The adjacent braid product contracts to the common evaluation complex #
The volume component of the lowest differential and the vertex component of the next differential are negative identity blocks. Split the self-tensor and reorder the other terms to cancel these blocks, with the corrected projection and inclusion supplied by the four-term contraction criterion.
References #
- R. S. Huerfano and M. Khovanov, A category for the adjoint representation, arXiv:math/0002060, Sections 3 and 4.6, Proposition 7.
- M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions, arXiv:math/0006056, Theorem 2.5 for the type-A relation.
noncomputable def
TauCeti.zigzagBraidAdjacentContraction
(k : Type w)
[CommRing k]
{V : Type u}
(G : SimpleGraph V)
[Finite V]
{i j : V}
(h : G.Adj j i)
:
Contraction (zigzagBraidComplexTriple k G i j i) (zigzagBraidNormalComplex k G i j)
Cancel the volume and repeated vertex blocks in the actual adjacent triple product.
Instances For
noncomputable def
TauCeti.zigzagBraidComplexAdjacentHomotopyEquiv
(k : Type w)
[CommRing k]
{V : Type u}
(G : SimpleGraph V)
[Finite V]
{i j : V}
(h : G.Adj j i)
:
HomotopyEquiv (zigzagBraidComplexTriple k G i j i) (zigzagBraidComplexTriple k G j i j)
The actual graded braid complexes satisfy the adjacent three-generator braid relation.