Fadno.Braids.Internal

Generators

data Gen a

Braid generator pairing position (absolute or relative) and polarity.

Constructors

Gen

Fields

• _gPos :: a
• _gPol :: Polarity

Instances

Functor Gen

Methods

fmap :: (a -> b) -> Gen a -> Gen b

(<$) :: a -> Gen b -> Gen a

Eq a => Eq (Gen a)

Methods

(==) :: Gen a -> Gen a -> Bool

(/=) :: Gen a -> Gen a -> Bool

Ord a => Ord (Gen a)

Methods

compare :: Gen a -> Gen a -> Ordering

(<) :: Gen a -> Gen a -> Bool

(<=) :: Gen a -> Gen a -> Bool

(>) :: Gen a -> Gen a -> Bool

(>=) :: Gen a -> Gen a -> Bool

max :: Gen a -> Gen a -> Gen a

min :: Gen a -> Gen a -> Gen a

Show a => Show (Gen a)

Methods

showsPrec :: Int -> Gen a -> ShowS

show :: Gen a -> String

showList :: [Gen a] -> ShowS

gPos :: forall a a. Lens (Gen a) (Gen a) a a

gPol :: forall a. Lens' (Gen a) Polarity

data Polarity

Braid generator "power", as (i + 1) "over/under" i. O[ver] == power 1 (i + 1 "over" i) U[nder] = power -1 (i + 1 "under" i)

Constructors

U

O

Instances

Enum Polarity

Methods

succ :: Polarity -> Polarity

pred :: Polarity -> Polarity

toEnum :: Int -> Polarity

fromEnum :: Polarity -> Int

enumFrom :: Polarity -> [Polarity]

enumFromThen :: Polarity -> Polarity -> [Polarity]

enumFromTo :: Polarity -> Polarity -> [Polarity]

enumFromThenTo :: Polarity -> Polarity -> Polarity -> [Polarity]

Eq Polarity

Methods

(==) :: Polarity -> Polarity -> Bool

(/=) :: Polarity -> Polarity -> Bool

Ord Polarity

Methods

compare :: Polarity -> Polarity -> Ordering

(<) :: Polarity -> Polarity -> Bool

(<=) :: Polarity -> Polarity -> Bool

(>) :: Polarity -> Polarity -> Bool

(>=) :: Polarity -> Polarity -> Bool

max :: Polarity -> Polarity -> Polarity

min :: Polarity -> Polarity -> Polarity

Show Polarity

Methods

showsPrec :: Int -> Polarity -> ShowS

show :: Polarity -> String

showList :: [Polarity] -> ShowS

ToWeaves [Weave a] a

Methods

toWeaves :: [Weave a] -> [Weave a]

power :: Integral a => Polarity -> a

Polarity to signum or "power" in literature.

complement :: Polarity -> Polarity

Flip polarity.

Representations

class (Integral a, Monoid (br a)) => Braid br a where

Braid representations.

Minimal complete definition

toGens, minIndex, maxIndex, invert

Methods

stepCount :: br a -> Int

Length, number of "steps"columnsartin generators.

strandCount :: br a -> a

N, braid group index, number of strandsrows"i"s.

toGens :: br a -> [[Gen a]]

Common format is br series of "steps" of absolute-indexed generators.

minIndex :: br a -> a

Minimum index (i) value

maxIndex :: br a -> a

Maximum index (i) value. Note this means values of (i+1) obtain, per generators.

invert :: br a -> br a

Invert indices

toArtin :: br a -> Artin a

convert to single-gen

toMultiGen :: br a -> MultiGen a

convert to multi-gen

Instances

Integral a => Braid Artin a

Methods

stepCount :: Artin a -> Int

strandCount :: Artin a -> a

toGens :: Artin a -> [[Gen a]]

minIndex :: Artin a -> a

maxIndex :: Artin a -> a

invert :: Artin a -> Artin a

toArtin :: Artin a -> Artin a

toMultiGen :: Artin a -> MultiGen a

Integral a => Braid MultiGen a

Methods

stepCount :: MultiGen a -> Int

strandCount :: MultiGen a -> a

toGens :: MultiGen a -> [[Gen a]]

minIndex :: MultiGen a -> a

maxIndex :: MultiGen a -> a

invert :: MultiGen a -> MultiGen a

toArtin :: MultiGen a -> Artin a

toMultiGen :: MultiGen a -> MultiGen a

(Integral a, Braid b a) => Braid (DimBraid b) a

Methods

stepCount :: DimBraid b a -> Int

strandCount :: DimBraid b a -> a

toGens :: DimBraid b a -> [[Gen a]]

minIndex :: DimBraid b a -> a

maxIndex :: DimBraid b a -> a

invert :: DimBraid b a -> DimBraid b a

toArtin :: DimBraid b a -> Artin a

toMultiGen :: DimBraid b a -> MultiGen a

newtype Artin a

Braid as "Artin generators" (one-at-a-time).

Constructors

Artin

Fields

• _aGens :: [Gen a]

Instances

Functor Artin

Methods

fmap :: (a -> b) -> Artin a -> Artin b

(<$) :: a -> Artin b -> Artin a

Foldable Artin

Methods

fold :: Monoid m => Artin m -> m

foldMap :: Monoid m => (a -> m) -> Artin a -> m

foldr :: (a -> b -> b) -> b -> Artin a -> b

foldr' :: (a -> b -> b) -> b -> Artin a -> b

foldl :: (b -> a -> b) -> b -> Artin a -> b

foldl' :: (b -> a -> b) -> b -> Artin a -> b

foldr1 :: (a -> a -> a) -> Artin a -> a

foldl1 :: (a -> a -> a) -> Artin a -> a

toList :: Artin a -> [a]

null :: Artin a -> Bool

length :: Artin a -> Int

elem :: Eq a => a -> Artin a -> Bool

maximum :: Ord a => Artin a -> a

minimum :: Ord a => Artin a -> a

sum :: Num a => Artin a -> a

product :: Num a => Artin a -> a

Integral a => Braid Artin a

Methods

stepCount :: Artin a -> Int

strandCount :: Artin a -> a

toGens :: Artin a -> [[Gen a]]

minIndex :: Artin a -> a

maxIndex :: Artin a -> a

invert :: Artin a -> Artin a

toArtin :: Artin a -> Artin a

toMultiGen :: Artin a -> MultiGen a

Eq a => Eq (Artin a)

Methods

(==) :: Artin a -> Artin a -> Bool

(/=) :: Artin a -> Artin a -> Bool

Show a => Show (Artin a)

Methods

showsPrec :: Int -> Artin a -> ShowS

show :: Artin a -> String

showList :: [Artin a] -> ShowS

Monoid (Artin a)

Methods

mempty :: Artin a

mappend :: Artin a -> Artin a -> Artin a

mconcat :: [Artin a] -> Artin a

aGens :: forall a a. Iso (Artin a) (Artin a) [Gen a] [Gen a]

newtype MultiGen a

Steps of many-at-a-time generators.

Constructors

MultiGen

Fields

• _mSteps :: [Step a]

Instances

Integral a => Braid MultiGen a

Methods

stepCount :: MultiGen a -> Int

strandCount :: MultiGen a -> a

toGens :: MultiGen a -> [[Gen a]]

minIndex :: MultiGen a -> a

maxIndex :: MultiGen a -> a

invert :: MultiGen a -> MultiGen a

toArtin :: MultiGen a -> Artin a

toMultiGen :: MultiGen a -> MultiGen a

Eq a => Eq (MultiGen a)

Methods

(==) :: MultiGen a -> MultiGen a -> Bool

(/=) :: MultiGen a -> MultiGen a -> Bool

Show a => Show (MultiGen a)

Methods

showsPrec :: Int -> MultiGen a -> ShowS

show :: MultiGen a -> String

showList :: [MultiGen a] -> ShowS

Monoid (MultiGen a)

Methods

mempty :: MultiGen a

mappend :: MultiGen a -> MultiGen a -> MultiGen a

mconcat :: [MultiGen a] -> MultiGen a

data Step a

Braid "step" of many-at-a-time generators. Absolute-head-offset-tail structure disallows invalid adjacent generators. Example: 'Step (Gen 1 U) [Gen 0 O]' translates to [s1,s3^-1].

Constructors

Empty

Step

Fields

• _sHead :: Gen a

  Absolute-indexed "top" generator

• _sOffsets :: [Gen Natural]

  (offset + 2)-indexed tail generators.

Instances

Eq a => Eq (Step a)

Methods

(==) :: Step a -> Step a -> Bool

(/=) :: Step a -> Step a -> Bool

Show a => Show (Step a)

Methods

showsPrec :: Int -> Step a -> ShowS

show :: Step a -> String

showList :: [Step a] -> ShowS

Integral a => Monoid (Step a)

Methods

mempty :: Step a

mappend :: Step a -> Step a -> Step a

mconcat :: [Step a] -> Step a

Integral a => Ixed (Step a)

Methods

ix :: Index (Step a) -> Traversal' (Step a) (IxValue (Step a))

type IxValue (Step a)

type IxValue (Step a) = Polarity

type Index (Step a)

type Index (Step a) = a

mSteps :: forall a a. Iso (MultiGen a) (MultiGen a) [Step a] [Step a]

insertWithS :: forall a. Integral a => (Polarity -> Polarity -> Polarity) -> Gen a -> Step a -> Step a

Insert a gen at absolute index into a Step. Ignores invalid indices, uses function with new, old value for update.

insertS :: Integral a => Gen a -> Step a -> Step a

Insert a gen at absolute index into a Step. Ignores invalid indices, overwrites on update.

lookupS :: Integral a => a -> Step a -> Maybe Polarity

Lookup by absolute index in a Step.

deleteS :: Integral a => a -> Step a -> Step a

Delete/clear a gen at absolute index.

stepGens :: Integral a => Iso' (Step a) [Gen a]

Iso for valid constructions.

stepToGens :: Integral a => Step a -> [Gen a]

translate Step to absolute-indexed gens.

gensToStep :: Integral a => [Gen a] -> Step a

translate absolute-indexed gens to Step. Drops invalid values.

data DimBraid b a

Braid with explicit dimensions (mainly for empty steps/strands)

Constructors

DimBraid

Fields

• _dBraid :: b a
• _dSteps :: Int
• _dStrands :: a

Instances

(Integral a, Braid b a) => Braid (DimBraid b) a

Methods

stepCount :: DimBraid b a -> Int

strandCount :: DimBraid b a -> a

toGens :: DimBraid b a -> [[Gen a]]

minIndex :: DimBraid b a -> a

maxIndex :: DimBraid b a -> a

invert :: DimBraid b a -> DimBraid b a

toArtin :: DimBraid b a -> Artin a

toMultiGen :: DimBraid b a -> MultiGen a

(Eq a, Eq (b a)) => Eq (DimBraid b a)

Methods

(==) :: DimBraid b a -> DimBraid b a -> Bool

(/=) :: DimBraid b a -> DimBraid b a -> Bool

(Show a, Show (b a)) => Show (DimBraid b a)

Methods

showsPrec :: Int -> DimBraid b a -> ShowS

show :: DimBraid b a -> String

showList :: [DimBraid b a] -> ShowS

(Monoid (b a), Integral a) => Monoid (DimBraid b a)

Methods

mempty :: DimBraid b a

mappend :: DimBraid b a -> DimBraid b a -> DimBraid b a

mconcat :: [DimBraid b a] -> DimBraid b a

dim :: Braid b a => b a -> DimBraid b a

Make DimBraid using braid's dimensions.

dBraid :: forall b a b. Lens (DimBraid b a) (DimBraid b a) (b a) (b a)

dSteps :: forall b a. Lens' (DimBraid b a) Int

dStrands :: forall b a. Lens' (DimBraid b a) a

Strands, loops, weaves

type Weave a = (a, Polarity)

Instruction to send the value "over" or "under" to the next value in a Strand or Loop. Newtyping is undesirable, want to keep Pair instances.

class ToWeaves w a where

Extract a list of weaves.

Minimal complete definition

toWeaves

Methods

toWeaves :: w -> [Weave a]

Instances

ToWeaves [Weave a] a

Methods

toWeaves :: [Weave a] -> [Weave a]

ToWeaves (Strand a) a

Methods

toWeaves :: Strand a -> [Weave a]

ToWeaves (Loop a) a

Methods

toWeaves :: Loop a -> [Weave a]

data Strand a

Concrete braid strand presentation as values delimited by polarities.

Constructors

Strand

Fields

• _sWeaves :: [Weave a]
• _sLast :: a

Instances

Functor Strand

Methods

fmap :: (a -> b) -> Strand a -> Strand b

(<$) :: a -> Strand b -> Strand a

Foldable Strand

Methods

fold :: Monoid m => Strand m -> m

foldMap :: Monoid m => (a -> m) -> Strand a -> m

foldr :: (a -> b -> b) -> b -> Strand a -> b

foldr' :: (a -> b -> b) -> b -> Strand a -> b

foldl :: (b -> a -> b) -> b -> Strand a -> b

foldl' :: (b -> a -> b) -> b -> Strand a -> b

foldr1 :: (a -> a -> a) -> Strand a -> a

foldl1 :: (a -> a -> a) -> Strand a -> a

toList :: Strand a -> [a]

null :: Strand a -> Bool

length :: Strand a -> Int

elem :: Eq a => a -> Strand a -> Bool

maximum :: Ord a => Strand a -> a

minimum :: Ord a => Strand a -> a

sum :: Num a => Strand a -> a

product :: Num a => Strand a -> a

Traversable Strand

Methods

traverse :: Applicative f => (a -> f b) -> Strand a -> f (Strand b)

sequenceA :: Applicative f => Strand (f a) -> f (Strand a)

mapM :: Monad m => (a -> m b) -> Strand a -> m (Strand b)

sequence :: Monad m => Strand (m a) -> m (Strand a)

Eq a => Eq (Strand a)

Methods

(==) :: Strand a -> Strand a -> Bool

(/=) :: Strand a -> Strand a -> Bool

Show a => Show (Strand a)

Methods

showsPrec :: Int -> Strand a -> ShowS

show :: Strand a -> String

showList :: [Strand a] -> ShowS

ToWeaves (Strand a) a

Methods

toWeaves :: Strand a -> [Weave a]

strand :: (Integral a, Braid b a) => a -> b a -> Strand a

Extract a single strand from a braid.

strand' :: Integral a => a -> [[Gen a]] -> Strand a

Strand from gen matrix.

strands :: (Integral a, Braid b a) => b a -> [Strand a]

Extract all strands from a braid.

sWeaves :: forall a. Lens' (Strand a) [Weave a]

sLast :: forall a. Lens' (Strand a) a

newtype Loop a

Capture strands into a loop, where _sLast of one strand is the first value of the next. Foldable instance ignores "last" values of strands (since they will equal the next head).

Constructors

Loop

Fields

• _lStrands :: [Strand a]

Instances

Functor Loop

Methods

fmap :: (a -> b) -> Loop a -> Loop b

(<$) :: a -> Loop b -> Loop a

Foldable Loop

Methods

fold :: Monoid m => Loop m -> m

foldMap :: Monoid m => (a -> m) -> Loop a -> m

foldr :: (a -> b -> b) -> b -> Loop a -> b

foldr' :: (a -> b -> b) -> b -> Loop a -> b

foldl :: (b -> a -> b) -> b -> Loop a -> b

foldl' :: (b -> a -> b) -> b -> Loop a -> b

foldr1 :: (a -> a -> a) -> Loop a -> a

foldl1 :: (a -> a -> a) -> Loop a -> a

toList :: Loop a -> [a]

null :: Loop a -> Bool

length :: Loop a -> Int

elem :: Eq a => a -> Loop a -> Bool

maximum :: Ord a => Loop a -> a

minimum :: Ord a => Loop a -> a

sum :: Num a => Loop a -> a

product :: Num a => Loop a -> a

Eq a => Eq (Loop a)

Methods

(==) :: Loop a -> Loop a -> Bool

(/=) :: Loop a -> Loop a -> Bool

Show a => Show (Loop a)

Methods

showsPrec :: Int -> Loop a -> ShowS

show :: Loop a -> String

showList :: [Loop a] -> ShowS

Monoid (Loop a)

Methods

mempty :: Loop a

mappend :: Loop a -> Loop a -> Loop a

mconcat :: [Loop a] -> Loop a

ToWeaves (Loop a) a

Methods

toWeaves :: Loop a -> [Weave a]

toLoops :: (Eq a, Show a) => [Strand a] -> [Loop a]

Find loops in strands.

lStrands :: forall a a. Iso (Loop a) (Loop a) [Strand a] [Strand a]

Moves/isotopy

data Move b i

A la Reidemeister.

Constructors

Move (b i) (b i)

Instances

Eq (b i) => Eq (Move b i)

Methods

(==) :: Move b i -> Move b i -> Bool

(/=) :: Move b i -> Move b i -> Bool

Show (b i) => Show (Move b i)

Methods

showsPrec :: Int -> Move b i -> ShowS

show :: Move b i -> String

showList :: [Move b i] -> ShowS

Field2 (Move b i) (Move b i) (b i) (b i)

Methods

_2 :: Lens (Move b i) (Move b i) (b i) (b i)

Field1 (Move b i) (Move b i) (b i) (b i)

Methods

_1 :: Lens (Move b i) (Move b i) (b i) (b i)

inverse :: Move b i -> Move b i

Flip a move

moveH :: Braid a i => Move a i -> i

Move "height" or strand count

moveW :: Braid a i => Move a i -> Int

Move "width" or step count

data Loc a

Coordinate in braid.

Constructors

Loc

Fields

• _lx :: Int
• _ly :: a

Instances

Eq a => Eq (Loc a)

Methods

(==) :: Loc a -> Loc a -> Bool

(/=) :: Loc a -> Loc a -> Bool

Ord a => Ord (Loc a)

Methods

compare :: Loc a -> Loc a -> Ordering

(<) :: Loc a -> Loc a -> Bool

(<=) :: Loc a -> Loc a -> Bool

(>) :: Loc a -> Loc a -> Bool

(>=) :: Loc a -> Loc a -> Bool

max :: Loc a -> Loc a -> Loc a

min :: Loc a -> Loc a -> Loc a

Show a => Show (Loc a)

Methods

showsPrec :: Int -> Loc a -> ShowS

show :: Loc a -> String

showList :: [Loc a] -> ShowS

Field2 (Loc a) (Loc a) a a

Methods

_2 :: Lens (Loc a) (Loc a) a a

Field1 (Loc a) (Loc a) Int Int

Methods

_1 :: Lens (Loc a) (Loc a) Int Int

lx :: forall a. Lens' (Loc a) Int

ly :: forall a a. Lens (Loc a) (Loc a) a a