On discrete symmetries of the cube of smoothings

TL;DR

This work analyzes discrete cube-of-smoothings symmetries for closed polygonal links in $\mathbb{R}^{3}$ and connects them to Khovanov homology. It builds a combinatorial framework in which each smoothing state $\boldsymbol{v}\in\{0,1\}^{k}$ carries a vertex group $G_{\boldsymbol{v}} \cong \prod_i D_{d_i}$ and a permutation $\sigma_{\boldsymbol{v}}^{\omega}$ that encodes the oriented smoothing, with cobordisms enforcing explicit relations among vertex groups. The authors study how local deformations (polygonal Reidemeister moves) alter the cube via transpositions and conjugations, providing moves between equivalent structures and a pathway to potential new link invariants. The approach yields a foundation for new combinatorial, group-theoretic invariants of links and suggests practical ways to compute Khovanov homology through symmetry-aware cube structures, including explicit examples and constructions of the cube of permutations.

Abstract

We study the Khovanov complex of closed piecewise linear curves in the 3-space. A polygonal link representation endows the cube of resolutions with an additional combinatorial structure. The set of symmetries preserving this structure and its quotient under link equivalence are studied. Our results offer new combinatorial ways of computing Khovanov homology and might lead to other group-theoretic invariants of links.

Figures (13)

• Figure 1: A good diagram of a trefoil knot
• Figure 2: A positive (left) and a negative crossing (right) of a good diagram
• Figure 3: The 1-smoothing (left) and the 0-smoothing (right) of a crossing
• Figure 4: The cube of smoothings for a diagram with 3 crossings
• Figure 5: A labeled polygonal link and one of its smoothings

Theorems & Definitions (22)

• Definition
• Definition
• Definition
• Lemma 1.1
• proof
• Proposition 2.1
• Remark 2.2
• Example 2.3
• Example 2.4
• Lemma 2.5