Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Twist positivity, L-space knots, and concordance

Siddhi Krishna, Hugh Morton

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.

Twist positivity, L-space knots, and concordance

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 is a twist positive L-space knot, the braid index and bridge index of 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 , 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.
Paper Structure (12 sections, 21 theorems, 42 equations, 12 figures)

This paper contains 12 sections, 21 theorems, 42 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Organization
  4. Acknowledgements
  5. Twist Positivity and the Alexander Polynomial
  6. Proof of \ref{['thm:Lorenz']}
  7. Twist positive L-space knots
  8. Proof of \ref{['thm:DistinctConcordanceClasses']}
  9. Preliminaries
  10. Computing the signature via the Goeritz matrix
  11. Signatures and the Seifert genus
  12. Discussion

Key Result

Theorem 1.2

Suppose $K$ is twist positive on $n$ strands. Then the Alexander polynomial of $K$ has the form $\Delta_K(t) = 1 - t + t^n +t^{n+1}R(t)$, where $R(t) \in \mathbb Z[t]$. In particular, the braid index of $K$ appears as an exponent in $\Delta_K(t)$.

Figures (12)

  • Figure 1:
  • Figure 2: The template for the matrices $\sigma_i(t)$, the building blocks of the matrix $B(t)$.
  • Figure 3: Conventions for $\eta(D)$ are on the left, and assignments for Type I and II double points are on the right.
  • Figure 4: The twisted torus knot $T(3,7;4)$ and a well-adapted spanning surface used to identify a Goeritz matrix. The label for the unbounded region, $X_0$, is suppressed. In this figure, $\sigma_1$ is closer to the bottom of the page, and $\sigma_2$ is closer to the top of the page.
  • Figure 5: Defining the matrices $N_{\ell, \epsilon}$ and $P_{\ell, \epsilon}$.
  • ...and 7 more figures

Theorems & Definitions (51)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Conjecture 1.8
  • Conjecture 1.9
  • Conjecture 1.10
  • ...and 41 more