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

KnotsBraids

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.

KnotsBraids

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:

KnotsBraids

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 ′.

KnotsBraids

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 ).

KnotsBraids

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

KnotsBraids

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

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

KnotsBraids

One powerful invariant is the fundamental group

π1(R3 − K ) = GK .

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π1(R3 − K ) = GK .

Equivalent knots have isomorphic groups.

KnotsBraids

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.

KnotsBraids

π1(R3 − K ) = GK .

GK 6∼= GK ′ =⇒ K 6= K ′.

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

KnotsBraids

π1(R3 − K ) = GK .

GK 6∼= GK ′ =⇒ K 6= K ′.

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

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

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

Trefoil ∆ = t−1 − 1 + t

Figure eight ∆ = −t−1 + 3 − t

Trivial ∆ = 1

KnotsBraids

Trefoil ∆ = t−1 − 1 + t

Trivial ∆ = 1

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

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

To show that two things are similar you need geometry.

KnotsBraids

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

KnotsBraids

This is generally the case for knots.

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

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.

KnotsBraids

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

disjoint strings running monotonically from one plane to anotherparallel plane.

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

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i i + 1

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i i + 1

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

KnotsBraids

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,

KnotsBraids

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

= σ2σ−13 σ2

4σ−12 σ3σ

−11 σ−1

5 σ2σ4

KnotsBraids

1 Knots

2 BraidsKnots from braidsFinding braid presentations for knots

3 Invariants from braidsThe coloured Burau representation

KnotsBraids

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

One is by making a plat.

KnotsBraids

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

KnotsBraids

The second way is to make a closed braid.

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

Look at a braid β from the front.

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

a−1ba a

KnotsBraids

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

a−1ba a

b bab−1

KnotsBraids

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

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

KnotsBraids

< 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),

KnotsBraids

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

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

x1x2 = x3x4 = . . . x2k−1x2k = e,

KnotsBraids

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

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

x1x2 = x3x4 = . . . x2k−1x2k = e,

y1y2 = . . . y2k−1y2k = e >

KnotsBraids

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)

KnotsBraids

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).

KnotsBraids

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.

KnotsBraids

A similar count can be made for a plat.

KnotsBraids

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 =

KnotsBraids

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))

KnotsBraids

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

KnotsBraids

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

(13)(45)

(14)(23)

KnotsBraids

(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)

KnotsBraids

(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.

KnotsBraids

(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.

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

1 Knots

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

The result is a plat presentation of K .

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

β = A

KnotsBraids

β = A

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

KnotsBraids

β = A

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

KnotsBraids

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

KnotsBraids

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,

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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.

KnotsBraids

1 Knots

KnotsBraids

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.

KnotsBraids

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

KnotsBraids

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. . .

KnotsBraids

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. . .

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

KnotsBraids

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

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

KnotsBraids

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

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

Bβ(t1, . . . , tn) =

(Cir (ar ))εr ,

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

KnotsBraids

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).

KnotsBraids

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.

KnotsBraids

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 .

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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

KnotsBraids

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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