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Using the Jones Polynomial to Prove Infinite Families of Knots Satisfy the Cosmetic Surgery Conjecture

F. M. Brady

TL;DR

The paper develops a computational framework to assess the purely cosmetic surgery conjecture by examining derivatives of the Jones polynomial at $1$ for infinite knot families generated by twists. Using a general twist-perturbation formula and Seifert-matrix analysis, the authors derive necessary derivative obstructions $V_K''(1)=0$ and relate higher derivatives to the Alexander polynomial constraints, enabling case-by-case elimination of potential counterexamples. Applying the method to the twist families generated from $7_6$, $8_{12}$, and $10_{58}$, they show all nontrivial knots within these families cannot admit cosmetic surgeries, with the $8_{12}$ case following as a corollary from the $10_{58}$ analysis and the remaining knots shown to be unknots where exceptions arise. The results provide a scalable computational approach for verifying the Cosmetic Surgery Conjecture across broad knot families and illustrate how higher Jones-derivative restrictions can be exploited in practice.

Abstract

This paper computes the Jones polynomial and the invariants obstructing cosmetic surgery which are derived from it for two infinite families of knots, proving they satisfy the Purely Cosmetic Surgery Conjecture. Both the method of computation and the method for generating families of knots extend.

Using the Jones Polynomial to Prove Infinite Families of Knots Satisfy the Cosmetic Surgery Conjecture

TL;DR

The paper develops a computational framework to assess the purely cosmetic surgery conjecture by examining derivatives of the Jones polynomial at for infinite knot families generated by twists. Using a general twist-perturbation formula and Seifert-matrix analysis, the authors derive necessary derivative obstructions and relate higher derivatives to the Alexander polynomial constraints, enabling case-by-case elimination of potential counterexamples. Applying the method to the twist families generated from , , and , they show all nontrivial knots within these families cannot admit cosmetic surgeries, with the case following as a corollary from the analysis and the remaining knots shown to be unknots where exceptions arise. The results provide a scalable computational approach for verifying the Cosmetic Surgery Conjecture across broad knot families and illustrate how higher Jones-derivative restrictions can be exploited in practice.

Abstract

This paper computes the Jones polynomial and the invariants obstructing cosmetic surgery which are derived from it for two infinite families of knots, proving they satisfy the Purely Cosmetic Surgery Conjecture. Both the method of computation and the method for generating families of knots extend.
Paper Structure (12 sections, 5 theorems, 69 equations, 16 figures)

This paper contains 12 sections, 5 theorems, 69 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Relation to Previous Work
  4. Jones Polynomial Restrictions
  5. General Jones Polynomial Computation
  6. Base Cases for $7_6$
  7. Alexander Polynomial Computation for $7_6$
  8. Computations for the Knot Family Generated by $7_6$
  9. Showing the Remaining $7_6$ Knots are Unknots
  10. Base Cases for $10_{58}$
  11. Alexander Polynomial Computations for $10_{58}$ and Analysis of $8_{12}$
  12. Computations for the Knot Family Generated by $10_{58}$

Key Result

Proposition 1

If a knot $K$ has cosmetic surgeries, then $V_K""(1) = 0$.

Figures (16)

  • Figure 1: A $\overline{t_2'}$ move. Every $\overline{t_2'}$ move is of this form or its mirror is.
  • Figure 2: The skeletons for the knot family built from $6_2$ and (o1) Montesinos knots.
  • Figure 3: The skeletons for the knot family built from $7_7$ and (o2) Montesinos knots.
  • Figure 4: The skeletons for the knot family built from $7_5$ and (o3) Montesinos knots, and the generalization of (o3) knots which connects the two.
  • Figure 5: The skeletons for the knot family built from $7_5$ and (o4) Montesinos knots, and the generalization of (o4) knots which connects the two.
  • ...and 11 more figures

Theorems & Definitions (8)

  • Conjecture 1: Purely Cosmetic Surgery Conjecture
  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Corollary 1
  • proof
  • Theorem 2
  • proof