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A polynomial upper bound on Reidemeister moves for each link type

Marc Lackenby

TL;DR

The paper proves that for any fixed link type K in S^3, there exists a polynomial p_K such that any two diagrams with c1,c2 crossings differ by at most p_K(c1)+p_K(c2) Reidemeister moves. The approach extends Lackenby’s polyhedral-unfolding techniques from the unknot to general non-split links by constructing exponentially controlled hierarchies for the link exterior, organizing the exterior via normal surfaces, JSJ decompositions, and Seifert-fibered pieces, and then using arc-presentations and branched surfaces to bound the required Reidemeister moves. A key contribution is a detailed framework for controlling surface complexity through weakly fundamental/essential surfaces, envelopes, and generalized admissible forms, culminating in explicit polynomial bounds p_K for various link classes (e.g., figure-eight and torus knots). The results place the K-recognition problem in NP and show determinism in exponential time, with practical polynomials computable for specific K. The methods offer a principled path toward a universal polynomial p that would place Cohen–Kuperberg type knot equivalence problems into NP, marking a significant advance in the complexity of 3-manifold knot theory.

Abstract

For each link type $K$ in the 3-sphere, we show that there is a polynomial $p_K$ such that any two diagrams of $K$ with $c_1$ and $c_2$ crossings differ by at most $p_K(c_1) + p_K(c_2)$ Reidemeister moves. As a consequence, the problem of recognising whether a given link diagram represents $K$ is in the complexity class NP and hence can be completed deterministically in exponential time. We calculate this polynomial $p_K$ explicitly for various classes of links.

A polynomial upper bound on Reidemeister moves for each link type

TL;DR

The paper proves that for any fixed link type K in S^3, there exists a polynomial p_K such that any two diagrams with c1,c2 crossings differ by at most p_K(c1)+p_K(c2) Reidemeister moves. The approach extends Lackenby’s polyhedral-unfolding techniques from the unknot to general non-split links by constructing exponentially controlled hierarchies for the link exterior, organizing the exterior via normal surfaces, JSJ decompositions, and Seifert-fibered pieces, and then using arc-presentations and branched surfaces to bound the required Reidemeister moves. A key contribution is a detailed framework for controlling surface complexity through weakly fundamental/essential surfaces, envelopes, and generalized admissible forms, culminating in explicit polynomial bounds p_K for various link classes (e.g., figure-eight and torus knots). The results place the K-recognition problem in NP and show determinism in exponential time, with practical polynomials computable for specific K. The methods offer a principled path toward a universal polynomial p that would place Cohen–Kuperberg type knot equivalence problems into NP, marking a significant advance in the complexity of 3-manifold knot theory.

Abstract

For each link type in the 3-sphere, we show that there is a polynomial such that any two diagrams of with and crossings differ by at most Reidemeister moves. As a consequence, the problem of recognising whether a given link diagram represents is in the complexity class NP and hence can be completed deterministically in exponential time. We calculate this polynomial explicitly for various classes of links.
Paper Structure (143 sections, 105 theorems, 91 equations, 56 figures)

This paper contains 143 sections, 105 theorems, 91 equations, 56 figures.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Structure of the paper
  4. Notation
  5. Normal Surfaces
  6. Definition
  7. The weight of a surface
  8. Boundary patterns
  9. Boundary patterns and normal surfaces
  10. Annuli and tori in $M$
  11. Normal summation and fundamental surfaces
  12. A topological interpretation of normal summation
  13. Components of least weight surfaces
  14. Exponentially controlled and weakly fundamental surfaces
  15. Making surfaces weakly fundamental
  16. ...and 128 more sections

Key Result

Theorem 1.1

For each link type $K$ in the 3-sphere, there is a polynomial $p_K$ with following property. If $D_1$ and $D_2$ are two diagrams for $K$ with $c_1$ and $c_2$ crossings respectively, then they differ by a sequence of at most $p_K(c_1) + p_K(c_2)$ Reidemeister moves.

Figures (56)

  • Figure 1: Elementary normal discs in a tetrahedron
  • Figure 2: Almost clean annulus and necklace annulus in $\partial M$
  • Figure 3: Normal summation
  • Figure 4: Isotopy of $S$ when a summand has a boundary-parallel torus component $S_2'$
  • Figure 5: A generalised product region is obtained from a product by collapsing certain arcs in the boundary
  • ...and 51 more figures

Theorems & Definitions (187)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • proof
  • Proposition 2.1
  • Proposition 2.2
  • ...and 177 more