Embedded and Lagrangian Knotted Tori in $\BR^4$ and Hypercube Homology

TL;DR

This paper introduces hypercube diagrams for embedded and Lagrangian knotted tori in four-dimensional space, establishing a new invariant called hypercube homology based on knot Floer homology, with applications to distinguishing complex knotted tori.

Contribution

It develops hypercube diagrams and homology as novel tools for studying knotted tori in four dimensions, extending knot invariants to higher-dimensional embeddings.

Findings

Hypercube homology is invariant under 4D cube diagram moves.

The new invariant distinguishes between different knotted tori.

Examples include the Hopf linked tori and a knotted torus combining features of the 5_2 and trefoil knots.

Abstract

In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology. We provide examples of hypercube diagrams and hypercube homology, including using the new invariant to distinguish (up to cube moves) two "Hopf linked" tori. We also give examples of a "Trefoil" torus and an immersed knotted torus that is an amalgamation of the $5_2$ knot and a trefoil knot.