Maps on Surfaces as a Structural Framework for Genus-One Virtual Knot Classification

TL;DR

This paper introduces a combinatorial surface map framework for classifying genus-one virtual knots, enabling systematic enumeration and analysis without traditional diagram drawing, validated up to crossing number 8.

Contribution

It develops a novel permutation-based combinatorial model for genus-one virtual knots, allowing complete enumeration and classification without duplication or drawing diagrams.

Findings

Validated against existing classifications up to N=5 crossings.

Extended enumeration to crossing number N=8.

Provided open-source implementation and datasets for virtual knot analysis.

Abstract

We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus $T^2\times I$. Using the theory of maps on surfaces, cellular $4$--regular torus projections are encoded by permutation pairs $(\alpha,\sigma)$, and unsensed projection classes are enumerated completely and without duplication via canonical representatives. For a fixed projection, crossing assignments are encoded by bit data, and an immediate Reidemeister~II reduction supported by a bigon face is characterized directly in terms of these bits. The genus-one generalized Kauffman-type bracket is then evaluated as a state sum entirely within the permutation model, without drawing diagrams in a fundamental polygon. The implementation is validated against published genus-one classifications for $N\le 5$ under explicit comparison conventions, with remaining discrepancies explained at the level of global conventions. Beyond the published range, we compute projection and diagram data for crossing numbers up to $N=8$ and provide a public reference implementation together with machine-readable datasets. Via the standard correspondence between virtual knots and knots in thickened surfaces, this yields a canonical and fully reproducible genus-one framework for virtual knot tabulation.