braid-0.1.0.0: Types and functions to work with braids and Khovanov homology.

Kh

Description

Longer description to come.

Synopsis

# Documentation

Return the diagram underlying a `Generator`.

Compute the homological grading of a `Generator`.

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

data Triviality Source

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

 Source Source Source

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.

Compute the k-grading of a `Generator`.

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

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

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

data ElMo Source

An elementary morphism is determined by `Components`. We distinguish between `Merge` and `Split` `ElMo`s

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

 Source

Take two diagrams and returns the `ElMo`s between them, if there is one.

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

Compute the difference in k-grading between two `Generator`s.

Compute the filtration level of an `AlgGen`.

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.

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

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

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

psiCube :: Braid -> [BDiagram] Source

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.