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<p>We study the skein relation that governs the HOMFLYPT invariant of links colored by one-column Young diagrams. Our main result is a categorification of this colored skein relation. This takes the form of a homotopy equivalence between two one-sided twisted complexes constructed from Rickard complexes of singular Soergel bimodules associated to braided webs. Along the way, we prove a conjecture of Beliakova–Habiro relating the colored 2-strand full twist complex with the categorical ribbon element for quantum <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathfrak {sl}_{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">sl</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

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A skein relation for singular Soergel bimodules

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月24日

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  • Matthew Hogancamp,
  • David E. V. Rose,
  • Paul Wedrich

摘要

We study the skein relation that governs the HOMFLYPT invariant of links colored by one-column Young diagrams. Our main result is a categorification of this colored skein relation. This takes the form of a homotopy equivalence between two one-sided twisted complexes constructed from Rickard complexes of singular Soergel bimodules associated to braided webs. Along the way, we prove a conjecture of Beliakova–Habiro relating the colored 2-strand full twist complex with the categorical ribbon element for quantum \(\mathfrak {sl}_{2}\) sl 2 .