The rigidity classes of the skein category

The evaluation and coevaluation classes — the single strand read as a (2,0)- or (0,2)-fragment — together with the braiding class on two strands, and the supersymmetry of the evaluation: precomposing the evaluation with the braiding (or postcomposing the coevaluation) is absorbed, because the strand is symmetric under any boundary relabelling. These are the data that the Deligne fibre functor sends to the standard form and copairing.

noncomputable def RS.strandRelabelEquiv (e : Fin 2 ≃ Fin 2) : (Fragment.strand.relabel e).Equiv Fragment.strand

The strand is invariant under every boundary relabelling: any permutation of Fin 2 commutes with the end swap.

Equations

• RS.strandRelabelEquiv e = { flagEquiv := e, vertexEquiv := Equiv.refl Empty, attach_comm := ⋯, pairing_comm := ⋯, circles_eq := ⋯ }

noncomputable def RS.evFrag : Fragment (Fin (2 + 0))

The evaluation fragment: the strand as a (2,0)-morphism.

Equations

• RS.evFrag = RS.Fragment.strand.relabel (finCongr RS.evFrag._proof_2)

noncomputable def RS.coevFrag : Fragment (Fin (0 + 2))

The coevaluation fragment: the strand as a (0,2)-morphism.

Equations

• RS.coevFrag = RS.Fragment.strand.relabel (finCongr RS.coevFrag._proof_2)

noncomputable def RS.evClass {R : ℕ} (f : EdgeRankParameter R) : HomSpace f.val (2 + 0)

The evaluation class.

Equations

• RS.evClass f = RS.HomSpace.ofFragment f.val RS.evFrag

noncomputable def RS.coevClass {R : ℕ} (f : EdgeRankParameter R) : HomSpace f.val (0 + 2)

The coevaluation class.

Equations

• RS.coevClass f = RS.HomSpace.ofFragment f.val RS.coevFrag

noncomputable def RS.braidClass {R : ℕ} (f : EdgeRankParameter R) : HomSpace f.val (2 + 2)

The braiding class on two strands.

Equations

• RS.braidClass f = RS.HomSpace.ofFragment f.val (RS.permFragment (Equiv.swap 0 1))

theorem RS.braid_comp_evClass {R : ℕ} (f : EdgeRankParameter R) : ((HomSpace.comp f 2 2 0) (braidClass f)) (evClass f) = evClass f

Supersymmetry of the evaluation: the braiding is absorbed by the evaluation class.