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The knot complement problem for null-homotopic knots

Aliakbar Daemi, Tye Lidman

TL;DR

The paper advances Boileau's nullhomotopic knot complement problem by combining Dehn-surgery arguments with $SU(2)$-representation theory and instanton Floer homology. It shows that for nullhomotopic knots in a rational homology sphere, any surgery preserving the ambient fundamental group must have slope $\pm1$, and in the $SU(2)$-non-degenerate case, such a surgery would force an orientation-preserving equivalence that is incompatible with the instanton-dimensional analysis. The authors develop a Morse–Bott perturbation framework and a robust energy filtration to compare cobordism-induced maps, culminating in a main theorem that nontrivial surgery cannot preserve oriented homeomorphism type under the stated hypotheses. These results provide a significant step toward Boileau's conjecture by narrowing the allowed surgery data and highlighting the role of orientation in knot complement problems.

Abstract

We prove that for three-manifolds satisfying a certain algebraic condition on their fundamental group, null-homotopic knots are determined by their complements. This answers a Kirby Problem posed by Boileau for this special case of 3-manifolds. The argument uses techniques in instanton Floer homology and SU(2)-representation varieties.

The knot complement problem for null-homotopic knots

TL;DR

The paper advances Boileau's nullhomotopic knot complement problem by combining Dehn-surgery arguments with -representation theory and instanton Floer homology. It shows that for nullhomotopic knots in a rational homology sphere, any surgery preserving the ambient fundamental group must have slope , and in the -non-degenerate case, such a surgery would force an orientation-preserving equivalence that is incompatible with the instanton-dimensional analysis. The authors develop a Morse–Bott perturbation framework and a robust energy filtration to compare cobordism-induced maps, culminating in a main theorem that nontrivial surgery cannot preserve oriented homeomorphism type under the stated hypotheses. These results provide a significant step toward Boileau's conjecture by narrowing the allowed surgery data and highlighting the role of orientation in knot complement problems.

Abstract

We prove that for three-manifolds satisfying a certain algebraic condition on their fundamental group, null-homotopic knots are determined by their complements. This answers a Kirby Problem posed by Boileau for this special case of 3-manifolds. The argument uses techniques in instanton Floer homology and SU(2)-representation varieties.

Paper Structure

This paper contains 10 sections, 23 theorems, 58 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Instanton homology and detection of the trivial knot
  3. Controlling the surgery coefficient
  4. The case of $|m|\neq 1$
  5. The case of $|n|\neq 1$
  6. Surgery slope $\pm 1$ and the surgery exact triangle
  7. More background on instanton Floer homology of admissible pairs
  8. Gradient-like perturbations of the ASD equation and perturbed flat instantons
  9. Proof of the main theorem
  10. An example illustrating why orientations matter

Key Result

Theorem 1

If two knots $K_1$ and $K_2$ in $S^3$ have homeomorphic exteriors, then there is a homeomorphism of $S^3$ taking $K_1$ to $K_2$.

Figures (6)

  • Figure 1: The left figure shows the knot $K$, a nullhomotopic knot in $L(5,1) = S^3_{5}(U)$. (It is nullhomotopic since $K$ has linking number zero with the unknot, and in a lens space, nullhomologous knots are automatically nullhomotopic.) Performing $1$-surgery on $K$ yields the figure on the right, where $K'$ is the core curve of the surgery. The ambient manifold is $S^3_{5}(T_{2,3}) = -L(5,1)$, and this knot is again nullhomotopic. It follows that $K$ and $K'$ have the same exterior, but the knots are not equivalent, being in different oriented three-manifolds.
  • Figure 2: The pillowcase is drawn in black. In red is $\iota^*\chi(E_{T_{2,3}})$, the image of the character variety for the right-handed trefoil under restriction. The $x$-axis is given by the value of a character on $\mu$ and the $y$-axis is given by the value of a character on $\lambda$.
  • Figure 3: The path $\gamma_3$ in the pillowcase.
  • Figure 4: The correspondence associated to the character varieties
  • Figure 5: The 2-handle cobordism $W$ from $-L(5,1)$ to $L(5,1)$.
  • ...and 1 more figures

Theorems & Definitions (51)

  • Theorem 1: GL
  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Theorem 2
  • Corollary 1
  • proof
  • Definition 1
  • Theorem 3
  • Remark 1
  • ...and 41 more