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The search for exotic knot traces

Marc Kegel, Jonathan Spreer

Abstract

Two distinct knots are said to be friends if their complements, filled along the 0-slope, produce diffeomorphic 3-manifolds. In this article, we develop a practical algorithm, implemented using SnapPy and Regina, to search for a friend of a given knot. As an application, we construct a census of simple knots that admit friends and use these data to formulate conjectures about knot friends.

The search for exotic knot traces

Abstract

Two distinct knots are said to be friends if their complements, filled along the 0-slope, produce diffeomorphic 3-manifolds. In this article, we develop a practical algorithm, implemented using SnapPy and Regina, to search for a friend of a given knot. As an application, we construct a census of simple knots that admit friends and use these data to formulate conjectures about knot friends.
Paper Structure (17 sections, 6 theorems, 2 equations, 13 figures, 3 tables, 6 algorithms)

This paper contains 17 sections, 6 theorems, 2 equations, 13 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Algorithmic search for friends
  3. A census of friends
  4. Piccirillo friends
  5. Concordance friends
  6. The friend search
  7. Correctness and caveats of \ref{['alg:friend_search']}
  8. Other methods to find friends
  9. Notable results of running \ref{['alg:friend_search']}
  10. A census of friends
  11. Experiments and running times
  12. Piccirillo friends
  13. The concordance friend search
  14. The concordance search
  15. The concordance friend search
  16. ...and 2 more sections

Key Result

Corollary 1.5

If there exist concordance friends with different sliceness statuses, then there exists a smooth $4$-manifold that is homotopy equivalent to $S^4$ but not diffeomorphic to $S^4$.∎

Figures (13)

  • Figure 1: Left: A diagram of $K6a2$. Right: A diagram of a $20$-crossing friend of $K6a2$.
  • Figure 2: Two more friends of the Conway knot.
  • Figure 3: Running times of all four methods compared. For randomised methods "exhaustive" and "MCMC", results from multiple runs are presented.
  • Figure 4: Number of candidate curves to check until a friend is found, ranging over all four methods.
  • Figure 5: Constructing a Piccirillo friend from an unknotting crossing. Note that the red knot is an unknot and thus the surgery on red and blue produces a green knot $K_G$ in $S^3$.
  • ...and 8 more figures

Theorems & Definitions (20)

  • Conjecture 1.3
  • Remark 1.4
  • Corollary 1.5
  • Example 2.1
  • Example 2.2
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 5.1: Piccirillo friend Piccirillo_Conway_knot
  • proof
  • ...and 10 more