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

Braids

Description

Longer description to come.

Synopsis

Documentation

data Braid Source

A `Braid` is a width and word. Integers in the word represent Artin generators and their invereses. E.g. `[1,3,-2]` represents the word sigma_1sigma_3sigma_2^{-1}.

Constructors

 Braid FieldsbraidWidth :: Int braidWord :: [Int]

Instances

 Source Source Source

data Node a Source

A braid can also be written as a collection of `Node`s. See knotatlas for more info on `Cross` and `Join`.

Constructors

 Cross a a a a Join a a

Instances

 Source Eq a => Eq (Node a) Source Ord a => Ord (Node a) Source Read a => Read (Node a) Source Show a => Show (Node a) Source

type Resolution = [Int] Source

A `Resolution` is a collection of integers. These should all be 0 or 1. At some point I will change this to `type Resolution = [Resolution']` and `data Resolution' = One | Zero deriving (Show, Eq, Ord)` or somesuch.

A `Component` is represented by a set of integer labels for its arcs.

data BDiagram Source

A resolved `BDiagram` has a `Resolution` and a set of `Component`s.

Constructors

 BDiagram Fieldsresolution' :: Resolution components' :: Set Component

Instances

 Source Source

braidToPD :: Braid -> PD Source

Turn a `Braid` into a `PD`.

crossingToNodes :: Int -> Int -> Int -> [Node Int] Source

Turn a crossing into a `Node`.

resolve :: PD -> Resolution -> PD Source

Take a `PD` and a `Resolution` and returns the resolved `PD`. Note that the output is always a list of `Join`s.

resolutions :: PD -> [PD] Source

Compute all resolutions of a `PD`.

This is the only use for Data.Graph.

cubeOfResolutions :: PD -> [BDiagram] Source

Take a `PD` and returns a list of all the `BDiagram`s of its `Resolution`s.

braidCube :: Braid -> [BDiagram] Source

The total cube of resolutions for a braid.

Returns the mirror of `b`.

allRes :: PD -> [Resolution] Source

Utility function to return all the resolutions of a planar diagram. This amounts to all sequences of 0s and 1s of some length.