Kh

Longer description to come.

Documentation

gToD :: Generator -> BDiagram

Return the diagram underlying a Generator.

homGrading :: Generator -> Int

Compute the homological grading of a Generator.

qGrading :: Generator -> Int

Compute the q-grading of a Generator. In this convention, the Khovanov differential *lowers* the q-grading by 1.

data Triviality

Data type to represent whether a circle is trivial or not. Used to be Bool' but I got confused about which was True and which was False.

Constructors

Trivial

NonTrivial

Instances

Eq Triviality

Ord Triviality

Show Triviality

nonTrivialCircle :: Int -> Component -> Triviality

Determine if a component is non-trivial. To compute the mod 2 winding number about the braid axis, check how many of the 'top arcs' live in a component.

kGrading :: (Resolution, Set Component, Map Component Sign) -> Int -> Int

Compute the k-grading of a Generator.

khovanov :: Int -> BDiagram -> [Generator]

"Apply the filtered Khovanov functor to a diagram." We still need the braid width as input to get the k-grading.

khovanovComplex :: Int -> [BDiagram] -> Map Int (Set Generator)

The next three functions apply the Khovanov functor to the vertices of a cube of resolutions.

data ElMo

An elementary morphism is determined by Components. We distinguish between Merge and Split ElMos

Constructors

Merge (Set Component) (Set Component)

merges the first set to the second

Split (Set Component) (Set Component)

splits the first set into the second

Instances

Show ElMo

whichMorphism :: BDiagram -> BDiagram -> Maybe ElMo

Take two diagrams and returns the ElMos between them, if there is one.

morphismAction :: Maybe ElMo -> Generator -> Generator -> Bool

Take a morphism and two generators and returns True if there should be a 'Morphism from one to the other.

kDrop :: Generator -> Generator -> Int

Compute the difference in k-grading between two Generators.

kgrade' :: AlgGen -> Int

Compute the filtration level of an AlgGen.

kDrop' :: AlgGen -> AlgGen -> Int

filteredMorphismAction :: Int -> Maybe ElMo -> Generator -> Generator -> Bool

Like morphismAction, but only connects two Generators if the drop in k-grading from one to the other is less than or equal to a.

filteredMorphismsFrom :: Int -> Generator -> Set Generator -> Morphisms

Return Morphisms from the Generator into the set with kDrop less than or equal to k.

filteredComplexLevel :: Int -> Map Int (Set Generator) -> Int -> Morphisms

Applies filteredMorphismsFrom to every Generator in a list into the same list.

psi :: Braid -> Generator

Produces the Generator corresponding to the transverse invariant of a braid.

psiCube :: Braid -> [BDiagram]

Produces the portion of the cube of resolutions of a braid which is relevant for computing kappa. This means only using resolutions whose weights are less than or equal to psi's. Note that this only uses the homological grading of psi, so we don't need a separate function for the quotient.