Algebraic invariants of multi-virtual links

TL;DR

This work initiates algebraic invariants for multi-virtual links by developing operator quandle colorings and operator quandle 2-cocycle invariants, anchored in a generating set for multi-virtual Reidemeister moves. It establishes both the theoretical framework (including permutation-based equivalences and projection considerations) and practical tools (R-cliques and explicit infinite quandle families) to distinguish multi-virtual knots, including infinite families with a single classical crossing. The paper also demonstrates the applicability of these invariants to classify and distinguish small multi-virtual knots, corroborated by computations and explicit examples, while highlighting open problems and a rich algebraic structure worth further exploration. Overall, it extends classical knot invariants to the multi-virtual setting and reveals new phenomena, such as unbounded total crossing numbers and diverse commuting-right-translation quandles, with potential implications for both topology and nonassociative algebra. The results lay groundwork for further cataloging and classification of multi-virtual knots and links using operator-quandle techniques.

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. Many open problems are presented throughout the paper.

Figures (28)

• Figure 1: Two conventions for drawing multi-virtual link diagrams.
• Figure 2: A multi-virtual detour move.
• Figure 3: The multi-virtual detour move is equivalent to the four moves $v1$, $v2$, $cv3$ and $mv3$.
• Figure 4: The $v2$ and $mv3$ moves realized as multi-virtual detour moves.
• Figure 5: Example of a multi-virtual detour move realized as a sequence of the moves $v1$, $v2$, $v3$, $cv3$, and $mv3$.

Theorems & Definitions (62)

• Proposition 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• Theorem 2.5
• proof
• Corollary 2.6
• Proposition 2.7
• proof