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<p>In the last decades, finite type invariants and categorification have been active research directions in geometry and topology, and the two theories meet at quantum invariants. Finite type behavior of categorified invariants is hence a natural question. In this paper, we focus on Khovanov homology, which is a categorification of the Jones polynomial. We aim to study its finite type behavior, in comparison with the case of the Jones polynomial. Specifically, the goal of this paper is to prove a categorified analogue of Kontsevich’s 4T relation on Vassiliev derivatives of Khovanov homology. This categorification is described as higher commutativity of a hexagon-prism diagram. Indeed, unwinding the proof of the 4T relation on classical polynomial invariants, one sees that the relation follows from a relationship between isotopy moves and crossing changes. In the case of Khovanov homology, this relationship becomes commutativity of a diagram consisting of morphisms corresponding to these moves. The statements and the proofs are given in terms of cobordisms following Bar-Natan’s formulation. Our formulation of the categorified FI and 4T relations assert that the Khovanov homology kills the cycles of the singular knot space which bound the singular loci of cusps and triple points respectively. The result reveals the relationship between categorified relations on quantum invariants and geometry of singularities.</p>

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On the four-term relation on Khovanov homology

  • 期刊论文

  • 发表日期：

    2026年2月10日

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  • Noboru Ito,
  • Jun Yoshida

摘要

In the last decades, finite type invariants and categorification have been active research directions in geometry and topology, and the two theories meet at quantum invariants. Finite type behavior of categorified invariants is hence a natural question. In this paper, we focus on Khovanov homology, which is a categorification of the Jones polynomial. We aim to study its finite type behavior, in comparison with the case of the Jones polynomial. Specifically, the goal of this paper is to prove a categorified analogue of Kontsevich’s 4T relation on Vassiliev derivatives of Khovanov homology. This categorification is described as higher commutativity of a hexagon-prism diagram. Indeed, unwinding the proof of the 4T relation on classical polynomial invariants, one sees that the relation follows from a relationship between isotopy moves and crossing changes. In the case of Khovanov homology, this relationship becomes commutativity of a diagram consisting of morphisms corresponding to these moves. The statements and the proofs are given in terms of cobordisms following Bar-Natan’s formulation. Our formulation of the categorified FI and 4T relations assert that the Khovanov homology kills the cycles of the singular knot space which bound the singular loci of cusps and triple points respectively. The result reveals the relationship between categorified relations on quantum invariants and geometry of singularities.