Minimal grid diagrams of the prime alternating knots with 13 crossings

TL;DR

This paper presents minimal grid diagrams for all 4878 prime alternating knots with 13 crossings, extending previous work on knots with fewer crossings and providing a comprehensive set of minimal representations.

Contribution

The authors computed minimal grid diagrams with 15 vertical segments for all prime alternating knots with 13 crossings, advancing the understanding of knot representations.

Findings

Minimal grid diagrams for 4878 prime alternating knots with 13 crossings

Extension of previous work on knots with fewer crossings

Provides a complete set of minimal representations for these knots

Abstract

A knot is a closed loop in space without self-intersection. Two knots are equivalent if there is a self homeomorphism of space bringing one onto the other. An arc presentation is an embedding of a knot in the union of finitely many half planes with a common boundary line such that each half plane contains a simple arc of the knot. The minimal number of such half planes among all arc presentations of a given knot is called the arc index of the knot. A knot is usually presented as a planar diagram with finitely many crossings of two strands where one of the strands goes over the other. A grid diagram is a planar diagram which is a non-simple rectilinear polygon such that vertical edges always cross over horizontal edges at all crossings. It is easily seen that an arc presentation gives rise to a grid diagram and vice versa. It is known that the arc index of an alternating knot is two plus its minimal crossing number. There are 4878 prime alternating knots with minimal crossing number 13. We obtained minimal arc presentations of them in the form of grid diagrams having 15 vertical segments. This is a continuation of the works on prime alternating knots of 11 crossings and 12 crossings.