An exploration of low crossing and chiral cosmetic bands with grid diagrams

TL;DR

The paper investigates non-coherent $H(2)$-moves between low-crossing knots using grid diagrams as a computational framework, significantly expanding the known $H(2)$-distance table for knots up to eight crossings. It develops GridPym-based workflows to systematically apply all feasible $H(2)$-attachments, yielding 322 distance-one pairs (including 43 cosmetic) and identifying 33 previously unknown adjacencies, including two new sub-$7$-crossing cases. The work also analyzes chirally cosmetic bandings, confirming a new example for $7_3$ and connecting these results to Montesinos’ trick and lens-space lifts, with an appendix detailing an infinite family construction. All code and data are openly available, underscoring the utility of grid-diagram methods for computational knot theory and related topological questions.

Abstract

We computationally explore non-coherent band attachments between low crossing number knots, using grid diagrams. We significantly improve the current H(2)-distance table. In particular, we find two new distance one pairs with fewer than seven crossings: one between $3_1\#3_1$ and $7_4m$, and a chirally cosmetic one for $7_3$. We further determine a total of 33 previously unknown H(2)-distance one pairs for knots with up to $8$ crossings. The appendix by Kazuhiro Ichihara, In Dae Jong and Masakazu Teragaito contains a construction explaining the existence of chirally cosmetic bands for an infinite family of knots, including $5_1,\, 7_3$ and $8_8$.

Figures (8)

• Figure 1: A grid diagram of size $9$, representing the knot $7_3$.
• Figure 2: (a) Depicts a single portion of a grid diagram (on the left) being changed in each of the five possible ways (on the right). The top row shows that the link diagram on the left changes by a planar isotopy if both pairs of markings are moved. The mid row shows that swapping markings of the same type in the left diagram results in a coherent band attachment. Note that the two possibilities produce equivalent diagrams. The bottom row shows that swapping two markings of different types in the left diagram results in a non-coherent band attachment. Note that in this case the two possibilities might produce non-equivalent links. (b) The two moves on the left diagram correspond to non-coherent band attachments yielding equivalent links.
• Figure 3: Distribution of crossing number and grid number for grids generated by a standard randomisation run: $1000$ random moves and $400$ distinct randomisations per seed diagram.
• Figure 4: Top: A non-coherent band attachment from $7_4$ to $3_1m\# 3_1m$. Bottom: an explicit example of the band attachment realised in the grids model found by our computations. The band (in red) is attached on row 28.
• Figure 5: A chiral cosmetic band attachment on $7_3m$

Theorems & Definitions (2)

• Proposition 4.1
• Remark 4.2