Knots, Spiders & the ISA

The BIND opcode is the trivalent vertex. The Fano plane is the right coefficient ring. Khovanov homology falls out of the opcode chain complex.

Table of contents

  1. Start here: Kauffman’s problems through the ISA lens
  2. The starting point: you already know the objects
  3. The G₂ spider is the BIND calculus
    1. Trivalent vertices and the BIND opcode
  4. Khovanov homology from the opcode chain complex
    1. The ISA chain complex
    2. Recovery of Khovanov homology
  5. The Grassmannian as common parent
    1. Bonding and scattering in the same space
  6. The Fano plane as coefficient ring
    1. Why GF(2) is the right field
  7. Weyl chamber homology
  8. What the ISA adds for topologists
  9. Papers
  10. What to read first

Start here: Kauffman’s problems through the ISA lens

Kauffman’s Problems Through an Origami ISA Lens is the paper written directly for this audience.

Over forty years, Louis Kauffman developed a sequence of knot invariants — bracket, Jones polynomial, Khovanov homology, virtual knots, slice concordance, loop braids — each more powerful than the last, yet none completing the classification of knots. The ISA reading: this progression is not a sequence of failures but a traversal of topological tiers. Each invariant lives at a specific level of the H^k cohomological hierarchy, and the reason no single invariant suffices is that knots occupy multiple tiers simultaneously.

Invariant ISA tier What it detects
Bracket / Jones polynomial H⁰ Writhe; framing; stabiliser-level topology
HOMFLY-PT H⁰–H¹ boundary Two-variable interpolation across the tier boundary
Alexander polynomial H¹ Seifert surface genus; H¹ obstruction
Khovanov homology H¹ → H² differential Categorification = the ISA chain complex differential ∂
Knot Floer homology H² Fibred knots; concordance; deep entanglement
Virtual knots / loop braids H²+ Higher-tier structure; open questions

The ISA is a lens, not a solver. The paper does not resolve Kauffman’s open problems; it organises them by tier and identifies what a higher-tier invariant would need to detect.

The starting point: you already know the objects

If you work in low-dimensional topology or representation theory, the relevant objects are ones you know well:

  • The Kuperberg G₂ spider — a planar diagram calculus for G₂ representations, with trivalent vertices and the 14-dimensional fundamental representation
  • Khovanov homology — a categorification of the Jones polynomial; the differential ∂ on the chain complex satisfies ∂² = 0 by a direct algebraic argument
  • The Grassmannian Gr(k,n) — the space of k-dimensional subspaces of ℂⁿ, carrying Schubert calculus, Plücker coordinates, and the amplituhedron construction
  • The Fano plane PG(2,2) — the unique projective plane over GF(2); 7 points, 7 lines, automorphism group PSL(2,7) of order 168

The ISA claim is that these are not four separate structures imported into physics from different corners of mathematics. They are four faces of a single underlying algebra: the opcode algebra of the ISA, whose tier structure (H⁰, H¹, H²) is the natural home for each.

The G₂ spider is the BIND calculus

Trivalent vertices and the BIND opcode

The Kuperberg G₂ spider~[572] has two generators: trivalent vertices (for the 14-dimensional fundamental of G₂) and crossings. The spider relations — the planar isotopy moves — determine when two diagrams are equal as morphisms in the G₂ representation category.

The ISA BIND opcode is an H² operation: it takes two systems and creates an irreducible entanglement between them that cannot be removed by any H⁰ or H¹ operation alone. In the diagrammatic language, BIND is represented by a trivalent vertex — one wire in, two wires out, with the three legs carrying the three colour indices of SU(3) (or the three legs of the G₂ fundamental decomposition).

Theorem (Paper 572): The Kuperberg G₂ spider is isomorphic to the BIND calculus: there is a functor from the spider category to the ISA opcode category that sends each trivalent vertex to a BIND operation and each spider relation to an ISA identity. The BIND theorem — that any closed BIND diagram evaluates to a scalar in GF(2) — follows from the spider evaluation formula.

The practical consequence: every G₂ spider identity is an ISA tautology, and vice versa. Computations in the spider calculus can be mechanically verified by checking ISA opcode sequences; conversely, every ISA proof involving BIND produces a valid G₂ spider identity.

Khovanov homology from the opcode chain complex

The ISA chain complex

The ISA has a natural chain complex structure~[571]. Assign to each opcode a cohomological degree:

Opcode Degree Tier
ORBIT 0 H⁰
TWIST, SNAP↑, SNAP↓ 1 H¹
BIND, MERGE, LINK 2 H²

The differential ∂: C^k → C^{k+1} is defined by the ISA composition law: ∂(f) = sum over all ways of promoting f by one tier, weighted by the Fano incidence matrix over GF(2).

Theorem (Paper 571, x571a): ∂² = 0. The ISA chain complex (C•, ∂) is a cochain complex.

The proof is direct: ∂²(f) counts paths of length 2 in the Fano incidence graph, weighted over GF(2). The Fano plane has the property that any two points determine a unique line, so each such path is counted exactly twice — and 2 = 0 over GF(2).

Recovery of Khovanov homology

Khovanov’s original construction~[Kho00] assigns to a link diagram a chain complex of graded abelian groups whose Euler characteristic is the Jones polynomial. The differential is defined by saddle cobordisms between resolutions of crossings.

The ISA chain complex recovers this: the two resolutions of a crossing are the two SNAP states (SNAP↑ and SNAP↓), and the saddle cobordism between them is the BIND differential. The grading on Khovanov’s complex is the tier grading (H⁰ = 0-smoothing, H¹ = 1-smoothing, H² = two-smoothing connected by a cobordism).

The coefficient ring GF(2) — the Fano plane’s arithmetic — is exactly the coefficient ring of Khovanov homology over F₂. The ∂² = 0 proof via Fano incidence is a one-line alternative to the standard proof by commutativity of saddle cobordisms.

The Grassmannian as common parent

Bonding and scattering in the same space

The Grassmannian Gr(k,n) carries two structures that have historically been studied separately:

  • Schubert calculus / amplituhedron: the positive Grassmannian Gr+(k,n) parametrises scattering amplitudes in N=4 SYM via the BCFW recursion; the amplituhedron is a region in Gr(k,k+4)
  • Chemical bonding: the occupied orbital subspace of a molecule is a point in Gr(k,n) (k occupied orbitals in n basis functions); the molecular Hamiltonian acts on this space by Schubert intersection

Paper 574 shows these are the same object: the Plücker embedding gives a unified parametrisation of both the bonding tier (H², irreducible entanglement between orbitals) and the scattering amplitude (H², irreducible entanglement between external legs). The ISA tier structure is the Schubert cell decomposition of Gr(k,n):

Schubert cell Dimension ISA tier
Grassmannian open cell k(n−k) H²
Codimension-1 Schubert divisor k(n−k)−1 H¹ boundary
Fixed points of torus action 0 H⁰

The snap threshold β* is the codimension-1 Schubert divisor: the locus in Gr(k,n) where the bonding (or scattering) transitions from one Schubert cell to another.

The Fano plane as coefficient ring

Why GF(2) is the right field

The Fano plane PG(2,2) appears in three distinct roles across the ISA:

  1. Colour geometry: the 7 off-diagonal SU(3) generators correspond to the 7 Fano points; the 7 Fano lines give the 7 quark colour-charge combinations; the colour-singlet condition (baryon) is a closed directed 3-cycle on the Fano plane (Paper 545)

  2. Magic state structure: the Wigner function negativity of a magic state is determined by its Fano orbit — which of the 7 Fano lines its Bloch vector projects onto (Paper 361)

  3. Chain complex coefficient ring: the ∂² = 0 proof for the ISA Khovanov complex uses the Fano incidence matrix over GF(2) — the unique field with 2 elements, whose projective plane is the Fano plane

These are not three independent uses of the same combinatorial object. They are three projections of a single fact: GF(2) is the natural coefficient field for ISA computations, because the ISA distinguishes only two states for each binary invariant (present/absent, in/out, up/down), and GF(2) is the unique field with that property.

The Fano plane is the projective geometry of GF(2)³ — the simplest non-trivial projective plane — and the ISA inherits its structure from it.

Paper 366 makes this explicit: the valence of a quantum magic state is its position in the Fano orbit decomposition, and the T-count lower bound follows from the Fano orbit structure over GF(2).

Weyl chamber homology

The Weyl chamber decomposition of a Lie algebra provides a canonical cell structure. For SU(2) acting on two qubits, the Weyl chambers are the orbits of the two-qubit exchange group, and the chamber walls are where entanglement changes tier.

Paper 595 computes the Bredon cohomology of the two-qubit Weyl chamber complex. The result: the H¹ chamber boundary has a non-trivial Bredon cohomology class — a topological obstruction to continuously deforming an H¹ state into an H² state without crossing a snap threshold. This is the cohomological certificate for the tier hierarchy.

The Weyl chamber complex is a CW complex; its cells are the ISA tiers; the Bredon differential is the ISA ∂. The cohomology is the ISA cohomology.

What the ISA adds for topologists

The ISA framework is not a redescription of known topology. It adds:

  1. A physical interpretation of the coefficient ring. Khovanov homology over GF(2) is usually presented as a computational convenience (the signs cancel). The ISA explains why GF(2): it is the natural arithmetic of binary physical invariants, derived from the Fano plane geometry of the qubit.

  2. A tier interpretation of the chain complex grading. The cohomological degree in the ISA complex is not just a bookkeeping integer — it is the H^k tier, which has a direct physical meaning (computational complexity of the corresponding process).

  3. A connection between knot invariants and scattering amplitudes. The Grassmannian common parent (Paper 574) suggests that the HOMFLY polynomial and scattering amplitudes in gauge theory share a Grassmannian Schubert cell decomposition — a precise statement that is open as a conjecture.

  4. Proton stability as a topological theorem. The colour-singlet condition for a baryon is a closed Fano orbit (Paper 545); baryon number conservation is the winding number of this orbit; proton stability below β*_QCD is a topological theorem, not an accidental symmetry.

Papers

Core topology papers

# Title DOI Notes
572 The Kuperberg G₂ Spider is the BIND Calculus 21278538 BIND theorem; spider = opcode calculus · Explainer
571 The ISA Chain Complex: Khovanov Homology from Opcode Projections 21278536 ∂²=0 proved; Khovanov recovery
574 The Grassmannian as Common Parent of Bonding and Scattering 21279006 Gr(k,n) unifies amplituhedron and chemistry · Explainer
595 Weyl Chamber Homology 21345107 Bredon cohomology; tier obstruction · Explainer
568 Schrödinger’s Equation on the Grassmannian 21277819 Correct Schmidt decomposition; Gr(k,n) frame

Fano plane and magic

# Title DOI Notes
361 Fano Orbit Decomposition of Magic 20541583 Wigner negativity from Fano orbit · Explainer
366 A Valence Theory of Quantum Magic 20541665 T-count lower bound from Fano structure · Explainer
408 The Fano Plane is the Right Way to Think About Qubits 20667176 Accessible entry point · Explainer
545 Topological Protection of the Proton 21515760 Baryon number = Fano winding number

ISA foundations (for context)

# Title DOI Notes
469 ISA Completeness: Nine Normal Forms 21219699 Completeness theorem; 9 opcode classes
607 Diagrammatic QEC as ISA 21372998 ZX-calculus ↔ ISA; footprints and fibres
468 Resource Theory as Circuit Syntax 20955514 ISA as symmetric monoidal category · Explainer

What to read first

If you work on spider calculi / planar algebras: Start with The Kuperberg G₂ Spider is the BIND Calculus. It shows exactly how the G₂ spider relations map to BIND opcode identities, with explicit diagrammatic proofs. The ISA formalism is introduced only as needed.

If you work on Khovanov homology / link homology: Start with The ISA Chain Complex. The ∂² = 0 proof via Fano incidence is in Section 2; the recovery of Khovanov homology over GF(2) is in Section 4.

If you work on the amplituhedron / positive Grassmannian: Start with The Grassmannian as Common Parent. It identifies the Schubert cell decomposition with the ISA tier structure and states the open conjecture connecting HOMFLY to scattering amplitudes.

If you want the one-page entry point: Read The Fano Plane is the Right Way to Think About Qubits first. It needs no ISA background and makes the Fano geometry concrete in 8 pages.

See also: Non-Associative Frontier — G₂ geometry and the octonions · PT Symmetry & Exceptional Points — a parallel entry-point page written for non-Hermitian physicists · Opcodes reference — canonical definitions of all 8 ISA opcodes