1 Introduction↩︎

Tabulation of knots and links has a long tradition going back to Tait [@Tait1877], culminating in very large classical tables [@HosteThistlethwaiteWeeks1998]. Virtual knots [@Kauffman1999] and links in thickened surfaces \(\Sigma_g\times I\) provide a natural extension of this program beyond \(S^3\). A fundamental result of Kuperberg [@Kuperberg2003] asserts that every virtual link admits a unique minimal-genus surface representative. We work throughout with this surface-based viewpoint.

For the thickened torus \(T^2\times I\), explicit genus-one tables at small crossing numbers appeared in [@Omelchenko2007; @Grishanov2009a; @Grishanov2009b; @AkimovaMatveev2012; @AkimovaMatveev2014; @AkimovaMatveevTarkaev2020]. These works typically start from abstract \(4\)-regular graphs, then classify inequivalent embeddings into the surface, and only afterwards assign crossing data. Controlling equivalence under surface homeomorphisms is the main bottleneck.

The central idea of this paper is that the embedding should be treated as the primary combinatorial object. This is precisely the subject of the classical theory of maps on surfaces. A cellular embedding of a connected graph into a closed orientable surface is a map, and a cellular \(4\)-regular knot projection is simply a \(4\)-regular map. Maps admit a compact permutation encoding [@LandoZvonkin2004]: an involution \(\alpha\) pairs the two darts of each edge, and a permutation \(\sigma\) records the cyclic order of darts at each vertex.

This approach is related to general-purpose map and graph generators, but it is not a direct invocation of them. The program plantri is optimized for many classes of plane maps and planar graphs [@BrinkmannMcKay2007]. More recent work of Brinkmann gives a general generator for maps on oriented surfaces by combining abstract graph generators with embedding and isomorphism rejection routines [@Brinkmann2025]. Our setting is more specialized: the generated objects are loopless cellular \(4\)-regular torus maps satisfying knot-theoretic filters, and the subsequent crossing assignments and state-sum invariants are computed in the same dart-permutation model. Thus Brinkmann’s framework provides a natural independent benchmark or alternative generator, whereas the present implementation is designed as an integrated pipeline for genus-one knot and link diagrams.

In this encoding, unsensed equivalence of projections reduces to a finite conjugacy relation on permutation pairs, so that canonical representatives yield a duplication-free enumeration without a separate classification of embeddings. Moreover, the number of straight-ahead components (hence the knot/link distinction) is read off directly from the permutations.

Building on the map-enumeration line developed for degree-\(4\) structures [@Krasko; @KraskoOmelchenko2019PartI; @KraskoOmelchenko2019PartII; @KraskoOmelchenko2021NoPlanarLoops], we turn these map-theoretic tools into a structural engine for knot and link tabulation on \(T^2\). The implementation is validated against published genus-one tables for \(N\le 5\) and then extended to \(N=6,7,8\). A public reference implementation and datasets are available [@OmelchenkoTorusMaps].

2 The Permutation Model for Maps↩︎

Let \(\Sigma_g\) be the closed orientable surface of genus \(g\). A map on \(\Sigma_g\) is a cellular embedding of a connected graph into \(\Sigma_g\). Permutations act on the right: \((pq)(h)=q(p(h))\).

Two encodings define the same sensed map if one is obtained from the other by simultaneous conjugation; they define the same unsensed map if, additionally, \(\sigma\mapsto\sigma^{-1}\) (orientation reversal) is allowed.

For \(4\)-regular maps, each \(\sigma\)-cycle has length \(4\), and the involution \(\sigma^2\) pairs opposite darts at each vertex. Define \(\rho:=\sigma^2\alpha\). Then the number of straight-ahead components of the corresponding projection (i.e. the components of the projected curve) is \[c(P)=\tfrac12\,\#\mathrm{cyc}(\rho).\] This detects the knot/link distinction directly from the permutation data.

Monogon faces (fixed points of \(\varphi\)) are the projection-level shadow of immediate Reidemeister I reductions and are excluded from our enumeration.

2.0.0.1 Example.

Consider the dart set \(H=\{1,\dots,8\}\) with \(\sigma_0=(1\,2\,3\,4)(5\,6\,7\,8)\) (\(N=2\) vertices) and \(\alpha=(1\,6)(2\,8)(3\,5)(4\,7)\). Then \(\varphi=\sigma_0\alpha=(1\,7)(2\,5)(3\,8)(4\,6)\), giving \(F=4\) cycles (each of length \(2\)) and \(V-E+F=2-4+4=2\neq 0\); this is a sphere map, not a torus one. The torus condition \(F=N\) is therefore a genuine constraint on the matching \(\alpha\).

3 Canonical Enumeration of Projections on the Torus↩︎

Fix \(N\ge 1\) and let \(H=\{1,\dots,4N\}\). We fix a standard vertex rotation \[\sigma_0=(1\,2\,3\,4)(5\,6\,7\,8)\cdots(4N{-}3\;4N{-}2\;4N{-}1\;4N).\] Every unlabelled \(4\)-regular map admits a labelled encoding of the form \((\alpha,\sigma_0)\) for some fixed-point-free involution \(\alpha\).

3.0.0.1 Canonical representatives and rooted normalization.

Fix a total order \(\prec\) on labelled pairs. The unsensed canonical representative of \((\alpha,\sigma)\) is the \(\prec\)-minimal element of the union of the conjugacy orbits of \((\alpha,\sigma)\) and \((\alpha,\sigma^{-1})\) under \(S_H\). Two encodings represent the same unsensed map if and only if their canonical representatives coincide.

In practice this minimum is computed by rooted normalization rather than by enumerating \(S_H\). For each root dart \(r\) and for each of the two orientations \(\sigma\) and \(\sigma^{-1}\), we perform a deterministic traversal of the dart graph, inspecting from a dart \(h\) the neighbours in the fixed order \((\sigma(h),\alpha(h))\). Darts are relabelled \(1,2,\dots,4N\) in order of first discovery, and the resulting labelled pair is recorded. The canonical form is the lexicographically smallest normal form over all \(4N\) roots and both orientations. A rooted connected map has no non-trivial label ambiguity once the root and orientation are fixed, so this procedure gives the same complete invariant as the abstract orbit minimum. Its cost is \(O(N)\) per root and hence \(O(N^2)\) per accepted labelled encoding.

3.0.0.2 Enumeration algorithm and crude complexity bounds.

The conceptual algorithm enumerates fixed-point-free involutions \(\alpha\) on \(H\); for each one it tests the defining conditions, computes the canonical representative, and stores it. After deduplication, this yields exactly one representative per unsensed class. The naive worst-case number of matchings is \[(4N-1)!!=\frac{(4N)!}{2^{2N}(2N)!},\] so the corresponding crude time bound is \(O(N^2(4N-1)!!)\) and the storage needed after deduplication is \(O(N|\mathcal{C}_N|)\). This bound is intentionally pessimistic. The implementation uses a canonical construction path with activated vertices and early rejection of partial matchings; it does not materialize all matchings. The lower bound is the output size: any complete enumeration must take \(\Omega(|\mathcal{C}_N|)\) time.

3.0.0.3 Primeness filters.

A projection is called prime in the computational sense used here if its underlying multigraph has no \(2\)-edge-cut and no splitness witness based on mixed vertices. These are explicit combinatorial witnesses, stable under unsensed equivalence. They are sufficient for matching the projection-level conventions used in the comparison tables below, but they are not presented as a complete topological characterization of primeness in every possible convention.

3.0.0.4 Benchmark data.

Table 1 records the sizes of the enumerated sets. Table 2 shows the distribution of prime link projections by component count.

4 From Projections to Diagrams↩︎

For a fixed projection \(P\) with \(N\) crossings, a diagram \(D=(P,b)\) is obtained by choosing a crossing assignment \(b:\mathrm{Vert}(P)\to\{0,1\}\), selecting which strand is overpassing at each vertex. Thus \(b\) is the over/under datum of the diagram. This gives \(2^N\) diagrams per projection, or half as many when the comparison convention identifies the global crossing switch \(b\mapsto 1-b\).

4.0.0.1 Bigon reduction.

A bigon face is a \(2\)-cycle \((i\,j)\) of \(\varphi=\sigma\alpha\). Suppose that it is incident to two distinct vertices \(u\) and \(v\). Choose local cyclic orders \((h_0\,h_1\,h_2\,h_3)\) at \(u\) and \((k_0\,k_1\,k_2\,k_3)\) at \(v\), where \(b(u)=0\) means that \(\{h_0,h_2\}\) is overpassing and \(b(v)=0\) means that \(\{k_0,k_2\}\) is overpassing. If \(i=h_r\) and \(j=k_s\), put \(p_u(i)=r\pmod 2\) and \(p_v(j)=s\pmod 2\). Then the bigon supports an immediate Reidemeister II reduction if and only if \[b(u)+b(v)+p_u(i)+p_v(j)\equiv 1 \pmod 2.\] The reduced-diagram convention used in the comparison tables keeps precisely those assignments for which this congruence fails for every two-vertex bigon. The simpler rule \(b(u)\ne b(v)\) is valid only in local labellings where \(p_u(i)=p_v(j)=0\).

4.0.0.2 State-sum invariants.

The generalized Kauffman bracket on the torus [@AkimovaMatveev2012] is formulated entirely in the permutation model. At a vertex, the crossing assignment \(b\) selects the overpassing local strand. An \(A\)- or \(B\)-smoothing replaces the crossing by one of the two non-crossing pairings of the four incident darts; the product over all vertices is a smoothing involution \(\tau_{b,s}\), where the state \(s\) records the \(A/B\) choice at each vertex. The state circles are the connected components of the \(2\)-regular dart graph whose edges are the \(\alpha\)-pairs and the \(\tau_{b,s}\)-pairs. Equivalently, each state circle gives two cycles of the permutation \(\pi_{b,s}:=\alpha\tau_{b,s}\).

The torus bracket distinguishes contractible and essential state circles. Let \(C_1\) be the \(\mathbb{F}_2\)-vector space generated by the projection edges and let \(\partial_2:C_2\to C_1\) be the cellular boundary map sending each face to its mod-\(2\) boundary vector. A state circle is contractible if and only if its edge incidence vector lies in \(\operatorname{im}\partial_2\); otherwise it is essential. Thus no drawing in a fundamental polygon is required. The bracket is \[\label{eq:bracket} \langle D\rangle(a,x)=\sum_{s\in\{A,B\}^N} a^{A(s)-B(s)}(-a^2-a^{-2})^{\gamma(s)}x^{\delta(s)},\tag{1}\] where \(A(s)\) and \(B(s)\) are the numbers of \(A\)- and \(B\)-smoothings in the state, and \(\gamma(s)\) and \(\delta(s)\) count contractible and essential state circles. For knots, \(w(D)\) denotes the usual writhe, and the normalized polynomial \[X_D(a,x)=(-a)^{-3w(D)}\langle D\rangle(a,x)\] is invariant under all Reidemeister moves.

An efficient implementation precomputes the pair \((\gamma(t),\delta(t))\) for every smoothing vector \(t\in\{0,1\}^N\) once per projection, then reuses the data for all crossing assignments via the bijection \(s=b\oplus t\).

4.0.0.3 Validation.

For \(N\le 5\) we compare raw locally reduced diagram classes with the published lists. The comparison is summarized in Table 3.

The discrepancies are convention-dependent and are not treated as contradictions with the published classifications. For \(N=4\) knots, the additional class in our raw list is supported by the unique \(N=4\) projection without bigon faces; it is excluded in the published table by a global primeness convention not captured by our local bigon rule. For \(N=4\) links, the published count \(21\) is obtained from an original list of \(22\) by a global equivalence already noted in the link classification. For \(N=5\) knots, one of the two extra raw classes is equivalent to a smaller-crossing class under the comparison conventions. The remaining difference reflects an additional global convention in the published knot table, such as the central-fragment convention related to \(\Omega_3\) moves, rather than a failure of the permutation enumeration itself. For \(N=5\) links the count agrees exactly with the published table when the optional over/under participation prefilter is not imposed.

5 Tabulation Data and Structural Observations↩︎

The computation is extended to \(N=6,7,8\). Table 4 records the new diagram types at each crossing number.

A log-linear fit over \(N=3,\dots,8\) is numerically close to \(0.052\cdot 5.14^N\), but this is only a descriptive summary of the available data. With six data points we make no asymptotic claim about the true exponential growth constant.

5.1 Maximum number of straight-ahead components↩︎

For each \(N\), let \(c_{\max}(N)\) denote the maximum number of straight-ahead components among prime projections. In the computed range \(3\le N\le 8\):

The proved even-\(N\) lattice construction gives the lower bound. The remaining problem is the upper bound. A possible approach is to study the intersection pattern of straight-ahead components as a collection of essential curves on the torus and to show that primeness forces enough mixed vertices between them. For odd \(N\), the computed maximizers suggest a triangle-hexagon defect inserted into the even lattice family; constructing and analyzing this family explicitly would be a natural first step.

5.2 Equi-quadrilateral projections are always links↩︎

A \(4\)-regular torus map is equi-quadrilateral if every face has degree \(4\). In the computed range, every prime equi-quadrilateral projection has \(c(P)\ge 2\): no equi-quadrilateral projection supports a knot diagram.

This admits a purely permutation-theoretic reformulation: if \(\sigma\) and \(\varphi=\sigma\alpha\) both have cycle type \((4^N)\) and \(\langle\sigma,\alpha\rangle\) acts transitively, must \(\rho=\sigma^2\alpha=\sigma\varphi\) have at least four cycles? Equivalently, can the straight-ahead component count be one? The computation suggests that homological restrictions on the straight-ahead cycles may rule out this case.

5.3 The \(a\)-span of the genus-one bracket↩︎

For a Laurent polynomial \(Q(a,x)\), define \[\operatorname{span}_a(Q)= \max\{k:[a^k]Q\ne0\}-\min\{k:[a^k]Q\ne0\}.\]

This is motivated by the classical Kauffman–Murasugi–Thistlethwaite span theorem, but the present statement is only conjectural in genus one. A possible route to a proof would be to adapt the state-graph argument behind the classical span inequality, while controlling cancellations between states with contractible and essential circles. Table 6 confirms the bound in the computed range.

5.4 Further observations↩︎

5.4.0.1 Purely essential knots.

A genus-one knot is purely essential if \(X_D(a,x)\) has no \(x^0\) term. Equivalently at the level of the invariant, all purely contractible contributions cancel or are absent. This phenomenon, vacuous on \(S^2\), accounts for a large fraction of genus-one knot types (Table 7).

5.4.0.2 Bigon prevalence.

A \(4\)-regular torus map with \(N\) vertices has at most \(N-1\) bigon faces: since Euler’s formula gives \(F=N\), not all faces can be bigons, and if \(N-1\) faces are bigons then the remaining face has degree \(2N+2\). This bound is sharp at every \(N\le 8\) in our data. The fraction of prime projections containing at least one bigon reaches \(97\%\) at \(N=8\). This trend is qualitatively consistent with general \(0\)–\(1\) laws for submaps: in broad map classes, an admissible fixed planar submap appears linearly often with probability tending to one [@BenderGaoRichmond1992]. Our loopless, \(4\)-regular, prime torus projections form a more constrained family, so this theorem does not by itself prove the observed statement here; it explains why rapid appearance of bigons should not be surprising.

6 Outlook↩︎

The permutation framework applies without modification to any \(\Sigma_g\). On the torus, the contractible/essential dichotomy for state circles reduces to an \(\mathbb{F}_2\) boundary test, and a single variable \(x\) suffices for the bracket. For \(g\ge 2\), separating essential circles can be null-homologous, requiring integral homology classes and mapping-class-group canonicalization; this is the main obstacle to scaling the pipeline beyond genus one.