Slides Morton

KnotsBraids

Invariants from braids

Knots, braids and invariants

H.R.Morton

University of Liverpool

Barcelona October 2010

H.R.Morton Knots, braids and invariants

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Knot theory spans some 150 years, going back to Maxwell, andeven Gauss.


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The most basic idea: Take a piece of rope and tie a knot in it.


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A trefoil (overhand) knot,


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A trefoil (overhand) knot,

A figure eight knot.


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Join the two ends of the rope to make a closed curve.


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Trefoil


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Trefoil

Figure eight


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Trefoil

Figure eight

Unknotted (trivial) curve.


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A knot K means a simple closed curve in Euclidean space R3.


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Think of two knots as essentially the same if we can manipulateone as a piece of elastic rope to become the other.


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A classic combinatorial result of Reidemeister shows that allmanipulations can be realised by combinations of just 3 basicmoves on diagrams:


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A classic combinatorial result of Reidemeister shows that allmanipulations can be realised by combinations of just 3 basicmoves on diagrams:

RI : ↔ , RII : ↔ , RIII : ↔


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We can similarly use several closed curves - termed a link - such as

the Hopf link


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the Hopf link

the Borromean rings.


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the Hopf link

the Borromean rings.

While these can’t be separated, if any one ring is removed theresult is


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Basic problem.Given knots K ,K ′ decide if it is possible to manipulate K to K ′.


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An invariant of K is some algebraic object I (K ) (maybe a numberor function) which, typically, can be calculated from a diagram ofK , in such a way that different pictures of K give the same objectI (K ).


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Compare K ,K ′ by calculating I (K ) and I (K ′).


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Compare K ,K ′ by calculating I (K ) and I (K ′).

If I (K ′) 6= I (K ) then K ,K ′ really are different knots.


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One powerful invariant is the fundamental group

π1(R3 − K ) = GK .


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π1(R3 − K ) = GK .

Equivalent knots have isomorphic groups.


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π1(R3 − K ) = GK .

Equivalent knots have isomorphic groups.So

GK 6∼= GK ′ =⇒ K 6= K ′.


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π1(R3 − K ) = GK .


GK 6∼= GK ′ =⇒ K 6= K ′.

If K has the same group as the trivial knot then K itself is trivial.


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π1(R3 − K ) = GK .


GK 6∼= GK ′ =⇒ K 6= K ′.


GK∼= GK ′ + conditions on special subgroup =⇒ K = K ′.


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π1(R3 − K ) = GK .


GK 6∼= GK ′ =⇒ K 6= K ′.



It is easy to find a presentation for GK from any diagram of K .


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π1(R3 − K ) = GK .


GK 6∼= GK ′ =⇒ K 6= K ′.



It is easy to find a presentation for GK from any diagram of K .The problem with this invariant comes in deciding when the twogroups are isomorphic.


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The Alexander polynomial ∆K (t) ∈ Z[t±1] is a classical invariant(1920) which can be derived from the knot or directly from GK .


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It is easy to check when two Laurent polynomials are equal.

Trefoil ∆ = t−1 − 1 + t

Figure eight ∆ = −t−1 + 3 − t

Trivial ∆ = 1


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Trefoil ∆ = t−1 − 1 + t


Trivial ∆ = 1

But there are lots of knots with the same ∆ and even with trivial∆.


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Trefoil ∆ = t−1 − 1 + t


Trivial ∆ = 1

But there are lots of knots with the same ∆ and even with trivial∆.Hence there is quite limited discrimination.


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There has been a long involvement of algebra in geometry andtopology.


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To show that two things are similar you need geometry.


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To show that two things are similar you need geometry.To show they are different you need algebra.


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This is generally the case for knots.


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This is generally the case for knots.In rare instances algebra is enough to establish similarity.


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Knot groups are residually finite. (Any non-trivial element can bedetected by a homomorphism to some finite group.)


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There is a key element (the longitude of K ) in GK which is e ifand only if K is trivial.


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A non-trivial knot K then always has a homomorphism from GK tosome permutation group Sn with non-abelian image.


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Analysing representations of GK to quite small permutation groupsgives a surprising amount of information.


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Analysing representations of GK to quite small permutation groupsgives a surprising amount of information.

Thistlethwaite used this technique systematically to ensure thatthere were no duplicates in his extensive tables of knots with up to16 crossings.


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Knots from braidsFinding braid presentations for knots

Braids provide an important interface between knots and algebra,dating back to work of Artin in 1925.


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From the geometric viewpoint an n-string braid is a union of n

disjoint strings running monotonically from one plane to anotherparallel plane.


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Braids are studied up to ambient isotopy keeping the ends fixed-strings may move around between the planes, but may not passthrough each other.


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With a fixed choice of endpoints on the plane they can becomposed by placing one above another.


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They form a group Bn, generated by the elementary braids σi witha single crossing between strings i and i + 1


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σi =

i i + 1


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σi =

i i + 1

Artin gave the classic presentation for Bn with these generatorsand the relations


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σi =

i i + 1

Artin gave the classic presentation for Bn with these generatorsand the relations

σiσj = σjσi , |i − j | > 1,

σiσi+1σi = σi+1σiσi+1,


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The braid shown earlier can be written in Artin’s notation as

= σ2σ−13 σ2

4σ−12 σ3σ

−11 σ−1

5 σ2σ4


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1 Knots

2 BraidsKnots from braidsFinding braid presentations for knots

3 Invariants from braidsThe coloured Burau representation


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A braid can be used to determine a knot in two classical ways.

One is by making a plat.


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Take a braid with 2k strings and join up the strings with k localmaxima at the top and k local minima at the bottom.


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The second way is to make a closed braid.


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Join the top points to the corresponding bottom points withoutfurther crossings.


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Both of these methods come with theorems to say that every knothas such a presentation,


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Both of these methods come with theorems to say that every knothas such a presentation, along with simple moves on braids relatingany two braids which result in the same knot.


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Algebraic constructions starting from a braid which are invariantunder these moves then depend only on the resulting knot.


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Algebraic constructions starting from a braid which are invariantunder these moves then depend only on the resulting knot.

These give a useful source of knot invariants. The Jonespolynomial was originally defined in this way using closed braidpresentations.


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Braids give a convenient way of presenting the group of a knot,and of handling its representations to other groups.


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Braids give a convenient way of presenting the group of a knot,and of handling its representations to other groups.

At the heart of this is the action of the braid group Bn on the freegroup Fn of rank n.


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Look at a braid β from the front.


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Look at a braid β from the front.With the eye as base point we can draw loops once round eachstring at the bottom of the braid, representing elements x1, . . . , xn.


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x1 xn


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x1 xn

These generate the fundamental group of the complement of thebraid strings in R2 × I , which is free of rank n.


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The elements y1, . . . , yn represented by loops round the strings atthe top of the braid are determined by β in terms of x1, . . . , xn.

x1 xn

y1

yn


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x1 xn

y1

yn

Write (y1, . . . , yn) = β(x1, . . . , xn).


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x1 xn

y1

yn

Write (y1, . . . , yn) = β(x1, . . . , xn).The resulting automorphism β of the free group can be foundreadily by composing the automorphisms for the elementary braidsσ±1

i which make up the braid β.


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In this way the braid group can be viewed as an explicit subgroupof the automorphism group of the free group Fn.


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The essential part of the automorphism is the action of theelementary braid σi , determining σi(x1, . . . , xn) = (y1, . . . , yn).


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This is given by yi+1 = xi , yi = x−1i xi+1xi , with yj = xj otherwise.


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The effect at each crossing comes from the simple Wirtingerrelations shown.

a b

a−1ba a


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The effect at each crossing comes from the simple Wirtingerrelations shown.

a b

a−1ba a

a b

b bab−1


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The presentation for the group of the closure of a braid β is

< x1, . . . , xn | β(x1, . . . , xn) = (x1, . . . , xn) >


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< x1, . . . , xn | β(x1, . . . , xn) = (x1, . . . , xn) >

For the plat closure of an n = 2k braid we have

< x1, . . . , xn | β(x1, . . . , xn) = (y1, . . . , yn),


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< x1, . . . , xn | β(x1, . . . , xn) = (x1, . . . , xn) >


< x1, . . . , xn | β(x1, . . . , xn) = (y1, . . . , yn),

x1x2 = x3x4 = . . . x2k−1x2k = e,


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< x1, . . . , xn | β(x1, . . . , xn) = (x1, . . . , xn) >


< x1, . . . , xn | β(x1, . . . , xn) = (y1, . . . , yn),

x1x2 = x3x4 = . . . x2k−1x2k = e,

y1y2 = . . . y2k−1y2k = e >


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It is then possible to test out representations of the knot group toa finite group G by going through all sequences of elements(g1, . . . , gn)


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It is then possible to test out representations of the knot group toa finite group G by going through all sequences of elements(g1, . . . , gn) and counting (for a closed braid) those for whichβ(g1, . . . , gn) = (g1, . . . , gn).


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It is then possible to test out representations of the knot group toa finite group G by going through all sequences of elements(g1, . . . , gn) and counting (for a closed braid) those for whichβ(g1, . . . , gn) = (g1, . . . , gn).

The count can be refined by fixing the conjugacy class of all theelements gi , and by fixing the subgroup of G that they generate.


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A similar count can be made for a plat.


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A similar count can be made for a plat.

For example, the figure eight knot is the plat closure of the 4-braid

σ22σ

−11 σ2 =


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A non-trivial representation of its group using products of twodisjoint 2-cycles in S5 is given by the starting sequence

(g1, g2, g3, g4) = ((25)(34), (25)(34), (13)(45), (13)(45))


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(25)(34) (25)(34) (13)(45) (13)(45)


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(25)(34) (25)(34) (13)(45) (13)(45)

(13)(45)

(14)(23)


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(25)(34) (25)(34) (13)(45) (13)(45)

(13)(45)(14)(23)

(12)(35)


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(25)(34) (25)(34) (13)(45) (13)(45)

(13)(45)(14)(23)

(12)(35)(13)(45)


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(25)(34) (25)(34) (13)(45) (13)(45)

(13)(45)(14)(23)

(12)(35)

(13)(45)(14)(23)


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(25)(34) (25)(34) (13)(45) (13)(45)

(13)(45)(14)(23) (13)(45)(14)(23)

The final sequence isβ(g1, g2, g3, g4) = ((14)(23), (14)(23), (13)(45), (13)(45)), whichsatisfies the relations.


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(25)(34) (25)(34) (13)(45) (13)(45)

(13)(45)(14)(23) (13)(45)(14)(23)

The final sequence isβ(g1, g2, g3, g4) = ((14)(23), (14)(23), (13)(45), (13)(45)), whichsatisfies the relations.The image of the knot group in this case is the group D(5) ofsymmetries of a pentagon.


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Taking G = S3 and using transpositions counts the classical3-colourings of a knot.


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Taking G = S3 and using transpositions counts the classical3-colourings of a knot. A quick check shows that the figure eightknot only admits trivial 3-colourings - those with the sametransposition in G used throughout.


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1 Knots




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It is very easy to find a plat presentation from a diagram of a knotK .


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It is very easy to find a plat presentation from a diagram of a knotK .Choose a direction in the plane.

↑


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It is very easy to find a plat presentation from a diagram of a knotK .Choose a direction in the plane.

↑

Move the local maxima and minima of the diagram to take all thelocal maxima to the top and all the local minima to the bottom.


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↑


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↑


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↑

The result is a plat presentation of K .


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In a closed braid diagram the curves run monotonically around apoint.


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In a closed braid diagram the curves run monotonically around apoint.


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In R3 this corresponds to an axis around which the curves run.


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Compactifying to work in S3 the axis becomes an unknotted curveA ⊂ S3 −K with K running monotonically around the complementS3 − A ∼= S1 × R2.


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A


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A

The closure of a braid determines a link K ∪ A made up of theclosed braid K and axis A.


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Conjugate braids determine isotopic links.


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A

α

β = A

β

α


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A

α

β = A

β

α

Conversely, any two braids determining isotopic links K ∪ A areconjugate.


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A

α

β = A

β

α

Conversely, any two braids determining isotopic links K ∪ A areconjugate.So closed braid + axis is a nice geometric counterpart of aconjugacy class in Bn.


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A closed braid presentation for K corresponds to finding a suitableaxis A ⊂ S3 − K .


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The essential part is to ensure that K runs monotonically around A.


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The most efficient method in terms of preserving features of thediagram of K is due to Yamada.


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A more indirect approach of mine gives a quick existence proof,


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A more indirect approach of mine gives a quick existence proof,along with the extra moves needed to pass between any two closedbraid presentations of the same knot.


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A more indirect approach of mine gives a quick existence proof,along with the extra moves needed to pass between any two closedbraid presentations of the same knot.

See Threading knot diagrams 1986


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Invariants from braidsThe coloured Burau representation

As demonstrated earlier, it is quite easy to use braids to deal withrepresentations of the group of a knot K , via plats or closed braids.


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Closed braids have also been used considerably in defining andcalculating polynomial invariants of K .


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There are nice recursive methods for finding the Jones polynomial,and its extension the Homfly polynomial, given a closed braidpresentation of K .


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There are nice recursive methods for finding the Jones polynomial,and its extension the Homfly polynomial, given a closed braidpresentation of K .

In this last case the braid is expressed as an element of thefinite-dimensional Hecke algebra, before the effect of closing it istaken into account.


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The Homfly polynomial is also an extension of the classicalAlexander polynomial.


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The Burau representation of the braid group gives an establishedway to calculate the Alexander polynomial of a closed braid.


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A less known variant of the Burau representation can be used todeal neatly with the multivariable Alexander polynomial.


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A less known variant of the Burau representation can be used todeal neatly with the multivariable Alexander polynomial. Thisworks most naturally for the link consisting of the closed braid andits axis.


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1 Knots




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Label the individual strings of β ∈ Bn by t1, . . . , tn, putting thelabel tj on the string which starts from the point j at the bottom.


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Label the individual strings of β ∈ Bn by t1, . . . , tn, putting thelabel tj on the string which starts from the point j at the bottom.

For example when β = σ1σ−12 σ1σ

−12 σ1σ

−12 σ3 we have

t1 t2 t3 t4

t4

t2

t1

t2

t1

t4 t3


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Write Ci(a) for the (n − 1) × (n − 1) matrix which differs from theunit matrix only in the three places shown on row i , for1 ≤ i ≤ n − 1.

Ci (a) =

1. . .

1a −a 1

1. . .

1

.


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Write Ci(a) for the (n − 1) × (n − 1) matrix which differs from theunit matrix only in the three places shown on row i , for1 ≤ i ≤ n − 1.

Ci (a) =

1. . .

1a −a 1

1. . .

1

.

When i = 1 or i = n − 1 the matrix is truncated appropriately togive two non-zero entries in row i .


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Now construct the coloured reduced Burau matrix Bβ(t1, . . . , tn)of the general braid

β =

l∏

r=1

σεr

ir

as a product of matrices Ci (a), in which a is the label of thecurrent undercrossing string.


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Now construct the coloured reduced Burau matrix Bβ(t1, . . . , tn)of the general braid

β =

l∏

r=1

σεr

ir

as a product of matrices Ci (a), in which a is the label of thecurrent undercrossing string.

Thus

Bβ(t1, . . . , tn) =

l∏

r=1

(Cir (ar ))εr ,

where ar is the label of the undercrossing string at crossing r ,counted from the top of the braid.


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In the example shown, where β = σ1σ−12 σ1σ

−12 σ1σ

−12 σ3, the labels

a1, . . . , a7 are t1, t4, t2, t1, t4, t2, t4 respectively and Bβ is the3 × 3 matrix product

C1(t1)C2(t4)−1C1(t2)C2(t1)

−1C1(t4)C2(t2)−1C3(t4).


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The braid β determines a permutation π ∈ Sn where the stringsrun from position j at the bottom to position π(j) at the top. Inour example above π(1) = 1, π(2) = 2, π(3) = 4, π(4) = 3.


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The braid β determines a permutation π ∈ Sn where the stringsrun from position j at the bottom to position π(j) at the top. Inour example above π(1) = 1, π(2) = 2, π(3) = 4, π(4) = 3.

Theorem The multivariable Alexander polynomial ∆β̂∪A

, where A

is the axis of the closed n-braid β̂, is given by the characteristicpolynomial det(I − xBβ(t1, . . . , tn)) with the identifications ofvariables tπ(j) = tj .


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The polynomial for the closed braid itself, without the axis, can befound from the Torres-Fox formula.


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In this case we must set x = 1, to suppress the axis, and divide by1 − t1t2 . . . tn.


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A simple Maple procedure multiburau.maple implementing thiscan be found on the Liverpool knot theory website.


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An older program for calculating the Homfly polynomial of a closedbraid can also be found there, as well as details of other papers.


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An older program for calculating the Homfly polynomial of a closedbraid can also be found there, as well as details of other papers.

(Google Hugh Morton to find them)


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