Publication Date
3-25-2026
Document Type
Article
Organizational Units
College of Natural Science and Mathematics, Mathematics
Keywords
Multi-virtual knot theory, Link invariants, Operator quandles, Quandle 2-cocycles, Commuting quandle automorphisms, Bell number
Abstract
Multi-virtual knot theory was introduced in 2024 by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves, we discuss the equivalence of multi-virtual link diagrams, particularly those that have the same virtual projections. We introduce operator quandles (that is, quandles with a list of pairwise commuting automorphisms) and construct an infinite family of connected operator quandles in which at least one third of right translations are distinct and pairwise commute. Using our set of generating moves, we establish the operator quandle coloring invariant and the operator quandle 2-cocycle invariant for multi-virtual links, generalizing the well-known invariants for classical links. With these invariants at hand, we then classify certain small multi-virtual knots based on the existing tables of small virtual knots due to Bar-Natan and Green. Finally, to emphasize a key difference between virtual and multi-virtual knots, we construct an infinite family of pairwise nonequivalent multi-virtual knots, each with a single classical crossing. Open problems are presented throughout the paper.
Copyright Date
4-7-2026
Copyright Statement / License for Reuse
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Rights Holder
Louis H. Kauffman, Sujoy Mukherjee, and Petr Vojtěchovský
Provenance
Received from Elsevier
File Format
application/pdf
Language
English (eng)
Extent
40 pgs
File Size
1.44 MB
Publication Statement
Copyright is held by the Authors. User is responsible for all copyright compliance. This article was originally published as
Kauffman, L. H., Mukherjee, S., & Vojtěchovský, P. (2026). Algebraic Invariants of Multi-Virtual Links. Journal of Algebra, 698. https://doi.org/10.1016/j.jalgebra.2026.03.018
Publication Title
Journal of Algebra
Volume
698
First Page
493
Last Page
532
ISSN
0021-8693
Recommended Citation
Kauffman, Louis H.; Mukherjee, Sujoy; and Vojtěchovský, Petr, "Algebraic Invariants of Multi-Virtual Links" (2026). Mathematics: Faculty Scholarship. 61.
https://digitalcommons.du.edu/math_faculty/61
https://doi.org/10.1016/j.jalgebra.2026.03.018