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Equivariant unknotting numbers of strongly invertible knots

Keegan Boyle, Wenzhao Chen

TL;DR

The paper investigates equivariant unknotting numbers for strongly invertible knots under a 180-degree symmetry, introducing the total invariant $ ilde{u}(K)$ and three restricted variants $ ilde{u}_A(K), ilde{u}_B(K), ilde{u}_C(K)$ corresponding to type A, B, and C crossing changes. It establishes that $ ilde{u}(K)$ need not be additive under connected sum, and develops a framework based on symmetric crossings, quotient knots $ rak{q}_1(K), rak{q}_2(K)$, and a theta-graph perspective to derive lower bounds and structural results. The paper proves sharp results for torus knots via intravergent braids, shows unbounded growth of $ ilde{u}_A(K)$ in twist knots, and connects type-B estimates to 4-move invariants, while completely classifying type-C unknotting as finite exactly for $(1,2)$-knots with axis as a core. The main takeaway is that symmetry-enforced unknotting behaves subtly: while some restricted moves are always finite and additive in certain senses, the total equivariant unknotting number can fail additivity, highlighting rich interactions between symmetry, crossing changes, and knot invariants.

Abstract

We study symmetric crossing change operations for strongly invertible knots. Our main theorem is that the most natural notion of equivariant unknotting number is not additive under connected sum, in contrast with the longstanding conjecture that unknotting number is additive.

Equivariant unknotting numbers of strongly invertible knots

TL;DR

The paper investigates equivariant unknotting numbers for strongly invertible knots under a 180-degree symmetry, introducing the total invariant and three restricted variants corresponding to type A, B, and C crossing changes. It establishes that need not be additive under connected sum, and develops a framework based on symmetric crossings, quotient knots , and a theta-graph perspective to derive lower bounds and structural results. The paper proves sharp results for torus knots via intravergent braids, shows unbounded growth of in twist knots, and connects type-B estimates to 4-move invariants, while completely classifying type-C unknotting as finite exactly for -knots with axis as a core. The main takeaway is that symmetry-enforced unknotting behaves subtly: while some restricted moves are always finite and additive in certain senses, the total equivariant unknotting number can fail additivity, highlighting rich interactions between symmetry, crossing changes, and knot invariants.

Abstract

We study symmetric crossing change operations for strongly invertible knots. Our main theorem is that the most natural notion of equivariant unknotting number is not additive under connected sum, in contrast with the longstanding conjecture that unknotting number is additive.

Paper Structure

This paper contains 14 sections, 32 theorems, 16 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Unknotting operations
  3. Lower bounds
  4. Torus knots
  5. Questions
  6. Acknowledgements
  7. Equivariant crossing changes and unknotting numbers
  8. Torus knots
  9. Type A unknotting
  10. Type B unknotting
  11. Type C unknotting
  12. Type C unknottable knots
  13. Proof of Theorem \ref{['thm:1_2typeC']}
  14. The total equivariant unknotting number

Key Result

Theorem 1.2

There are strongly invertible knots $K_1$ and $K_2$ and an equivariant connected sum $K_1\#K_2$ such that $\widetilde{u}(K_1\#K_2) > \widetilde{u}(K_1) + \widetilde{u}(K_2)$. In particular, the total equivariant unknotting number is not additive or even sub-additive.

Figures (14)

  • Figure 1: Examples of the three types of equivariant transverse self-intersections, realized as a movie starting on the figure-eight knot. The first column is Type A ('off the axis'), the middle column is Type B ('through the axis'), and the right column is Type C ('along the axis').
  • Figure 2: An unknotting sequence of $3_1 \# r3_1$ consisting of equivariant isotopies indicated by arrows labelled with an $i$, and type B moves indicated by arrows labelled with a B$_x$, where $x$ is the number of type B moves applied. For compactness, the axis of symmetry is horizontal in each diagram. A total of 4 type B moves are used so that $\widetilde{u}_B(3_1\#r3_1) \leq 4$.
  • Figure 3: An intravergent braid on 3 strands given by the word $\sigma_1 \sigma_2$ when read left to right. The symmetry is given by $\pi$ rotation around the central marked point.
  • Figure 5: An example of unknotting a strongly invertible knot with only type A crossing changes, by ensuring that the black arc always passes over the gray arc, and that the black and gray arcs are unknotted.
  • Figure 6: A strongly invertible twist knot $K_n$ (left) and the quotient $\mathfrak{q}_1(K_n) = T(2,2n+1)$ (right) corresponding to the unbounded arc of the axis. The $n$ indicates $n$ full twists so that $K_n$ is alternating with $2n+2$ crossings.
  • ...and 9 more figures

Theorems & Definitions (73)

  • Conjecture 1.1: Additivity of unknotting number MR1545700
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 63 more