The Khovanov cube of resolutions of a planar code #
The vertices of the cube of resolutions of a PD-code D with n crossings are the subsets
S ⊆ Fin n, where S is the set of crossings given the B-smoothing; the corresponding native
state TauCeti.PDCode.cubeState S is false exactly on S. The edge from S to S ∪ {a}
changes the smoothing at the crossing a. When D is planar, every such edge merges two circles
or splits one, so it carries the saddle map TauCeti.PDCode.resolutionEdgeMap between the
resolved-circle tensors, of quantum degree -1, and every square face of saddle maps commutes.
Shifting the vertex S by its height |S|, the vertex V(S)⟨|S|⟩ in quantum degree j is the
homogeneous piece of degree j - |S| of the circle tensor of S. Along each edge the height
grows by one, which compensates the quantum degree -1 of the saddle map, so for each quantum
degree j the degree-j pieces form a commutative cube of R-modules,
TauCeti.PDCode.resolutionCube. Its signed totalization is the degree-j component of the
unnormalized Khovanov complex, Bar-Natan's bracket ⟦D⟧, whose term in cochain degree r is
⊕[|S| = r] V(S)⟨r⟩. The family of these cochain complexes, indexed by the quantum degree, is
TauCeti.PDCode.khovanovBracket. The normalized Khovanov complex of an oriented code with n₊
positive and n₋ negative crossings is obtained from it by lowering the cochain degree by n₋
and raising the quantum degree by n₊ - 2n₋.
Main definitions #
All in the namespace TauCeti.PDCode:
cubeState S: the state with theB-smoothing exactly at the crossings ofS.resolutionCube D hD R j: the degree-jquantum component of the cube of resolutions of the planar codeD, a functorFinset (Fin n) ⥤ ModuleCat R.resolutionCubeObjCongr: reindexing the circle tensors along a bijection of resolved circles, as a linear equivalence of vertices of cubes of resolutions with pieces of the same degree.khovanovBracket D hD R: the unnormalized Khovanov complex, as a graded object of cochain complexes indexed by the quantum degree.
Main results #
resolutionCube_obj,range_resolutionCubeι: the vertexSin quantum degreejis the degreej - |S|piece of the circle tensor of the resolution ofS, included byresolutionCubeι;resolutionCubeVertexEquivis this identification as a linear equivalence.resolutionCube_map_homOfLE_insert_comp_resolutionCubeι: along the edge fromStoS ∪ {a}the cube acts by the saddle map at the crossinga.
References #
- M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), §3.
- D. Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), §2.2.
The states of the vertices #
A cube vertex S has normalized cochain degree |S| - n₋.
A cube vertex S has quantum shift |S| + n₊ - 2n₋.
The vertex S is recovered from its state.
The vertices of the cube correspond bijectively to the states: the state s is the state of
the vertex formed by its B-smoothings.
The cube of resolutions #
The Khovanov cube of resolutions of a planar code D, in quantum degree j. Its vertex
S is the homogeneous piece of degree j - |S| of the circle tensor of the resolution
cubeState S, the degree-j piece of V(S)⟨|S|⟩, and along the edge from S to S ∪ {a} it
acts by the saddle map at the crossing a, merging two circles or splitting one.
Instances For
The vertex S of the degree-j cube of resolutions is the piece of degree j - |S| of the
circle tensor of the resolution of S.
The inclusion of a vertex of the cube of resolutions is injective.
The image of the vertex S of the degree-j cube of resolutions is the piece of degree
j - |S| of the circle tensor of the resolution of S.
The vertex S of the degree-j cube of resolutions as a linear equivalence with the piece
of degree j - |S| of the circle tensor of the resolution of S. This is the identification
resolutionCube_obj, available where the definition of the cube is not unfolded.
Instances For
Under resolutionCubeVertexEquiv, an element of a vertex lies over its image under the
inclusion resolutionCubeι.
Reindexing the circle tensors along a bijection e of resolved circles, as a linear
equivalence from the vertex S of the degree-j cube of resolutions of D to the vertex S'
of the degree-j' cube of resolutions of D', whose pieces have the same degree
j - |S| = j' - |S'|. It is characterized by
TauCeti.PDCode.resolutionCubeι_resolutionCubeObjCongr.
Instances For
TauCeti.PDCode.resolutionCubeObjCongr reindexes the circle tensor along the bijection of
resolved circles.
Along the edge from S to S ∪ {a}, the cube of resolutions acts by the saddle map
TauCeti.PDCode.resolutionEdgeMap at the crossing a: it merges the two circles through a or
splits the circle through a.
Along the edge from S to S ∪ {a}, the cube of resolutions acts by the saddle map
TauCeti.PDCode.resolutionEdgeMap at the crossing a: it merges the two circles through a or
splits the circle through a.
Along an edge S ⋖ T of the cube, the cube of resolutions acts by the saddle map at the
crossing added along the edge. This restates
resolutionCube_map_homOfLE_insert_comp_resolutionCubeι for a target which is only
propositionally insert a S.
The unnormalized Khovanov complex #
The unnormalized Khovanov complex of a planar code D, Bar-Natan's bracket ⟦D⟧, as a
family of cochain complexes indexed by the quantum degree. Its degree-j component is the signed
totalization of the degree-j cube of resolutions: in cochain degree r it is the biproduct of
the degree-j pieces of V(S)⟨r⟩ over the subsets S with |S| = r, and its differential is
the sum of the saddle maps along the edges from S to S ∪ {a}, with sign (-1) to the number of
crossings of S below a.