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Title: Knot polynomials of open and closed curves
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.  more » « less
Award ID(s):
1913180
PAR ID:
10279657
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume:
476
Issue:
2240
ISSN:
1364-5021
Page Range / eLocation ID:
20200124
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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