TY - JOUR

T1 - Vassiliev measures of complexity of open and closed curves in 3-space

AU - Panagiotou, Eleni

AU - Kauffman, Louis H.

N1 - Funding Information: E.P. was supported by NSF (grant no. DMS-1913180). L.H.K. was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation). Publisher Copyright: © 2021 The Author(s).

PY - 2021/10/1

Y1 - 2021/10/1

N2 - In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.

AB - In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.

KW - open knots

KW - links

KW - Vassiliev invariants

KW - knotoids

KW - linking number

KW - polygonal curves

KW - POLYNOMIAL INVARIANT

KW - RANDOM KNOTS

KW - ENTANGLEMENT

KW - LINKING

KW - SYSTEMS

KW - MODELS

KW - NUMBER

KW - DNA

UR - http://www.scopus.com/inward/record.url?scp=85119604296&partnerID=8YFLogxK

U2 - 10.1098/rspa.2021.0440

DO - 10.1098/rspa.2021.0440

M3 - Article

VL - 477

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 1364-5021

IS - 2254

M1 - 20210440

ER -