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Kauffman brackets and the Jones polynomial

Lecture 8

Daniel Bump

May 28, 2019

a

⟨ ⟩+ a−1

⟨ ⟩=

⟨ ⟩

Kauffman brackets and the Jones polynomial

Ambient isotopy

Two knots, links or tangles are equivalent by ambient isotopy ifthe ambient space R3 or (for tangles) R2 × [0,1] can becontinuously deformed taking the first knot to the second.

Knot invariants, or pre-invariants are described by recipes thatapply to two-dimensional link diagrams that show crossingssuch as:

Reidemeister proved that two such diagrams describe thesame knot (up to ambient isotopy) if and only if they areequivalent by Reidemeister moves.

Kauffman brackets and the Jones polynomial

Reidemeister moves of Type I

If the knots are represented by their two-dimensionalprojections, a necessary and sufficient condition is that theseprojections be related by a sequence of Reidemeister moves.There are three kinds. A Reidemeister move of Type I undoes atwist:

Kauffman brackets and the Jones polynomial

Reidemeister moves of Type II

A Reidemeister move of type II changes:

Kauffman brackets and the Jones polynomial

Reidemeister moves of Type III

A Reidemeister move of type III changes:

Kauffman brackets and the Jones polynomial

Framing of link diagrams

There is a natural framing of a knot diagram: we choose thenormal vector field of the 3-dimensional knot as pointingperpendicular to the plane of the 2-dimensional diagram. Thisis actually not quite the right framing for links: we may want totwist the framing on a component of a link depending on itswinding number with other components.

We will not worry about this point. Instead, we will proceed bydefining polynomials of link diagrams such as the Kauffmanbracket that are unchanged by Reidemeister moves of Types O,II and III, then note that these are invariants of framed links.

Ultimately we wish to modify these invariants to be independentof the framing. This means that we want invariance underType I moves.

Kauffman brackets and the Jones polynomial

Type O moves

For framed links, knots or tangles, which we can visualize asribbons, Type I Reidemeister moves no longer correspond toambient isotopy, because they introduce a twist. Instead wehave the following equivalence:

=

In a rigid braided category, the corresponding equivalencewould bethis would be vv−1 = 1. This type of move substitutesfor Type I Reidemeister moves in the category of framedtangles. We call it Type O.

Kauffman brackets and the Jones polynomial

Reidemeister’s theorem for framed links

PropositionA necessary and sufficient condition for two framed links(ribbons) to be equivalent by ambient isotopy is that they beequivalent by Type O, II and III moves.

Type O moves may be reduced to Type II and III if we adjoin apoint at infinity to the ambient plane to make it a sphere.

Knot =Knot

=Knot

Kauffman brackets and the Jones polynomial

The Kauffman bracket

Let L be a link diagram. The Kauffman bracket 〈L〉 is apolynomial in a variable a. We will denote

d = −a2 − a−2.

If L consists of n unlinked circles, 〈L〉 = dn.More generally, adding an unlinked circle O to a link L multipliesthe Kauffman bracket by d :

〈L ∪O〉 = d〈L〉.

To say the circle O embedded in R3 is unlinked from a link Lmeans that O bounds a disk that is disjoint from L. For the linkdiagram, we ask that the interior of the circle O doesn’t containany part of L.

Kauffman brackets and the Jones polynomial

The Kauffman bracket

Let L be a link diagram with n crossings. Now we assumerecursively that 〈L′〉 is defined for L′ with fewer than n crossings.We select a crossing in the diagram L and let L′ and L′′ be thediagrams replacing the crossing by parallel lines as follows:

L L′ L′′

The figures agree outside the circled portion. We define:

〈L〉 = a〈L′〉+ a−1〈L′′〉.

Kauffman brackets and the Jones polynomial

The Kauffman bracket is an invariant of framed links

The identity〈L〉 = a〈L′〉+ a−1〈L′′〉

is called the skein relation.

We must argue that this definition does not depend on thechoice of crossing. But it is actually clear that the order in whichwe resolve crossings into parallels is unimportant: for if weresolve all or any subset of k crossings, the 2k endconfigurations are the same regardless of what order we do itin, so the Kauffman bracket is well-defined.

TheoremThe Kauffman bracket is unchanged by Reidemeister moves ofTypes O, II or III. Therefore it is an invariant of the framed links.

Kauffman brackets and the Jones polynomial

Adding or removing a curl

First we compute what happens when we remove a curl from alink. Suppose that L and L′ agree outside the circled portion:

L L′

Then〈L〉 = −a3〈L′〉.

To see this, note that by the skein relation 〈L〉 equals

a

⟨ ⟩+ a−1

⟨ ⟩= ad〈L′〉+ a−1〈L′〉.

Since d = −a2 − a−2, the statement follows.

Kauffman brackets and the Jones polynomial

Adding or removing a curl (continued)

Similarly for a negative curl:

L L′

A similar calculation shows

〈L〉 = −a−3〈L′〉.

Kauffman brackets and the Jones polynomial

Type O moves

Now it is clear that the Kauffman bracket is invariant underType O moves, since adding curls of opposite parities multipliesand divides the bracket by −a3:

=

Kauffman brackets and the Jones polynomial

Reidemeister moves of Type II

Let us show that if link diagrams L and L′ agree except on theinterior of the circle, then 〈L〉 = 〈L′〉.

L L′

〈L〉 = a−2〈L1〉+ a2〈L2〉+ 〈L3〉+ 〈L4〉 .

L1 L2 L3 L4

Kauffman brackets and the Jones polynomial

Reidemeister moves of Type II (continued)

Now

a−2〈L1〉+ a2〈L2〉+ 〈L4〉 = (a−2 + a2 + d)

⟨ ⟩= 0.

Therefore〈L〉 = 〈L3〉 = 〈L′〉

proving that the Kauffman bracket is unchanged byReidemeister moves of Type II.

Kauffman brackets and the Jones polynomial

Reidemeister moves of Type III

⟨ ⟩= a

⟨ ⟩+ a−1

⟨ ⟩

Using Reidemeister moves of Type II on the second term thisequals

a

⟨ ⟩+ a−1

⟨ ⟩=

⟨ ⟩

Kauffman brackets and the Jones polynomial

The writhe

The writhe is a very simple invariant of oriented framed links.An orientation of a (smooth) knot or link assigns a unit tangentvector field. A knot has 2 orientations, so if a link has ncomponents it obviously has 2n possible orientations. Weassume an orientation is selected.

Let us consider an oriented link diagram. Every crossing in anoriented link is assigned a sign as follows.

+1 −1

The writhe is the sum of the signs over all crossings.

Kauffman brackets and the Jones polynomial

The wrihe of a knot does not depend on the orientation

The writhe of a knot does not depend on orientation. Indeed, ifwe change orientation, both arrows are switched and the signof a crossing is not changed.

+1 −1

becomes:

+1 −1

(For links, we have to be careful.)

Kauffman brackets and the Jones polynomial

The writhe is an invariant of framed, oriented links

It is clearly invariant under moves of Type O,II and III. We donot draw the orientations.

⇒ ⇒

⇒

Kauffman brackets and the Jones polynomial

The normalized Kauffman bracket

Let w(L) denote the writhe of an oriented link. By combining itwith the Kauffman bracket we can make an invariant ofunframed links. Let

f (L) = (−a)−3w(L)〈L〉.

This is unchanged under Reidemeister moves of Types O, IIand III. What about moves of Type I? If we add a twist:

L L′

〈L〉 = −a3〈L′〉, w(L) = w(L′) + 1, f (L) = f (L′).

Kauffman brackets and the Jones polynomial

The skein relations revisited

We may rewrite the skein relations.⟨ ⟩= a

⟨ ⟩+ a−1

⟨ ⟩

Rotating:⟨ ⟩= a−1

⟨ ⟩+ a

⟨ ⟩

Solving:

a

⟨ ⟩− a−1

⟨ ⟩= (a2 − a−2)

⟨ ⟩

Kauffman brackets and the Jones polynomial

Two kinds of skein relations

So there are two different types of skein relations. We may askfor linear relations between:

L L′ L′′

or (possibly contingent on an assumed orientation)

L+ L− L0

Kauffman brackets and the Jones polynomial

The skein relation for the normalized Kauffman bracket

We have seen that the Kauffman bracket satisfies the skeinrelation

a〈L+〉 − a−1〈L−〉 = (a2 − a−2)〈L0〉

L+ L− L0

Now recall that fL(a) = (−a)−3w(L)〈L〉 is the normalizedKauffman bracket, which is an isotopy invariant. We havew(L+) = w(L0) + 1, w(L−) = w(L0)− 1 and this implies

−a4fL+(a) + a−4fL−(a) = (a2 − a−2)fL0(a).

Kauffman brackets and the Jones polynomial

The Jones polynomial

The normalized Kauffman bracket is essentially the Jonespolynomial. Note that the Kauffman polynomial of a circle isd = −a2 − a−2. We replace a by t−1/4 and divide by d . Thus

VL(t) =1d

fL(t−1/2).

Dividing by d has the effect that the VO = 1 if O is an unknottedcircle. The skein relation becomes

t−1VL+ − tVL− = (t1/2 − t−1/2)VL0 .

It may be shown that VL involves only integer polynomials of t ifL is a knot, though if L is a link, it may involve odd powers oft1/2.

Kauffman brackets and the Jones polynomial

Skein relations and orientation

Skein relations are possible for invariants of oriented knots andlinks. Then we of course require the orientations to beconsistently matched.

L+ L− L0

The dependence is important for links since we could changethe orientation of one line in a crossing but not the other. Thiswould interchange L+ and L−. For knots, the distinction is notso important since both orientations would have to be changed,and L+ is never turned into L− by a change in orientation.

Kauffman brackets and the Jones polynomial

The Jones and Conway-Jones polynomials

In 1923 Alexander discovered a knot invariant known as theAlexander polynomial ∆(L). Conway defined a variant denoted∇(L) and showed that it satisfied a skein relation. If as usual

L+ L− L0

then∇L+(t)−∇L−(t) = t∇L0(t).

The Alexander polynomial (slightly modified from Alexander’snormalization) is then

∆L(t2) = ∇L(t − t−1).

Kauffman brackets and the Jones polynomial

The HOMFLYPT polynomial

The Jones polynomial was discovered in 1984 by VaughnJones through a study of the Temperley-Lieb algebra, analgebra similar to the Hecke algebra that is related to exactlysolvable models in statistical mechanics.

Meanwhile Conway had found the Conway-Alexanderpolynomial which utilized the skein relation in its definition. Inretrospect, the skein relation was known to Alexander, butConway showed its importance.

The HOMFLYPT polynomial was found independently andsimultaneously by 4 different groups: Freyd and Yetter,Ocneanu, Millet and Lickorish and Hoste. All three sent theirmanuscripts to the same journal (BAMS) in October 1984. Theeditors noticed and a joint paper resulted.

Kauffman brackets and the Jones polynomial

The HOMFLYPT polynomial (continued)

The HOMFLY polynomial was also found around the same timeby Przytycki and Traczyk and when this was recognized theirinitials were added to the acronym. With this addition theacronym becomes apt since “flyping,” a procedure usedextensively by the Peter Guthrie Tait (the founder of knottheory) is a generalization of Type III Reidemeister moves andhence is related to the Yang-Baxter equation.

We may define the HOMFLYPT polynomial P(α, z) by PO = 1where and the skein relation

αPL+ − α−1PL− = zPL0 .

The Alexander and Jones polynomials are then

∆L(t) = PL(1, t1/2 − t−1/2),VL(t) = PL(t−1, t1/2 − t−1/2).

Kauffman brackets and the Jones polynomial

Hecke algebras and quantum groups

In Jones’s original work, he made use of the Temperley-Liebalgebra. This is a close relative of the Hecke algebra that wehave already seen related to the theory of quantum groupsthrough Frobenius-Schur-Duality. After the HOMFLY paperJones explained the two-variable representations through theHecke algebra representations in his 1987 Annals paper.

Through Schur-Weyl-Jimbo duality, we understand that Heckealgebra representations are closely connected to quantumgroup representations, and also to solvable lattice models.Roughly we can say that the Jones polynomial is related toUq(sl2) and the Alexander polynomial is related to Uq(gl(1|1)),a quantized enveloping algebra of a Lie superalgebra; or ratheraffinizations of these quantum groups.

Kauffman brackets and the Jones polynomial

Witten’s approach

In his 1989 PJM paper on connections of solvable latticemodels, Jones complained about the lack of a direct3-dimensional connection; instead knots were studied bymeans of 2-dimensional projections. He compared the situationto Plato’s Allegory of the Cave.

A direct 3-dimensional approach was found by Witten (1989)who obtained the Jones polynomial from 3-dimensionalquantum field theories called Chern-Simons theories.Eventually (2016) modified Chern-Simons theories were foundby Costello that connect directly to 2-dimensional solvablelattice models.