Lecture 15 (16.11.2015)

To conclude Chapter 3, we will in this lecture discuss some further polynomial invariants. Recall that the Jones polynomial satisfies (among others) the three properties (Thm. 3.9):

  • Polynomial invariant: $V$ is an invariant of oriented links, taking values in the Laurent polynomials in $t^{1/2}$.
  • Normalisation: The Jones polynomial of the unknot is 1.
  • Skein Relation: There holds
    jones-skein

Theorem 3.9 furthermore states that these three properties uniquely fix the Jones polynomial. That is not hard to believe: If you take any oriented diagram and repeatedly apply the skein relation, you will simplify it (for the term where the crossing is resolved) or relate it to other, usually simpler diagrams (for the term where the crossing is switched, where typically a Reidemeister move II simplifies matters). Working your way up (as far as complexity is concerned) from the unknot, one can well imagine that the three properties above uniquely determine $V$ on all oriented links.

We now take this idea one step further: Instead of the above skein relation, we postulate a different one, for a new polynomial invariant, called $P$. We demand that $P$ is normalised in the sense that it returns 1 on the unknot, and satisfies a modified skein relation of the form

HOMFLY-skein

Here $l,m$ are the variables of $P$, which is thought of as a Laurent polynomial in these two variables (similar to how the bracket polynomial depends on the three variables $A,B,d$.)

Note that we did not give an explicit definition of $P$ — for now, we just imagine there would be a polynomial invariant of oriented links satisfying the above skein relation, and the normalisation $P_{\cal U}(l,m)=1$. Would that be sufficient information to calculate $P_{\cal L}$ for any oriented link ${\cal L}$?

For example, in the context of our other polynomial invariants, we also discussed a relation of the form $P({\cal L}\sqcup{\cal U})=…$, expressing the polynomial of the disconnected union of an arbitrary oriented link $\cal L$ and the unknot $\cal U$. But this information is actually already encoded in the properties of 1) invariance, 2) normalization, 3) skein relation, as we explain now:

Let $\cal L$ be an arbitrary oriented link, with some diagram like
homfly-l1Then, using Reidemeister moves 0 and I (and the inverse of I), we see that also the following three are diagrams of $\cal L$:
homfly-l2The last two of these diagrams appear in the skein relation of $P$ (diagram two has a negative crossing, and diagram three has a positive crossing on the right hand side). If we also add the diagram with that crossing resolved, we are led to consider

homfly-l3

The first two of these three diagrams are diagrams of $\cal L$, and the third one is a diagram of ${\cal L}\sqcup\cal U$. Moreover, they differ only locally, on the right hand side, in precisely the manner described in the skein relation. Hence application of the skein relation gives $$l^{-1}\cdot P_{\cal L}(l,m)+l\cdot P_{\cal L}(l,m)+m\cdot P_{{\cal L}\sqcup \cal U}(l,m)=0,$$ which is equivalent to $$P_{{\cal L}\sqcup\cal U}(l,m)=-\frac{l+l^{-1}}{m}\cdot P_{\cal  L}(l,m)\,.$$ This relation is a consequence of the invariance of $P$ and its skein relation. It follows directly that the $P$-invariant of the $n$-unlink ${\cal U}_n$ must be $$P_{{\cal U}_n}(l,m)=\left(-\frac{l+l^{-1}}{m}\right)^{n-1}\,.$$

Also the values of $P$ on non-trivial links are fixed by the three properties of invariance, normalisation, and the skein relation, as we now demonstrate at the example of a Hopf link

hopf-o1

The upper crossing of this diagram (let’s call it $H_-$) is negative. If we switch the upper crossing, we obtain a new diagram, called $H_+$, which can be pulled apart by a move II, and is thus a diagram of the $2$-unlink, with $P(H_+)=P({\cal U}_2)=-\frac{l+l^{-1}}{m}$ according to what we learned above. Finally, if we resolve the upper crossing, we get a diagram, called $H_0$, which represents the unknot and thus satisifies $P(H_0)=1$. If we insert these two polynomials into $$l^{-1}\cdot P_{H_-}+l\cdot P_{H_+}+m\cdot P_{H_0}=0,$$ we get $$P_{H_-}(l,m)=-l\cdot m+\frac{l^3}{m}+\frac{l}{m}\,.$$

As you might imagine, proceeding in this manner by applying the skein relation (maybe several times) and invariance, one can compute the $P$-invariant of any oriented link.

What is less clear is that this procedure does not depend on where in our diagrams we apply the skein relation. For example, in the above diagram of the Hopf link, we could have applied the skein relation to the lower instead of the upper crossing. Would we have arrived at the same result for $P$?

In this example, you can easily check that yes, we would have ended up with the same polynomial. This is in fact always the case. We write down these results in a theorem.

Theorem 3.12: There exists a unique map $P$ from the set of all oriented links to Laurent polynomials in two variables $l,m$, such that

  • $P$ is an invariant under ambient isotopy.
  • $P$ is normalised, i.e. $P_{\cal U}=1$ for the unknot $\cal U$.
  • $P$ satisfies the HOMFLY skein relation
    HOMFLY-skeinfor all oriented links $\cal L$.

This polynomial is called the HOMFLY polynomial after its discoverers Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, who found it in 1985.

We do not give a proof of Theorem 3.12 here. The main problem in the proof is to demonstrate the existence of $P$, i.e. the fact that applying the skein relation to different crossings does not result in different answers.

 

We next discuss the relation between the HOMFLY and the Jones polynomials. In this context, we use the imaginary unit $i\in\mathbb C$, and recall that $1/i=-i$. If we replace the variables $l,m$ of the HOMFLY polynomial by $$l:=\frac{i}{t}\,,\qquad m:=i(t^{-1/2}-t^{1/2}),$$ then one can easily check that the HOMFLY skein relation turns into the Jones skein relation. That is, the polynomial $W_{\cal L }(t):=P_{\cal L }(\frac{i}{t},i(t^{-1/2}-t^{1/2}))$ satisfies the skein relation of the Jones polynomial. Furthermore, $W$ is an invariant because $P$ is, and $W_{\cal U}=1$ because also $P$ is normalised. Hence, by applying the uniqueness statement of Theorem 3.9, we conclude that $W=V$ must coincide with the Jones polynomial. We have therefore proven the following:

Proposition 3.13: Let $\cal L$ be an oriented link. Then $$P_{\cal L}(\frac{i}{t},i(t^{-1/2}-t^{1/2}))=V_{\cal L}(t)\,.$$

By this choice of variables, we can recover the Jones polynomial from the HOMFLY polynomial. In particular, the HOMFLY polynomial is a stronger invariant than the Jones polynomial. It is, however, again not complete.

We do not discuss further properties of the HOMFLY polynomial here, but rather introduce yet another — our last — polynomial invariant: The Conway polynomial.

Definition 3.14: The Conway polynomial $\nabla_{\cal L}(z)$ of an oriented link $\cal L$ is defined by setting $$l=i\,,\qquad m=-i\,z$$ in the HOMFLY polynomial, i.e. $$\nabla_{\cal  L}(z):=P_{\cal L}(i,-i\,z)\,.$$

It is clear from this definition that

  • $\nabla$ is an invariant of oriented links
  • $\nabla$ is normalised, i.e. $\nabla_{\cal U}(z)=1$ for the unknot $\cal U$
  • $\nabla$ satisfies a skein relation.

To calculate the skein relation of the Conway polynomial, one simply has to make the substitutions $l=i$ and $m=-i\,z$ in the skein relation of the HOMFLY polynomial. This gives
conway-skeinwhich is probably the simplest skein relation: A factor $+1$ for the polynomial with the positive crossing, a factor $-1$ for the polynomial with the negative crossing, and a factor $z$ (on the right hand side of the equation) for the polynomial with the resolved crossing.

Again, one can also calculate the effect of adding an unknot as a disconnected component, i.e. ${\cal L}\to{\cal L}\sqcup{\cal U}$. From the corresponding relation for the HOMFLY polynomial, one finds $$\nabla_{{\cal L}\sqcup{\cal U}}=0$$ for any oriented link $\cal L$.

These relations then make it possible to calculate Conway polynomials of arbitrary links.

 

To conclude this chapter, we point out the relation between the Conway and the Alexander polynomial: If one sets $z=t^{-1/2}-t^{1/2}$ in the Conway polynomial, one obtains an Alexander polynomial, i.e. $$\nabla_{\cal L}(t^{-1/2}-t^{1/2})=\Delta_{\cal L}(t).$$ Recall that Alexander polynomials, computed from determinants, are only defined up to a factor $\pm t^m$, $m\in\mathbb Z$.

This relation between the Conway and Alexander polynomials shows that the two ideas underlying their construction — labeling arcs and crossings, and studying their relations via determinants, on the one hand, and breaking a diagram into small pieces, and studying the pattern of the resulting unlinks, on the other hand, are not independent of each other. Problem 12 reviews this relation in examples.