Integral geometry of plane curves and knot invariants

1994/11/30 by Xiao-Song Lin, Zhenghan Wang, Lin, Xiao-Song +1 · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th)

paper · pdf · doi:10.48550/arxiv.dg-ga/9411015

openalex publication_date 1994/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related with invariants of generic plane curves defined by Arnold recently with deep motivations in symplectic and contact geometry. Quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant.

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