IKMS Table: Prime Knot Classification in T×I

• The IKMS table is a comprehensive classification of prime knots in T×I, defined by minimal diagrams and a refined toroidal Kauffman bracket polynomial.
• It employs a three-stage process—enumerating 4-valent graphs, prime torus projections, and generating minimal diagrams—to ensure completeness and non-redundancy.
• The table serves as a benchmark for higher-genus knot classification, providing a methodological framework for invariant-based distinctions in knot theory.

The Ishii–Kishimoto–Moriuchi–Suzuki (IKMS) Table is a comprehensive classification of knots in the thickened torus $T \times I$ with at most four crossings, introduced by Ishii, Kishimoto, Moriuchi, and Suzuki. This table rigorously enumerates all prime knots under these constraints, establishes their minimal diagrams, and distinguishes each via a refined extension of the Kauffman bracket polynomial tailored to the toroidal context. The IKMS table is also foundational for subsequent census work in higher genus handlebody-knot classification and serves as a methodological reference for the combinatorial and algebraic invariants used in low-crossing knot tabulation within surfaces of positive genus.

1. Construction of the IKMS Table

The construction of the IKMS table employs a meticulous three-stage process to guarantee completeness and minimality for prime knot diagrams with up to four crossings in $T\times I$ (Akimova et al., 2012):

1. Enumeration of Abstract 4-Valent Graphs. All possible 4-valent connected graphs with $n \leq 4$ vertices are enumerated. It follows that such connected graphs must contain a loop or a multiple edge, restricting the set to exactly 15 combinatorial types, labeled $\mathbf{a}$–$\mathbf{o}$.
2. Prime Projections in the Torus. Each abstract graph is projected into the torus, enforcing the straight-ahead rule so that traversing each edge without backtracking yields a single cycle. A projection is prime if it lacks removable loops (no 1-gon faces) and cannot be split into two nontrivial arcs by a simple closed curve in the torus. Full torus automorphism equivalence is enforced, resulting in exactly 36 minimal prime projections, labeled $0_1$, $1_1$, $2_1$–$2_4$, $3_1$–$3_{11}$, and $4_1$–$4_{24}$.
3. Generation of Minimal Diagrams. For each prime projection with $n$ vertices, all possible over/under crossing assignments are considered. Reductions are applied:
   • Simultaneous switching of all crossings does not change knot type.
   • Biangle-cancellation (forced equivalences from 2-gon faces).
   • Exclusion of forbidden local patterns and elimination of diagrams reducible via Reidemeister III moves to already known types.

After applying all reductions and rigorous cross-checks, the final enumeration yields exactly 64 distinct minimal prime diagrams, each corresponding to a unique knot type in the torus with at most four crossings.

2. Table Structure, Notation, and Indexing

Each knot in the IKMS table is indexed by a symbol $n_i$, where $n$ is the crossing number and $i$ enumerates the distinct types at that crossing number. The full breakdown is:

Crossing Number $n$	Number of Knots	Labels
0	1	$0_1$
1	1	$1_1$
2	4	$2_1$–$2_4$
3	11	$3_1$–$3_{11}$
4	47	$4_1$–$4_{47}$

For each entry $n_i$, the table records:

• The underlying abstract graph ($\mathbf{a}$–$\mathbf{o}$).
• The projection label ($0_1$–$4_{24}$).
• A schematic of the minimal diagram (see Figure 1 in (Akimova et al., 2012)).
• The generalized Kauffman polynomial ($X(n_i)$), serving as a complete invariant for the table.

3. The Generalized Toroidal Kauffman Bracket

To establish isotopy distinction among the 64 tabulated knots, the IKMS table utilizes a two-variable Laurent polynomial invariant, the toroidal Kauffman bracket $X(D)$, defined as follows:

$$X(D) = (-a)^{-3w(D)} \sum_{s} a^{\alpha(s)-\beta(s)} \left( -a^2 - a^{-2} \right)^{\gamma(s)} x^{\delta(s)}$$

Where:

• The sum is over all states $s$ (A–/B–marker assignments for crossings).
• $\alpha(s)$, $\beta(s)$: counts of A- and B-markers.
• $\gamma(s)$: number of trivial circles in the state.
• $\delta(s)$: number of essential (non-contractible) circles in the state.
• $w(D)$: writhe of the diagram.
• $a$: bracket variable, $x$: records essential circles.

This invariant is normalized so that the unknotted circle satisfies $X(0_1) = -a^2 - a^{-2}$, and is proved to be invariant under all Reidemeister moves in the torus. The skein relation includes a term to account for essential circles:

$$X(\text{crossing}) = a\,X(\text{A-smoothing}) + a^{-1}\,X(\text{B-smoothing}) + x\,X(\text{essential circle})$$

All 64 knots in the table are proven to have distinct $X(n_i)$, ensuring isotopy invariance and completeness (Akimova et al., 2012).

4. Completeness and Distinguishing Invariants

The completeness of the IKMS table is established through the following facts:

• The combinatorial enumeration of all abstract graphs, minimal prime projections, and minimal diagrams (with exhaustive reductions) ensures all prime types with $\leq 4$ crossings are present.
• The generalized Kauffman polynomial $X(n_i)$ serves as a complete invariant within the census: all 64 tabulated Laurent polynomials are pairwise distinct.

A partial listing of these polynomials is found in §3 of (Akimova et al., 2012), with the full set given in the paper’s appendix. As a result, no further identifications (isotopic equivalences) are possible within the enumerated set.

5. Legacy and Extensions in Knot Theory

The IKMS table provides a model for systematic census methodology in knot theory on surfaces, serving as the foundation for subsequent enumerative efforts in higher-crossing and higher-genus settings. Notably, the table of genus-two handlebody-knots through seven crossings (Bellettini et al., 15 Nov 2025) builds directly on the IKMS methodology, adopting concepts of minimal diagrams, projection-type reductions, and algebraic invariants (including Kitano–Suzuki G-invariants and derived fundamental group presentations) to ensure completeness and refine distinctions among spatial genus-two knotted handlebodies.

Key patterns observed in further work include increased prevalence of composite knots and decomposable spines at higher crossings, and continued efficacy of algebraic invariants for distinguishing non-equivalent diagram types. Unique challenges and the need for refined invariants (including dynamic rooted-tree classifications and topological decompositions) have emerged in distinguishing chiral pairs and borderline cases, confirming the significance of the IKMS table as both template and benchmark for tabulation and invariant-based classification in spatial graph and knot theory (Bellettini et al., 15 Nov 2025).