Trivalent vertices and bordered knot Floer homology in the standard basis

TL;DR

This paper introduces new algebraic structures and bimodules in bordered knot Floer homology, categorifying quantum group representations and incorporating trivalent vertices, with novel connections to Soergel bimodules and skew Howe duality.

Contribution

It constructs new bimodules and algebras that categorify quantum group representations and relate to holomorphic disk counts, extending bordered Floer homology with trivalent vertices.

Findings

Bimodules expressed as mapping cones for crossings.

First expression of braiding bimodules via Rouquier complexes in Heegaard Floer.

Trivalent vertex bimodules introduced, not seen in previous approaches.

Abstract

We define new algebras, local bimodules, and bimodule maps in the spirit of Ozsvath-Szabo's bordered knot Floer homology. We equip them with the structure of 2-representations of the categorified negative half U^- of U_q(gl(1|1)), 1-morphisms of such, and 2-morphisms respectively, and show that they categorify representations of U_q(gl(1|1)^-) and maps between them. Unlike with Ozsvath-Szabo's algebras, the algebras considered here can be built from a higher tensor product operation recently introduced by Rouquier and the author. Our bimodules are all motivated by holomorphic disk counts in Heegaard diagrams; for positive and negative crossings, the bimodules can also be expressed as mapping cones involving a singular-crossing bimodule and the identity bimodule. In fact, they arise from an action of the monoidal category of Soergel bimodules via Rouquier complexes in the usual way, the first time (to the author's knowledge) such an expression has been obtained for braiding bimodules in Heegaard Floer homology. Furthermore, the singular crossing bimodule naturally factors into two bimodules for trivalent vertices; such bimodules have not appeared in previous bordered-Floer approaches to knot Floer homology. The action of the Soergel category comes from an action of categorified quantum gl(2) on the 2-representation 2-category of U^- in line with the ideas of skew Howe duality, where the trivalent vertex bimodules are associated to 1-morphisms E, F in categorified quantum gl(2).