## Lecture 12 (3.11.2015)

We begin with a number of comments on the Kauffman bracket, derived last time.

• The Kauffman bracket, in contrast to the bracket polynomial, depends only on a single variable, $A$. Because we substituted $B=1/A$ in the bracket polynomial, the Kauffman bracket may contain positive as well as negative powers of $A$. As the Alexander polynomial, it is a Laurent polynomial.
• Because we substituted $d=-A^2-A^{-2}$, the Kauffman bracket may, in constrast to the bracket polynomial, have negative coefficients. It is also no longer true for the Kauffman bracket that the coefficients need to sum up to a power of 2.
• The bracket polynomial of the unknot (in its standard diagram) is 1. Hence also the Kauffman bracket of the unknot (in its standard diagram) is 1.
• We may translate the recursion relation of the bracket polynomial to the Kauffman bracket, simply by replacing $B$ by $A^{-1}$ and $d$ by $-A^2-A^{-2}$. We get
• By construction, the Kauffman bracket is an invariant under regular isotopy.
• Under move I, it behaves as

As an example, we calculate the Kauffman bracket of the trefoil diagram
which we call $3_1$. This is easy because we already know the bracket polynomial $$\langle 3_1\rangle(A,B,d)=A^3d+3A^2B+3AB^2d+B^3d^2.$$ Hence the Kauffman bracket is \begin{align*}\langle 3_1\rangle (A) &= A^3(-A^2-A^{-2})+3A^2A^{-1}+3AA^{-2}(-A^2-A^{-2})+A^{-3}(-A^2-A^{-2})^2\\&=-A^5-A^{-3}+A^{-7}.\end{align*}

We now want to proceed from the Kauffman bracket to a “full” invariant (“full” meaning an invariant for all Reidemeister moves, including move I, that is, an invariant under ambient isotopy). We first recall properties of the writhe (see Def. 1.13 c). Writhe is defined for oriented diagrams, and, similarly to the Kauffman bracket, an invariant under regular isotopy. The behavior of writhe under Reidemeister moves of type I is

Definition 3.4: The normalized bracket polynomial $X_D$ of an oriented diagram $D$ is defined as $$X_D(A):=(-A^3)^{-w(D)}\cdot \langle D\rangle (A),$$ where $\langle D\rangle$ denotes the Kauffman bracket of $D$, and $w(D)$ the writhe of $D$.

In this definition, it is necessary to consider oriented diagrams because of the appearance of writhe, which is only defined for oriented diagrams. The second factor in $X_D$, the Kauffman bracket, does not depend on the orientation. Furthermore, for knots, also the normalized bracket polynomial does not depend on the orientation: That is so because changing the orientation of a knot preserves its writhe. Only if we consider links with several components, where we have an independent choice of orientation on each component, there is a true dependence of $X_D$ on the orientation of $D$.

There is no consistent notation for the normalized bracket polynomial. Often it is written $L$ or $\cal L$ (in honour of Louis Kauffman, I believe), but I will avoid the letter $L$ as I always use it to denote links.

We next show that Def. 3.4 is a good definition – $X_D$ does not depend on the chosen diagram, but is a property of the underlying link itself.

Theorem 3.5: The normalized bracket polynomial is an invariant of oriented links. The normalized bracket polynomial is also an invariant of knots (without orientation).

Proof. Both writhe and the Kauffman bracket are invariant under regular isotopy. Hence also the normalized bracket polynomial is invariant under moves 0,II,III – these moves do change neither $w(D)$ nor $\langle D\rangle$, and therefore also leave $X_D(A)=(-A^3)^{-w(D)}\cdot \langle D\rangle (A)$ invariant.

The interesting part is to investigate what happens under move I. For a “negative kink”, we have to consider
where we dropped the orientation of the partial diagram in the Kauffman bracket, as this does not depend on the orientation. We now use the known behaviour of writhe and Kauffman bracket under move I to compute

If the orientation is reversed, the calculation is completely analogous:
This shows that $X$ is invariant under move I (the invariance under the “positive kink” version of move I is a consequence of the others moves. Of course, one could also check explicitly that these moves do not change $X$.)

Hence $X$ is an invariant for oriented links. The fact that $X$ is also an invariant for knots (without orientation) follows from the fact that both writhe for knots, and the Kauffman bracket in general, do not depend on orientation. ■

As we shall see, the normalized bracket polynomial (which is the same as the Jones polynomial, up to a change of variables – we will discuss this later) is a powerful invariant. As a first example, we compute the normalized bracket polynomial of the trefoil we were considering earlier in this lecture. We have $$w(3_1)=3$$ (independent of orientation) and therefore $$X_{3_1}(A)=(-A^3)^{-3}\cdot(-A^5-A^{-3}+A^{-7})=A^{-4}+A^{-12}-A^{-16}.$$

Now that we know that the normalized bracket polynomial does not depend on the diagram that we use to compute it, but only on the underlying link, we will also write $X_L$ instead of $X_D$ when $D$ is a diagram of $L$.

One of our motivations to search for better invariants than, say, the Alexander polynomial, was the inability of that invariant to distinguish a knot from its mirror image (see problem 7). We now test what the situation looks like with the normalized bracket polynomial.

To study the effect of taking mirror images on the bracket polynomial and its descendants (Kauffman bracket and normalized bracket polynomial), we first observe the following simple fact: When a crossing is flipped (overcrossing to undercrossing and vice versa), the “A” regions turn into “B” regions and vice versa:

Hence the bracket polynomial of the mirror image $D^*$ of a diagram $D$ (i.e. $D^*$ differs from $D$ by flipping all crossings) is related to the bracket polynomial of $D$ by

$$\langle D^*\rangle (A,B,d)=\langle D\rangle (B,A,d).$$

Proceeding to the Kauffman bracket, we substitute $B=A^{-1}$. Hence exchanging $A$ and $B$ now results in exchanging $A$ with $A^{-1}$. Note that $d=-A^2-A^{-2}$ is invariant under the exchange $A\leftrightarrow A^{-1}$. Hence we find $$\langle D^*\rangle(A)=\langle D\rangle(A^{-1}).$$

To derive the effect of mirror images on the normalized bracket polynomial, we recall how writhe behaves under mirror images: $$w(D^*)=-w(D).$$ This was discussed when we introduced writhe, and can easily be seen to be true when considering how signs of crossings do not change when the orientation (of all components of the link) is flipped:

(As an aside: Do not confuse taking mirror images with changing the orientation. While the above picture shows the effect of taking the mirror image on signs of crossings, it looks different when changing the orientation instead:

)

Thus we find \begin{align*}X_{D^*}(A)&=(-A^3)^{-w(D^*)}\cdot\langle D^*\rangle(A)\\&=(-A^3)^{w(D)}\cdot\langle D\rangle(A^{-1})\\&=(-(A^{-1})^3)^{-w(D)}\cdot\langle D\rangle(A^{-1})\\&=X_D(A^{-1}).\end{align*}

We write down what we have just proven as a proposition.

Proposition 3.6: The bracket polynomial, Kauffman bracket, and normalized bracket polynomial behave under taking mirror images as follows:\begin{align*}\langle D^*\rangle (A,B,d)&=\langle D\rangle (B,A,d)\\\langle D^*\rangle(A)&=\langle D\rangle(A^{-1})\\X_{D^*}(A)&=X_D(A^{-1})\end{align*}

The following corollary is immediate.

Corollary 3.7:

• If $L$ is an oriented link such that $X_L(A)\neq X_L(A^{-1})$, then $L$ is chiral (i.e. $L$ is not equivalent to its mirror image $L^*$).
• The trefoil $3_1$ is chiral, i.e. as topological knots, $3_1^*\neq 3_1$; the trefoil exists in two inequivalent “left handed” and “right handed” versions.

Proof. If $L$ is a link such that $X_L(A)\neq X_L(A^{-1})$, then $X_L(A)\neq X_{L^*}(A)$ because $X_{L^*}(A)=X_L(A^{-1})$. Since $X$ is an invariant of oriented links, this implies that $L$ is not equivalent to $L^*$.

For the statement about the trefoil $3_1$, we recall $$X_{3_1}(A)=A^{-4}+A^{-12}-A^{-16}.$$ This polynomial is not the same when we exchange $A$ with $A^{-1}$. Hence the trefoil is chiral. Since the trefoil is a knot, we do not need to choose an orientation for this argument, i.e. the trefoil is inequivalent from its mirror image, independent of orientation.  ■

This shows that the normalized bracket polynomial is better than the Alexander polynomial in detecting mirror images.

Exercise: From Problem 3d) we know that the figure of eight knot $4_1$ is achiral, i.e.

Calculate the normalized bracket polynomial $X_{4_1}$ and check that indeed, $X_{4_1}(A)=X_{4_1}(A^{-1})$.