Crowell's state space is connected

TL;DR

This paper proves that the set of Crowell states for prime alternating knots is connected, providing new insights into knot state spaces and offering a novel proof of a classification result for certain torus knots.

Contribution

It establishes the connectedness of Crowell's state space for prime alternating knots and offers a new proof of a classification theorem for (2,2n+1) torus knots.

Findings

Crowell's state space for prime alternating knots is connected.

Connectedness similar to Kauffman states.

New proof of Ozsvath and Szabo's classification of (2,2n+1) torus knots.

Abstract

We study the set of Crowell states for alternating knot projections and show that for prime alternating knots the space of states for a reduced projection is connected, a result similar to that for Kauffman states. As an application we give a new proof of a result of Ozsvath and Szabo characterizing (2,2n+1) torus knots among alternating knots.