Evolution method and HOMFLY polynomials for virtual knots
Ludmila Bishler, Alexei Morozov, Andrey Morozov, Anton Morozov
TL;DR
This paper extends the calculation of HOMFLY polynomials to virtual knots using the evolution method, exploring their invariance and differential hierarchy, and paving the way for defining colored virtual knot polynomials.
Contribution
It introduces an evolution-based approach to compute HOMFLY polynomials for virtual knots, analyzing their properties and potential for colored polynomial definitions.
Findings
HOMFLY polynomials for virtual knots can be computed via evolution methods.
Topological invariance is confirmed within the studied family.
Differential hierarchy is modified in the virtual knot case.
Abstract
Following the suggestion of arXiv:1407.6319 to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel 2-strand links. Within this family one can check topological invariance and understand how differential hierarchy is modified in virtual case. This opens a way towards a definition of colored (not only cabled) knot polynomials, though problems still persist beyond the first symmetric representation.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.