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On transverse-universality of twist knots

Sebastian Zapata

TL;DR

This work advances the understanding of transverse-universality by classifying transversely braided twist knots $K_m$ according to the tightness of their contact branched coverings. It shows that for $m\ge 2$ or $m=2k-1\le -3$, all contact branched coverings branched along a transverse $K_m$ are overtwisted, ruling out transverse-universality in those families, while for certain even $m\le -4$ (notably $m=-4$) tight coverings exist, revealing a nuanced dichotomy within the twist-knot family. The authors employ branching-word techniques, left-veering monodromy arguments, and braid-coloration to prove overtwistedness in the first regimes, and they invoke Legendrian classifications and quasipositive braids to establish tight, Stein-fillable coverings in the $m=-2n\le -4$ case. The results sharpen the landscape of universal vs non-universal transverse knots in $(\mathbb{S}^3,\xi_{std})$ and provide a constructive route to identifying tight branched coverings via Legendrian data, with implications for contact topology and 3-manifold branching structures.

Abstract

In the search for transverse-universal knots in the standard contact structure on $\mathbb{S}^3$, we present a classification of the transverse twist knots with maximal self-linking number, that admit only overtwisted contact branched covers. As a direct consequence, we obtain an infinite family of transverse knots in $(\mathbb{S}^3,ξ_{std})$ that are not transverse-universal, although they are universal in the topological sense.

On transverse-universality of twist knots

TL;DR

This work advances the understanding of transverse-universality by classifying transversely braided twist knots according to the tightness of their contact branched coverings. It shows that for or , all contact branched coverings branched along a transverse are overtwisted, ruling out transverse-universality in those families, while for certain even (notably ) tight coverings exist, revealing a nuanced dichotomy within the twist-knot family. The authors employ branching-word techniques, left-veering monodromy arguments, and braid-coloration to prove overtwistedness in the first regimes, and they invoke Legendrian classifications and quasipositive braids to establish tight, Stein-fillable coverings in the case. The results sharpen the landscape of universal vs non-universal transverse knots in and provide a constructive route to identifying tight branched coverings via Legendrian data, with implications for contact topology and 3-manifold branching structures.

Abstract

In the search for transverse-universal knots in the standard contact structure on , we present a classification of the transverse twist knots with maximal self-linking number, that admit only overtwisted contact branched covers. As a direct consequence, we obtain an infinite family of transverse knots in that are not transverse-universal, although they are universal in the topological sense.
Paper Structure (8 sections, 9 theorems, 31 equations, 21 figures)

This paper contains 8 sections, 9 theorems, 31 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Branched coverings
  4. Contact 3-manifolds
  5. Contact branched coverings
  6. Twist knots
  7. Proof of the main theorem
  8. Case m=-2n<=-4

Key Result

Theorem 1.1

If $m\geq2$ or $m=2k-1\leq-3$, then every contact branched covering with branch locus a transverse representative of the twist knot $K_m$ is overtwisted.

Figures (21)

  • Figure 1: Forbidden portions and crosses for a front projection
  • Figure 2: Branch cuts
  • Figure 3: Twist knot $K_m$
  • Figure 4: Isotopy of $\tilde{\gamma}$ in the surface $\tilde{\Sigma}$
  • Figure 5: Different types of simplifications around a branching point $A$ with $s=1$. The red curve is the original, and the blue one is the one obtained after the isotopy. The enumerations correspond to the specific sheet
  • ...and 16 more figures

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 2.1: Cas
  • Lemma 2.1: Cas
  • Theorem 2.2: Pu
  • Theorem 2.3: Classification of transverse twist knots EtLenVer
  • Theorem 2.4: EtLenVer
  • Corollary 3.1
  • Theorem 4.1: EtLenVer
  • Proposition 4.1: EtLenVer