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Transverse knots determined by their cyclic branched covers

Marc Kegel, Isacco Nonino

Abstract

Harvey-Kawamuro-Plamenevskaya demonstrated the existence of (transversely) non-isotopic transverse knots such that for every $n>1$ their $n$-fold cyclic branched covers are contactomorphic. In this short note, we construct other examples of non-isotopic transverse knots that have contactomorphic cyclic branched covers. Conversely, we prove that the transverse isotopy classes of many transverse knots are actually determined by the contactomorphism type of their cyclic branched covers.

Transverse knots determined by their cyclic branched covers

Abstract

Harvey-Kawamuro-Plamenevskaya demonstrated the existence of (transversely) non-isotopic transverse knots such that for every their -fold cyclic branched covers are contactomorphic. In this short note, we construct other examples of non-isotopic transverse knots that have contactomorphic cyclic branched covers. Conversely, we prove that the transverse isotopy classes of many transverse knots are actually determined by the contactomorphism type of their cyclic branched covers.

Paper Structure

This paper contains 9 sections, 16 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Transverse knots determined by their cyclic branched covers
  3. Transverse knots with contactomorphic cyclic branched covers
  4. Questions and further remarks
  5. Branched covers along transverse knots
  6. The homotopical invariants of a cyclic contact branched cover
  7. Branched covers and connected sums
  8. Transverse knots determined by their cyclic covers
  9. The Nakanishi--Sakuma construction for transverse knots

Key Result

Theorem 1.1

Let $K$ in $S^3$ be a prime, transversely simple knot. Then for every transverse realization $T$ of $K$ in $(S^3,\xi_{std})$

Figures (1)

  • Figure 1: The Legendrian realization of the above link drawn in front projection. It is evident from the picture that both unknots are standard $tb=-1$ unknots.

Theorems & Definitions (30)

  • Theorem 1.1: Transverse knots determined by cyclic branched covers
  • Theorem 1.2: Transverse torus knots determined by cyclic branched covers
  • Theorem 1.3: Transverse figure-eight knots are determined by every cyclic branched cover
  • Theorem 1.4: Composite knots determined by cyclic branched covers
  • Theorem 1.5: Connected sums of positive torus knots
  • Theorem 1.6: Smoothly non-isotopic transverse knots that share a cyclic branched cover
  • Theorem 1.7: Overtwisted cyclic branched covers
  • Proposition 2.1: Gonzalo Gonzalo
  • proof : Proof sketch
  • Lemma 3.1: Euler class and $d_3$-invariant of cyclic branched covers
  • ...and 20 more