Petal projections, knot colorings and determinants

Henrich, Allison; Truax, Robin

doi:10.2140/involve.2022.15.207

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Petal projections, knot colorings and determinants

Allison Henrich and Robin Truax

Vol. 15 (2022), No. 2, 207–232

DOI: 10.2140/involve.2022.15.207

Abstract

An übercrossing diagram is a knot diagram with only one crossing that may involve more than two strands of the knot. Such a diagram without any nested loops is called a petal projection. Every knot has a petal projection from which the knot can be recovered using a permutation that represents strand heights. Using this permutation, we give an algorithm that determines the $p$ -colorability and the determinants of knots from their petal projections. In particular, we compute the determinants of all prime knots with crossing number less than $10$ from their petal permutations.

Keywords

petal projection, knot determinant, colorability

Mathematical Subject Classification

Primary: 57K10

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Received: 31 March 2020

Revised: 13 August 2021

Accepted: 9 October 2021

Published: 29 July 2022

Communicated by Kenneth S. Berenhaut

Authors

Allison Henrich
Department of Mathematics Seattle University Seattle, WA United States

Robin Truax
Department of Mathematics Stanford University Stanford, CA United States

Vol. 15, No. 2, 2022