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Constructing and Cataloging 2-Adjacent Knots

John Carney, Everett Meike

TL;DR

This work determines the complete set of $2$-adjacent knots through $12$ crossings and introduces a novel obstruction framework that blends the Montesinos trick, the Alexander polynomial, and Heegaard Floer $d$-invariants. When the branched double-cover $\\Sigma(K)$ is an $L$-space, the Ni–Wu formula tightly constrains possible Alexander polynomials of lift knots $J$, enabling rigorous exclusion of candidates via $d$-invariants. The authors apply these techniques to rule out several knots (e.g., $11a_{255}$, $12a_{358}$, $12n_{620}$, $12n_{656}$, $12n_{586}$) from being $2$-adjacent and provide a constructive catalog of all $2$-adjacent knots up to $12$ crossings. They also develop a Seifert surface/Conway polynomial construction via finger-moves to realize families of $2$-adjacent knots, and they discuss implications for higher crossing knots and alternating cases, highlighting the potential of $d$-invariants as a powerful obstruction tool in knot adjacency problems.

Abstract

Generalizing unknotting number, $n$-adjacent knots have $n$ crossings such that changing any non-empty subset of them results in the unknot. In this paper, we determine the 2-adjacent knots through 12 crossings. Using Heegaard Floer $d$-invariants and the Alexander polynomial, we develop a new technique to obstruct 2-adjacency, and we prove conjectures of Ito and Kato regarding 2-adjacent knots.

Constructing and Cataloging 2-Adjacent Knots

TL;DR

This work determines the complete set of -adjacent knots through crossings and introduces a novel obstruction framework that blends the Montesinos trick, the Alexander polynomial, and Heegaard Floer -invariants. When the branched double-cover is an -space, the Ni–Wu formula tightly constrains possible Alexander polynomials of lift knots , enabling rigorous exclusion of candidates via -invariants. The authors apply these techniques to rule out several knots (e.g., , , , , ) from being -adjacent and provide a constructive catalog of all -adjacent knots up to crossings. They also develop a Seifert surface/Conway polynomial construction via finger-moves to realize families of -adjacent knots, and they discuss implications for higher crossing knots and alternating cases, highlighting the potential of -invariants as a powerful obstruction tool in knot adjacency problems.

Abstract

Generalizing unknotting number, -adjacent knots have crossings such that changing any non-empty subset of them results in the unknot. In this paper, we determine the 2-adjacent knots through 12 crossings. Using Heegaard Floer -invariants and the Alexander polynomial, we develop a new technique to obstruct 2-adjacency, and we prove conjectures of Ito and Kato regarding 2-adjacent knots.

Paper Structure

This paper contains 22 sections, 14 theorems, 30 equations, 28 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. The Alexander polynomial
  4. The HOMFLY-PT polynomial
  5. Signed unknotting number
  6. Known invariants of 2-adjacent knots
  7. Dehn Surgery
  8. The linking matrix
  9. Branched double-covers and the Montesinos trick
  10. Heegaard Floer homology
  11. Heegaard Floer $d$-invariants
  12. Torres/Baker-Motegi obstruction
  13. Ruling out DJ via d-invariants when Sigma(K) is an L-space
  14. The Ni-Wu formula
  15. Results
  16. ...and 7 more sections

Key Result

Theorem 1.1

The following knots are $2$-adjacent: $3_1$, $4_1$, $8_{17}$, $8_{21}$, $9_{44}$, $10_{88}$, $10_{136}$, $10_{156}$, $11a_{289}$, $11n_{84}$, $11n_{125}$, $12a_{1008}$, $12a_{1249}$, $12n_{275}$, $12n_{392}$, $12n_{464}$, $12n_{482}$, $12n_{483}$, $12n_{650}$, and $12n_{831}$. No other knots with $1

Figures (28)

  • Figure 1: Twisting a knot $\kappa$ along an unknot $c$. This is a simplified diagram. We assume nontrivial linking between $c$ and $\kappa$, so the $n$ twists are applied to all strands passing through $c$, producing $\kappa_n$.
  • Figure 2: Finding the lift of an unknotting arc in $11a_{255}$. The top left figure is the original knot. To the right of that, we circle an unknotting crossing in purple, and in the next one, the marked crossing has been changed, recorded by a purple crossing arc. The following figures show the process of isotoping the black unknot until it looks like the standard unknot, while keeping track of the purple crossing arc until we reach the bottom left figure. The last image shows the lift $J$ of the arc in $\Sigma(K)$.
  • Figure 3: On the left, we see the knot $12n_{586}$ with a three half-twist tangle circled. In the middle diagram, we replace the circled tangle with an unknotted arc. On the right, we see the lift $J$ of the green arc from the middle diagram.
  • Figure 4: An arbitrary 2-string tangle and the knot $K_T$ resulting from the finger-move construction. (Bottom) The rational $(-2,1)$ tangle. After finger moves, it forms the $2$-adjacent knot $12n_{650}$ upon the choice of a $(+,-)$ clasp structure. A $(+,+)$ clasp structure would yield the $2$-adjacent knot $12n_{464}$, and $(-,-)$ would yield the $12n_{483}$ knot. Note that since the right arc of the tangle has writhe, twisting must be added to ensure the right band is zero-framed.
  • Figure 5: The knot $12n_{650}$ with a Seifert surface and the curves $\alpha_1, \alpha_2$ (red) and $\beta_1,\beta_2$ (blue) shown.
  • ...and 23 more figures

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 16 more