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Khovanov homology and rational unknotting

Damian Iltgen, Lukas Lewark, Laura Marino

TL;DR

This work defines a Knots invariant $\lambda$ derived from a simple universal Khovanov theory over $\mathbb{Z}[G]$, proving that $\lambda(K)$ is a nonnegative integer lower bound for the proper rational unknotting number and that every $n\ge0$ is realized by some knot. The authors construct and compare $\mathbb{Z}[G]$-homology with Khovanov’s universal theory, establish reductions to standard Khovanov theory, and develop a Bar-Natan framework for tangles that underpins the $\lambda$-invariant. A central technical advance is Thompson-style computation of Bar-Natan complexes for rational tangles, yielding a recursive, efficient algorithm for these objects and enabling explicit calculations of $\lambda$ for a broad class of knots. The paper also analyzes $\lambda$-properties, including knot- and tangle-level generalizations, torsion-order obstructions, and thin-knot behavior, and demonstrates that $\lambda$ can grow arbitrarily large via staircase constructions. Overall, the results sharpen our understanding of unknotting-related invariants within Khovanov-style homologies and reveal new links between rational tangles, Bar-Natan theory, and torsion phenomena.

Abstract

Building on work by Alishahi-Dowlin, we extract a new knot invariant $λ\ge 0$ from universal Khovanov homology. While $λ$ is a lower bound for the unknotting number, in fact more is true: $λ$ is a lower bound for the proper rational unknotting number (the minimal number of rational tangle replacements preserving connectivity necessary to relate a knot to the unknot). Moreover, we show that for all $n \ge 0$, there exists a knot K with $λ(K) = n$. Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles.

Khovanov homology and rational unknotting

TL;DR

This work defines a Knots invariant derived from a simple universal Khovanov theory over , proving that is a nonnegative integer lower bound for the proper rational unknotting number and that every is realized by some knot. The authors construct and compare -homology with Khovanov’s universal theory, establish reductions to standard Khovanov theory, and develop a Bar-Natan framework for tangles that underpins the -invariant. A central technical advance is Thompson-style computation of Bar-Natan complexes for rational tangles, yielding a recursive, efficient algorithm for these objects and enabling explicit calculations of for a broad class of knots. The paper also analyzes -properties, including knot- and tangle-level generalizations, torsion-order obstructions, and thin-knot behavior, and demonstrates that can grow arbitrarily large via staircase constructions. Overall, the results sharpen our understanding of unknotting-related invariants within Khovanov-style homologies and reveal new links between rational tangles, Bar-Natan theory, and torsion phenomena.

Abstract

Building on work by Alishahi-Dowlin, we extract a new knot invariant from universal Khovanov homology. While is a lower bound for the unknotting number, in fact more is true: is a lower bound for the proper rational unknotting number (the minimal number of rational tangle replacements preserving connectivity necessary to relate a knot to the unknot). Moreover, we show that for all , there exists a knot K with . Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles.

Paper Structure

This paper contains 31 sections, 31 theorems, 165 equations, 16 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Rational replacements and rational unknotting
  3. A simple universal Khovanov homology
  4. The definition of $\lambda$
  5. The Bar-Natan complex of rational tangles
  6. Further properties and generalizations of $\lambda$
  7. A comparison of $\lambda$ with previously known invariants
  8. Computations
  9. Structure of the paper
  10. $\mathbb{Z}[G]$-homology and other variations of Khovanov homology
  11. Tangles and tangle diagrams
  12. Categorical Framework
  13. Definition of the $\mathbb{Z}[G]$-complex
  14. Frobenius Systems and TQFTs
  15. Equivalence of the $\mathcal{F}_{\textrm{univ}}$- and $\mathbb{Z}[G]$-theory
  16. ...and 16 more sections

Key Result

Theorem 1.1

For all knots $K$, one has $\lambda(K) \leq u_q(K)$.

Figures (16)

  • Figure 1: In (i), an example of a proper rational replacement ($1/3$ by $-1$ in the language of \ref{['dfn:ratioreplacement2']}), showing that the $P(3,3,2)$ pretzel knot has proper rational unknotting number 1. In (ii), an example of a non-proper rational replacement ($1/3$ by $0$), showing that the $P(3,3,-2)$ pretzel knot, which is also the $T_{3,4}$ torus knot, has rational unknotting number 1. Since $\lambda(T_{3,4}) = 2$, it follows from \ref{['thm:rationalreplacement']} that there is no proper rational replacement transforming the $T_{3,4}$ pretzel knot into the unknot, i.e. $T_{3,4}$ has proper rational unknotting number at least 2 (and in fact equal to 2).
  • Figure 4: A curtain of genus one.
  • Figure 5: Delooping.
  • Figure 6: The functors and constructions figuring in the statement of \ref{['lem:alpha']}.
  • Figure 7: Functors used in the proof of \ref{['lem:alpha']}.
  • ...and 11 more figures

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 1.5
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.6
  • Example 1.7
  • ...and 106 more