Knot Mosaics with Corner Connection Tiles

Aaron Heap; Una Donovan; Riley Grossman; Nickolas Laine; Connor McDermott; Marcus Paone; Drew Southcott

Home/Archive/arXiv

arXiv · 2306.09276

Knot Mosaics with Corner Connection Tiles

Aaron Heap·Una Donovan·Riley Grossman·Nickolas Laine·Connor McDermott·Marcus Paone·Drew Southcott

View original source ↗Read source document ↗Export metadata ↓

Abstract

A knot mosaic is a representation of a knot or link on a square grid using a collection of tiles that are either blank or contain a portion of the knot diagram. Traditionally, a piece of the knot on one tile connects to a piece of the knot on an adjacent tile at a connection point that is located at the midpoint of a tile edge. We introduce a new set of tiles in which the connection points are located at corners of the tile. By doing this, we can create more efficient knot mosaics for knots with small crossing number. In particular, when using these corner connection tiles, it is possible to create knot mosaic diagrams for all knots with crossing number 8 or less on a mosaic that is no larger and uses fewer non-blank tiles than is possible using the traditional tiles.

Explore related subjects

math.GT

Keep this discovery

BibTeX RIS

Aaron Heap, Una Donovan, Riley Grossman, Nickolas Laine, Connor McDermott, Marcus Paone, Drew Southcott. 2023-06-15. Knot Mosaics with Corner Connection Tiles. https://arxiv.org/abs/2306.09276

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bar cohomology of links: beyond Milnor invariants

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

math.GT↗

arXiv2026

A counterexample to the wrapping number conjecture

We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.

math.GT↗

arXiv2026

Homological lifts of Arnold invariants $J^-$ and $J^+$

Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

math.GT↗