TY - JOUR

T1 - Knot polynomials of open and closed curves

T2 - Knot polynomials of open curves

AU - Panagiotou, Eleni

AU - Kauffman, Louis H.

N1 - Publisher Copyright: © 2020 The Author(s). Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

AB - In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

KW - Jones polynomial

KW - Kauffman bracket polynomial

KW - knotoids

KW - linking number

KW - links

KW - open knots

KW - SPACE

KW - MECHANICS

KW - ENTANGLEMENT

KW - INVARIANT

KW - SYSTEMS

UR - http://www.scopus.com/inward/record.url?scp=85096568095&partnerID=8YFLogxK

U2 - 10.1098/rspa.2020.0124rspa20200124

DO - 10.1098/rspa.2020.0124rspa20200124

M3 - Article

AN - SCOPUS:85096568095

VL - 476

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 - 2240

M1 - 20200124

ER -