Braids

Longer description to come.

Documentation

data Braid

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

Fields

braidWidth :: Int

braidWord :: [Int]

Instances

Eq Braid

Read Braid

Show Braid

data Node a

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

Constructors

Cross a a a a

Join a a

Instances

Foldable Node

Eq a => Eq (Node a)

Ord a => Ord (Node a)

Read a => Read (Node a)

Show a => Show (Node a)

type Resolution = [Int]

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.

type Component = Set Int

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

data BDiagram

A resolved BDiagram has a Resolution and a set of Components.

Constructors

BDiagram

Fields

resolution' :: Resolution

components' :: Set Component

Instances

Eq BDiagram

Show BDiagram

braidToPD :: Braid -> PD

Turn a Braid into a PD.

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

Turn a crossing into a Node.

resolve :: PD -> Resolution -> PD

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

resolutions :: PD -> [PD]

Compute all resolutions of a PD.

resolutionToComponents :: PD -> Set Component

This is the only use for Data.Graph.

cubeOfResolutions :: PD -> [BDiagram]

Take a PD and returns a list of all the BDiagrams of its Resolutions.

braidCube :: Braid -> [BDiagram]

The total cube of resolutions for a braid.

mirror :: Braid -> Braid

Returns the mirror of b.

allRes :: PD -> [Resolution]

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