

TL;DR

The paper defines the symmetric braid index for ribbon knots via braided symmetric-union diagrams and proves a Khovanov homology–based classification for knots with , linking them to either or certain Montesinos knots. It combines this with Lisca's 3-braid classification to show that knots with lie in two families, while a third family yields knots with larger symmetric braid index; the results yield obstructions showing, for example, that some knots have and that chiral slice knots with determinant have . The work also provides explicit Kh formulas for Montesinos-type knots, develops obstructions via and , and includes extensive data for ribbon knots up to crossings, highlighting both matches and disparities between symmetric and regular braid indices. Overall, the paper builds a bridge between knot homology, symmetric-union braidings, and the braiding complexity of ribbon knots, with concrete consequences for distinguishing knot types and guiding future classifications.

Abstract

We define the symmetric braid index of a ribbon knot to be the smallest index of a braid whose closure yields a symmetric union diagram of , and derive a Khovanov-homological characterisation of knots with at most three. As applications, we show that there exist knots whose symmetric braid index is strictly greater than the braid index, and deduce that every chiral slice knot with determinant one has braid index at least four. We also calculate bounds for for prime ribbon knots with at most 11 crossings.

Paper Structure

This paper contains 6 sections, 12 theorems, 23 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Symmetric unions and their braidings
  3. Classification of knots with symmetric braid index 3
  4. Khovanov homology of knots with symmetric braid index 3
  5. Symmetric braid indices of small ribbon knots
  6. Some future directions

Key Result

Theorem 1

Suppose that $K$ is a ribbon knot with $b_s(K) \leq 3$. Then there exists $K_{p,q}$ with $p^2 = \det(K)$ such that $K$ admits it as a partial knot and one of the following holds:

Figures (5)

  • Figure 1: Left: an SU diagram of the stevedore knot $6_1$ obtained by inserting a single twist region, shown in red, on the axis of symmetry of $3_1 \# -3_1$. Right: a braided SU diagram of $6_1$ with three twist regions and the braid axis perpendicular to the page, indicated by the green circle; the corresponding SU braid is given by $\gamma C_1 \gamma^{-1} C_2 \in B_4$ for $\gamma = \sigma_2 \sigma_1^{-1} \sigma_2$, $C_1 = \sigma_3$ and $C_2 = \sigma_1 \sigma_3^{-1}$.
  • Figure 2: The Kinoshita--Terasaka family $\{K_n\}_{n \in \mathbb{Z} \setminus \{0\}}$ is obtained by inserting $n$-tangles in the twist region $\tau$.
  • Figure 3: Construction of the symmetric union diagram $(D_J \sqcup -D_J)(\infty_\mu, n_1, \dots, n_l)$; rectangles labelled $n_1, \dots, n_l$ contain the respective number of 'vertical' half-twists.
  • Figure 4: A knot $K$ with $b_s(K) \in \{ 3, 4 \}$ admits an SU diagram with the partial knot $J$ given by the plat closure of $\gamma \in B_4$, where each twist region $\tau_i$ contains one crossing. If $\gamma$ does not contain any $\sigma_1$ letters, then the topmost free strand in the right diagram can be moved all the way down so that the $\tau_3$ region can be removed via the Reidemeister I move; this yields the closure of an SU 3-braid. (Cf. lamm:original.)
  • Figure 5: The knots $K_\gamma(T,U)$ on the left and $K_\gamma(T^\sigma, U^{\overline{\sigma}})$ on the right defined in watson:identicalkh. Here when $T$ and $U$ are the rational tangles $\frac{p}{q}$ for $p=1$, $K_\gamma(T,U)$ is an SU knot. In particular, if $T$ and $U$ are $(\pm 1)$-tangles, it is an SU knot with $b_s=3$.

Theorems & Definitions (24)

  • Conjecture : 'Ribbon--SU conjecture'
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Example 5
  • Corollary 6
  • Example 7
  • Definition 8
  • Theorem 9: lamm:original
  • ...and 14 more