Odd Order Group Actions on Alternating Knots

TL;DR

This paper studies symmetries of alternating prime knots, showing that odd prime order actions can be simplified to a single flype and that such knots have alternating periodic diagrams, with quotients also remaining alternating.

Contribution

It demonstrates that odd prime order symmetries of alternating knots are isotopic to a single flype and establishes the preservation of alternation in quotient knots.

Findings

Any odd prime order action on an alternating knot is isotopic to a single flype.

Alternating prime knots admit either a reduced alternating periodic diagram or a free periodic diagram.

The quotient of an odd periodic alternating knot remains alternating.

Abstract

Let K be a an alternating prime knot in the 3-sphere. We investigate the category of flypes between reduced alternating diagrams for K. As a consequence, we show that any odd prime order action on K is isotopic through maps of pairs to a single flype. This implies that for any odd prime order action on K there is either a reduced alternating periodic diagram or a reduced alternating free periodic diagram. Finally, we deduce that the quotient of an odd periodic alternating knot is also alternating.