Strongly positive amphicheiral knots with doubly symmetric diagrams

TL;DR

This paper classifies prime strongly positive amphicheiral knots up to 16 crossings, showing many admit doubly symmetric diagrams, and identifies the first such non-slice knot with period 7.

Contribution

It provides a comprehensive classification of these knots and introduces the concept of almost doubly symmetric diagrams, expanding understanding of their symmetry properties.

Findings

Most strongly positive amphicheiral knots up to 16 crossings have doubly symmetric diagrams.

The first prime strongly positive amphicheiral knot with period 7 is not slice.

Introduction of almost doubly symmetric diagrams for certain knots.

Abstract

We determine the prime strongly positive amphicheiral knots up to 16 crossings and show that a large fraction of them admit knot diagrams with a double symmetry (rotational symmetry for strongly positive amphicheirality and an additional mirror symmetry for the ribbon property). The remaining knots are presented as `almost doubly symmetric' diagrams, defined as diagrams with double symmetry where exactly one symmetric pair of crossings is switched. We also find that the rosette knot 14a19470, which has period 7, is the first prime strongly positive amphicheiral knot in the knot tables which is not slice.