Knots as Reidemeister tiles: GCTS + RL research plan

Research target: classify exact cubic-lattice knots using lattice realizations of Reidemeister tiles. Visualization presents saved certificates. See the current lattice classifier subgoals and 12-crossing reference corpus. The diagram-seam pilot below is a separate control.

Current viewer: the search now uses brute force + online GCTS, with no knot-specific search hints. See the online search contract. This earlier plan documents the specialized control, retained as solveSpecialized; its offline seam learner, coloring shortcut, and RL policy are not used by the current viewer.

The search object is a movie assembled from Reidemeister-move tiles. Input knots are piecewise-linear embedded circles with edges parallel to coordinate axes. Their diagrams come from projecting the lattice embedding; the extra coordinate records time. A successful tile sequence is a certificate of ambient isotopy. The objective is either equivalence to a second knot or equivalence to the unknot.

The first executable pilot is the Reidemeister Tile Lab. It searches exact signed planar diagrams using all three move types, including expansions. It implements whole-diagram seam markings, a cold RL proposal control, rational-direction source projections, and independently replayable certificates. It is a foundation for the local tile system, not a completed general knot classifier or a geometric animation of every frame.

1. Resolve the tile geometry

The proposed geometries are compatible:

\[ C=[0,1]^3\times[0,1]\subset\mathbb R^3\times\mathbb R \]

is a four-dimensional space-time cube. With the spatial projection

\[ \pi(x,y,z)=\left(\frac{x-y}{\sqrt2}, \frac{x+y-2z}{\sqrt6}\right), \]

the image of a spatial unit cube is a regular hexagon, and \((\pi,\mathrm{id})(C)\) is a three-dimensional hexagonal prism. The projected coordinate directions are three directions at equal angles.

Recommended architecture: author and display Reidemeister tiles in diagram space-time as hexagonal cylinders; retain rectilinear spatial lifts and four-dimensional verification cells as certificate data. One move may occupy a block of cells. A cube or cylinder is a support container; a Reidemeister tile is the decorated local transition carried by that support. Do not require every move to fit one unit cube.

Different spatial cubes can project onto overlapping hexagons. Projection forgets depth and is not an injective identification of cells. Depth labels, strand identities, and unchanged exterior collars must survive projection. An arbitrary tiling of cylinders is not automatically a valid knot movie.

The spatial history of a knot is a swept annulus in space-time. It must come from embeddings at every time. An arbitrary embedded annulus connecting two knots in four dimensions gives concordance, which is weaker than isotopy. Births, deaths, saddles, strand passages, cutting and reconnection are excluded. See Livingston's survey.

2. Subgoals, dependencies, and acceptance gates

Prioritize F, then H, then G. Whole-state seams already give a sound control, but meaningful GCTS research requires a local interface that transfers between different diagrams. A larger policy model should wait until that interface and the movie verifier are stable.

3. Exact representation and primitive moves

Each input crossing has an identifier, an over/under flag, and an oriented crossing sign. The knot traversal gives a signed Gauss word. The sign convention is the sign of the determinant of the projected over-strand tangent followed by the under-strand tangent. This is consistent across the projector and the diagram engine; the label convention itself does not affect equivalence.

Construct a ribbon graph using incoming/outgoing half-edges at every crossing. The cyclic rotation at a crossing is determined by its sign. Require the Euler characteristic of the associated connected surface to be that of the sphere. This rejects signed words representing higher-genus virtual diagrams.

The current local disk checks are:

• Type I deletes a monogon, or inserts its inverse on any edge interval.
• Type II deletes an empty bigon whose two strands have consistent over/under roles and opposite crossing signs. Insertion enumerates interval pairs, sign and strand-order variants, including both orders at a shared cyclic gap, and verifies the new crossings bound a cancellable bigon.
• Type III flips the three sides of an empty triangular face. The three strands must have consistent top/middle/bottom order. Cyclic over/under triangles are rejected. Crossing signs and overpass identities persist.
• An identity tile holds a diagram between time slots; it is stationary filler, not a Reidemeister move or a counted move in the exported certificate.

Canonicalization permits relabelling crossings, cyclic shifts and reversal of the entire knot traversal. It does not quotient by reflection. Equivalent candidate outcomes are deduplicated only as complete signed diagram states; the exported move is independently replayed from its canonical input.

No edge-length reduction or crossing reduction is used as a legality rule. Expansions remain available. This matters for diagrams that must become more complex before they simplify.

4. Current point-value reduction and its precise scope

For a horizon \(H\), crossing cap \(C\), source \(D_0\), and target \(D_H\), enumerate every reachable signed diagram at each layer, with every allowed Reidemeister transition and identity hold. Stop with unknown if catalog construction exceeds its state budget. A partial catalog cannot support dead or forced conclusions.

The required point domain is

\[ Q=\{q_0,\ldots,q_{H-1}\}. \]

A transition placement \(c=(i,D,D')\) has the exact point function

\[ t_c(q_j)=\begin{cases}1&j=i,\\0&j\ne i.\end{cases} \]

It assigns marking values on two additional seam points:

\[ m_c(s_i)=\operatorname{key}(D),\qquad m_c(s_{i+1})=\operatorname{key}(D'). \]

The fixed exterior marking assigns the source and target at the endpoint seams. Exact coverage selects one tile per slot; equality at shared seam points joins the transitions into a movie. Seam points are marking-only points, and their dependencies are checked even though they are outside the coverage frontier. The placement universe and transforms are finite and explicitly authored; the only placement transform here is identity. This is an exact finite point model, not a claim that every local geometrical hexagonal tiling has been encoded.

Every required slot is an activated root obligation with generation zero. Global degree-zero checks precede every singleton placement, and singleton propagation precedes branching. All generations are equal in this reduction, so minimum degree is a permitted tie-breaker. The present implementation reconstructs the entire forward/reverse incidence graph and marking state from an immutable selected-placement list. Rollback is exact; incremental indexing and nontrivial generation propagation are future work.

The whole knot must not be replaced by an isolated local arc when constructing a failure key. An exterior tangle can change whether a local simplification extends to the target. Include the boundary context in every future local rule.

5. Failure markings: implemented control and intended GCTS

The pilot starts with no learned guards. Exhaustive backward reachability on the complete layered transition graph labels each diagram seam by whether it can reach the target in its remaining horizon under the crossing cap. Terminal failure means the target boundary does not match; earlier failure requires every outgoing transition to lead to a failed suffix.

For a proved failed state \(D\) at time \(i\), introduce a marking-only guard point \(f_{i,D}\). The exterior assigns

\[ m_{\mathrm{ext}}(f_{i,D})=0, \]

and every transition ending at that seam assigns

\[ m_c(f_{i,D})=1. \]

The ordinary marking-agreement test removes those candidates from the shared point graph. There is no runtime failure-clause scanner replacing the marking comparison. Guards are frozen before marked search. Scope includes target, horizon, crossing cap, candidate rules and version. They are redundant for that exact finite problem; they say nothing about larger caps or horizons. The guard proof DAG records failed states and outgoing candidate identities. No learned assignments are bundled into source files or silently reused.

This is a specialized global seam control, analogous to finite-state dynamic programming. It is not the master contract's local neighboring-pair marking synthesis protocol, and no runtime advantage is claimed. Full backward labeling already resolves finite reachability; this pilot tests its encoding as genuine point values and supplies a correctness baseline.

The next learner should enumerate neighboring Reidemeister tiles and larger boundary-collared patches; label them using unmarked bounded extension search; synthesize sparse equality/mismatch channels after each resolved result; permit individual absent values; replay the full resolved catalog; and activate a frozen marking with explicit scope. A local rule supported only by samples is a hypothesis and may guide ordering, but must not certify inequivalence.

6. RL proposals and evaluation

The current policy is cold tabular Q-learning with a small state feature: crossing count and move type. Training starts from random legal scrambles of the unknot, with all allowed move types available. Reaching the unknot rewards the trajectory. The policy composes up to three moves; each proposal has a base expansion and is replayed before supplying candidate-order preferences. The aggregate point contributions and union of markings must also agree with the live graph, including fixed failure guards and any already selected shared placement. Proposals never change candidate degrees or bypass global propagation.

This is intentionally weak and is not a learned knot classifier. The current examples are correctness controls, not a held-out evaluation corpus. Failure labels use all resolved states of the current finite catalog; that learning protocol is distinct from RL policy evaluation.

Next, train a policy on boundary-decorated ribbon-graph observations and variable-length tile clusters, including irregular patches and temporary expansions. Export transformations, base expansions, boundary ports, marking versions and training provenance. Split RL evaluation by underlying knot and scrambling generator, not just by rendering or crossing names. Permit arbitrary legal cluster composition rather than a fixed untangling strategy vocabulary.

Compare the unmarked point solver, exact marking lane, marking-plus-RL lane, direct graph reachability and classical knot tools on identical inputs and bounds. Include candidate catalog construction, failure labeling, training, proposal construction and validation, certificate verification, and peak memory. Current timings include the first five compute stages and JS replay, but do not yet measure memory or separate independent Python verification time. Do not infer a speedup from more removed candidates. The small controls show no branch reduction; RL training adds overhead.

7. Lattice input, projection, and the movie

The importer accepts an array of integer spatial vertices, or an object with vertices. Consecutive vertices must differ along exactly one coordinate axis. Long edges are expanded to unit edges, and repeated unit vertices are rejected. For unit cubic-lattice edges, any intersection occurs at a lattice vertex, so this certifies an embedded single closed polygon.

The source projector first attempts the exact body diagonal. Degenerate overlaps, endpoint crossings and multiple crossings trigger a small rational direction perturbation. Integer determinants classify crossings; rational depth comparison determines the overpass. Arbitrary-precision rational comparisons order the crossing events. Floating point is used only for display coordinates. Coordinate and length limits keep the integer operations inside exact safe-integer range; coincident crossing checks use arbitrary-precision integers.

The browser shows the faithful source projection and a schematic signed ribbon graph for each verified movie seam. The ribbon layout is not a planar knot drawing: only the numbered crossing disks are knot crossings. Smooth geometric untangling and removal of incidental projection crossings remain subgoal H. Do not interpolate arbitrary vertex lists between frames and call that a certified isotopy.

For H, require each local Reidemeister tile to carry a regular planar patch and an axis-aligned spatial lift, plus a collision-free piecewise-linear isotopy. Subdivide space-time as needed. Match unchanged lateral collars and depths. Projection perturbations and planar rearrangements are recorded display operations; a strand passage cannot be disguised as moving away a crossing.

8. Classification semantics

There are three global outcomes:

1. Equivalent: a verified Reidemeister movie reaches the exact target signed diagram, including an empty target diagram for the unknot.
2. Inequivalent: an independently justifiable invariant or stronger negative certificate differs. The pilot computes the dimension of Fox colorings over \(\mathbb F_3\), using \(2x_{\mathrm{over}}-x_{\mathrm{in}}-x_{\mathrm{out}}=0\). Different dimensions distinguish the trefoil from the unknot; equal dimensions do not establish equivalence or identify a knot.
3. Unknown: unresolved bounds, resource exhaustion or equal invariant values without a positive movie. Complete finite exhaustion is reported separately, but general knot equivalence remains unknown.

Fox-coloring background: Knot Colorings: Coloring and Goeritz matrices. Mirror knots remain separate inputs, although the pilot's coloring obstruction does not distinguish mirror trefoils. A full classifier needs more invariants and a classical certified negative lane, potentially normal-surface methods and certified knot-complement comparisons. Expanding positive movie search alone is a semidecision procedure and does not guarantee termination on inequivalent pairs. None of the current finite bounds establishes general completeness. BFACF lattice isotopies can serve as geometric comparison controls, not replacements for the user's Reidemeister tiles; see Janse van Rensburg and Rechnitzer.

9. Reproduce and independently verify

From the repository root, using an installed Node runtime:

node tests/test_knot_movie.mjs
node scripts/benchmark-knot-movie.mjs --certificates=/tmp/knot-movies.json
python3 research/knot-classifier/verify_movie.py /tmp/knot-movies.json

Browser-exported JSON uses the same independent verifier. It reconstructs ribbon faces and checks expansion tiles by inverse disk deletion, without calling the JS move enumerator, candidate graph, policy or marking learner. Certificate replay currently establishes diagram equivalence, not a full spatial space-time mesh certificate.

Initial exhaustive smoke evidence: 223 reachable planar diagram keys; 688 move checks, including 18 type-III transitions; inverse existence and Fox-coloring preservation; tiny bounded searches agree with direct path recursion; every learned failed seam independently exhausts at its remaining depth; corrupted movies and nonplanar words are rejected; catalog/search budget checks return unknown. These are finite tests, not a proof that the move implementation is complete for arbitrary crossing counts.

10. Conformance record and open gaps

Read the master contract before extending this engine. The present finite reduction has exact integer point contributions, assigned marking values including explicit zero, all required roots activated, full point/candidate incidence, marking-only dependencies, global scheduler ordering, immutable rollback, frozen guards, candidate fallback and certificate replay. The browser and policy consume the same candidate engine.

Remaining gaps: local hexagonal point functions and faithful spatial reduction; incremental dependencies and trails; nontrivial growth-generation tests; fractional/shared-support tests for a future local engine; pair-marking synthesis and activation gate; externally cross-checked move completeness; independent negative-certificate export; full geometric movie playback; persistent proof and cluster libraries; memory accounting; held-out RL benchmarks; an actual complete classifier. Preserve the seam-control benchmark semantics while developing that local engine.