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Knot polynomials of open and closed curves
Eleni Panagiotou1, Louis H Kauffman2,3
1Department of Mathematics and SimCenter, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA.
Proceedings. Mathematical, Physical, and Engineering Sciences
|September 14, 2020
Summary
We introduce a new method to measure the entanglement of open curves in 3D space, extending the Jones polynomial to these structures. This new approach offers a continuous and real-valued measure applicable to both open and closed curves.
Area of Science:
- Topology
- Mathematical Physics
- Knot Theory
Background:
- Knot and link polynomials are fundamental invariants in knot theory.
- Existing polynomials primarily apply to closed curves (knots and links).
- A need exists to quantify entanglement for open curves in 3-space.
Purpose of the Study:
- To develop a method for measuring the entanglement of open curves in 3-space.
- To extend the concept of knot and link polynomials to open curves.
- To define a generalized Jones polynomial applicable to both open and closed curves.
Main Methods:
- Definition of a bracket polynomial for curves in 3-space.
- Demonstration that the bracket polynomial has real coefficients and is continuous.
- Application of the bracket polynomial to define a generalized Jones polynomial for open curves.
Main Results:
- The generalized Jones polynomial for open curves possesses real coefficients and is continuous.
- As endpoints of an open curve converge, its Jones polynomial approaches that of the resultant knot.
- For closed curves, the generalized polynomial acts as a topological invariant, akin to the classical Jones polynomial.
- Simplified expressions and finite computation methods are derived for polygonal curves with 3 and 4 edges.
Conclusions:
- The introduced method successfully extends the Jones polynomial to open curves in 3-space.
- This generalization provides a robust tool for analyzing curve entanglement in various scientific domains.
- The findings offer computational advantages for polygonal curve entanglement analysis.
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