On an alternative stratification of knots

TL;DR

This paper proposes a new way to classify knots based on the lattice size where they first appear, providing insights into their distribution and realizability within small lattices, supported by computational and theoretical analysis.

Contribution

It introduces an alternative stratification of knots by lattice size and analyzes the distribution of unknots and complex knots within small lattices.

Findings

Ratio of unknots decreases exponentially with lattice size

Certain knots can be realized within small 3x3 and 5x5 lattices

Computational results align with theoretical estimates

Abstract

We introduce an alternative stratification of knots: by the size of lattice on which a knot can be first met. Using this classification, we find ratio of unknots and knots with more than 10 minimal crossings inside different lattices and answer the question which knots can be realized inside $3\times 3$ and $5\times 5$ lattices. In accordance with previous research, the ratio of unknots decreases exponentially with the growth of the lattice size. Our computational results are approved with theoretical estimates for amounts of knots with fixed crossing number lying inside lattices of given size.