Compatibility in Ozsváth–Szabó's bordered HFK via higher representations

William Chang and Andrew Manion

Vol. 323 (2023), No. 2, 253–279

DOI: 10.2140/pjm.2023.323.253

Abstract

We equip the basic local crossing bimodules in Ozsváth–Szabó’s theory of bordered knot Floer homology with the structure of 1-morphisms of 2-representations, categorifying the $U_q(\mathfrak{𝔤}\mathfrak{𝔩}{(1|1)}^{+})$-intertwining property of the corresponding maps between ordinary representations. Besides yielding a new connection between bordered knot Floer homology and higher representation theory in line with work of Rouquier and Manion, this structure gives an algebraic reformulation of a “compatibility between summands” property for Ozsváth and Szabó’s bimodules that is important when building their theory up from local crossings to more global tangles and knots.

Keywords

Heegaard Floer, bordered Floer, bordered HFK, higher representation theory, categorification

Mathematical Subject Classification

Primary: 57K18

Secondary: 18N25, 57K16

Milestones

Received: 18 July 2022

Revised: 23 January 2023

Accepted: 1 April 2023

Published: 2 June 2023