Geodesic
A deformation of Asaeda–Przytycki–Sikora homology
Li, Zhenkun1 ; Xie, Yi2 ; Zhang, Boyu3
1School of Mathematics and Statistics, University of South Florida, Tampa, FL, United States
2Beijing International Center for Mathematical Research, Peking University, Beijing, China
3Department of Mathematics, University of Maryland at College Park, College Park, MD, United States
Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1545-1560
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We define a 1-parameter family of homology invariants for links in thickened oriented surfaces. It recovers the homology invariant of Asaeda, Przytycki and Sikora (Algebr. Geom. Topol. 4 (2004) 1177–1210) and the invariant defined by Winkeler (Michigan Math. J. 74 (2024) 1–31). The new invariant can be regarded as a deformation of Asaeda–Przytycki–Sikora homology; it is not a Lee-type deformation as the deformation is only nontrivial when the surface is not simply connected. Our construction is motivated by computations in singular instanton Floer homology. We also prove a detection property for the new invariant, which is a stronger result than our previous work (Selecta Math. 29 (2023) art. id. 84).

DOI : 10.2140/agt.2025.25.1545
Keywords: homology theories in knot theory

Li, Zhenkun  1   ; Xie, Yi  2   ; Zhang, Boyu  3

1 School of Mathematics and Statistics, University of South Florida, Tampa, FL, United States
2 Beijing International Center for Mathematical Research, Peking University, Beijing, China
3 Department of Mathematics, University of Maryland at College Park, College Park, MD, United States 
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Li, Zhenkun; Xie, Yi; Zhang, Boyu. A deformation of Asaeda–Przytycki–Sikora homology. Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1545-1560. doi: 10.2140/agt.2025.25.1545

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