2025/05/19 by Ryan Budney, Budney, Ryan
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · #Advanced Materials and Mechanics #Connective tissue disorders research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2505.13699
openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we describe what should perhaps be called a `type-2' Vassiliev invariant of knots S2 -> S4. We give a formula for an invariant of 2-knots, taking values in Z2 that can be computed in terms of the double-point diagram of the knot. The double-point diagram is a collection of curves and diffeomorphisms of curves, in the domain S2, that describe the crossing data with respect to a projection, analogous to a chord diagram for a projection of a classical knot S1 -> S3. Our formula turns the computation of the invariant into a planar geometry problem. More generally, we describe a numerical invariant of families of knots Sj -> Sn, for all n >= j+2 and j >= 1. In the co-dimension two case n=j+2 the invariant is an isotopy invariant, and either takes values in Z or Z2 depending on a parity issue.