Twist positivity, L-space knots, and concordance

TL;DR

This paper explores the properties of twist positive knots, especially L-space knots, revealing their braid and bridge indices' relationships, and provides new examples and conjectures in knot concordance and fibered knot uniqueness.

Contribution

It establishes a link between twist positivity and Alexander polynomial exponents, and shows that for twist positive L-space knots, braid and bridge indices coincide, supporting new conjectures.

Findings

Braid index appears as the third exponent in the Alexander polynomial for twist positive knots.

For twist positive L-space knots, braid index and bridge index are equal.

Constructs an infinite family of positive braid knots that are distinct in concordance with increasing genus.

Abstract

Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots, 1-bridge braids, algebraic knots, and Lorenz knots. We prove that if a knot is twist positive, the braid index appears as the third exponent in its Alexander polynomial. We provide a few applications of this result. After observing that most known examples of L-space knots are twist positive, we prove: if $K$ is a twist positive L-space knot, the braid index and bridge index of $K$ agree. This allows us to provide evidence for Baker's reinterpretation of the slice-ribbon conjecture: that every smooth concordance class contains at most one fibered, strongly quasipositive knot. In particular, we provide the first example of an infinite family of positive braid knots which are distinct in concordance, and where, as $g \to \infty$, the number of hyperbolic knots of genus g gets arbitrarily large. Finally, we collect some evidence for a few new conjectures, including the following: the braid and bridge indices agree for any L-space knot.