Vassiliev invariants and knots modulo pure braid subgroups

1998/05/20 by Theodore Stanford, Theodore B. Stanford, Stanford, Theodore B. · 3 citations
Mathematics · #57M25 #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.GT #math.QA #msc:57M25

paper · pdf · doi:10.48550/arxiv.math/9805092

21 pages, plain tex, 4 eps figures

arxiv created 1998/05/20 · openalex publication_date 1998/05/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in terms of the structure of the braid groups. We also prove some results about knots modulo the nth derived subgroups of the pure braid groups, and about knots modulo braid subgroups in general.

Cited by

• Goussarov-Polyak-Viro's n-equivalence and the pure virtual braid group
• Algebraic structures of Vassiliev invariants for knot families
• Embedding calculus and grope cobordism of knots

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• Vassiliev invariants I: Braid groups and rational homotopy theory
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