A table of genus two handlebody-knots with seven crossings

TL;DR

This work extends the seven-crossing classification of genus two handlebody-knots by constructing complete tables of irreducible and reducible cases up to mirror image, building on IshKisMorSuz:12 who treated up to six crossings. The authors partition seven-crossing diagrams by connectivity (3-connected, 2-connected, 1-connected) and enumerate via graph-theoretic and decomposition techniques, supplemented by a dynamic, invariant-based pruning pipeline (including ks_G and the G-image invariant) to identify equivalence classes. Irreducibility is established for most entries using rank-based criteria and modular invariant checks, while a targeted analysis confirms the distinctness of several hard pairs via P3-sphere decompositions and tangle arguments. The result is a pair of comprehensive tables (irreducible and reducible) for seven-crossing genus two handlebody-knots, with completeness proofs and discussions of subtleties such as non-uniqueness of minimal diagrams and potential duplicates across diagrams.

Abstract

We enumerate all genus two handlebody-knots with seven crossings, up to mirror image, extending the Ishii-Kishimoto-Moriuchi-Suzuki table.

Figures (11)

• Figure 2.1:
• Figure 2.2: Classical Reidemeister moves of type I, II, III.
• Figure 2.3: Reidemeister moves IV and V involving a trivalent vertex.
• Figure 2.4:
• Figure 3.1: Crossing-reducing moves.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: Ish:08
• Theorem 3.1: Completeness
• proof
• Remark 3.1
• Remark 3.2
• Theorem 5.1
• Theorem 6.1
• proof