-- Copyright (c) David Amos, 2008. All rights reserved. {-# LANGUAGE FlexibleInstances, TypeSynonymInstances #-} module Math.Projects.KnotTheory.Braid where import Data.List ( (\\) ) import Math.Algebra.Field.Base import Math.Algebra.NonCommutative.NCPoly import Math.Projects.KnotTheory.LaurentMPoly type LPQ = LaurentMPoly Q instance Invertible LPQ where inv = recip -- BRAID ALGEBRA data BraidGens = S Int deriving (Eq,Ord) -- Inverse of S n is S (-n) instance Show BraidGens where show (S i) | i > 0 = 's': show i | i < 0 = 's': show (-i) ++ "'" s_ i = NP [(M [S i], 1)] :: NPoly LPQ BraidGens s1 = s_ 1 s2 = s_ 2 s3 = s_ 3 s4 = s_ 4 instance Invertible (NPoly LPQ BraidGens) where inv (NP [(M [S i], 1)]) = s_ (-i) {- braidRelations n = [s_ j * s_ i - s_ i * s_ j | i <- [1..n-1], j <- [i+2..n-1] ] ++ [s_ (i+1) * s_ i * s_ (i+1) - s_ i * s_ (i+1) * s_ i | i <- [1..n-2] ] -- !! need relations for the inverses too !! -- (but we're not intending to work in the braid algebra - we're intending to map into Temperley-Lieb or Iwahori-Hecke) -} -- The writhe of a braid == the sum of the signs of the crossings writhe (NP [(M xs,c)]) = sum [signum i | S i <- xs] -- Some knots - Lickorish p5, p27 -- (Note: These knots/braids give the correct Homfly/Jones polynomials compared to Lickorish) -- (In general, that's not sufficient to prove that they are the claimed knots, although in these cases, they are.) k3_1 = s1^-3 k4_1 = s2^-1 * s1 * s2^-1 * s1 k5_1 = s1^-5 k7_1 = s1^-7