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Computations of quandle 2-cocycle knot invariants without explicit 2-cocycles
W Edwin Clark1, Larry A Dunning1, Masahico Saito1
1Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA.
Journal of Knot Theory and Its Ramifications
|April 8, 2021
Summary
This study introduces a new knot invariant using quandle colorings, which simplifies the construction of quandle 2-cocycle invariants. This method effectively distinguishes knots up to 12 crossings, aiding in topological classification.
Area of Science:
- Knot theory
- Algebraic topology
- Discrete mathematics
Background:
- Knot invariants are crucial for distinguishing topological knots.
- Quandles provide algebraic structures for defining knot invariants.
- Quandle 2-cocycle invariants offer a powerful tool in knot classification.
Purpose of the Study:
- To develop a novel knot invariant derived from quandle colorings.
- To establish a method for constructing quandle 2-cocycle invariants without explicit 2-cocycles.
- To apply the invariant for classifying oriented prime knots.
Main Methods:
- Coloring 1-tangles with arbitrary connected quandles.
- Utilizing abelian extensions of quandles, specifically generalized Alexander quandles.
- Equating the novel invariant with existing knot polynomials like Eisermann's knot coloring polynomial.
Main Results:
- The developed invariant is equivalent to the quandle 2-cocycle invariant for specific abelian quandles.
- Many abelian extensions are constructed using generalized Alexander quandles, enabling new 2-cocycle invariants.
- The invariant successfully distinguishes all oriented prime knots up to 11 crossings and most with 12 crossings, including symmetry classifications.
Conclusions:
- The study provides an efficient method for constructing and applying quandle 2-cocycle invariants.
- This approach enhances the capability of distinguishing complex knot structures.
- The invariant serves as a valuable tool for advancing the field of knot theory and topological classification.
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