2006/08/14 by Jonathan Fine, Fine, Jonathan
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Physics (math-ph) #Number Theory (math.NT) #Numerical Analysis (math.NA) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.math/0608359
openalex publication_date 2006/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Kontsevich integral Z associates to each braid b (or more generally knot k) invariants Zi(b) lying in finite dimensional vector spaces, for i = 0, 1, 2, .... These values are not yet known, except in special cases. The inverse problem is that of determining b from its invariants Zi(b). In this paper we study the case of braids on two strands, which is already sufficient to produce interesting and unexpected mathematics. In particular, we find connections with number theory, numerical analysis and field theory in physics. However, we will carry this study out with an eye to the more general case of braids on n strands. We expect that solving the inverse problem even for n=3 will present real difficulties. Most of the concepts in this paper also apply to knots, but to simplify the exposition we will rarely mention this. The organisation and bulk of the writing of this paper predates its most significant results. We hope later to present better and develop further these results.