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Alexander Polynomial Alexander Polynomial One of the earliest and still one of the most useful knot invariants is the Alexander polyno- mial. We will learn how to compute it in three different ways! Let's work with oriented diagrams from now on, because they actually tend to be easier. Plus, it can be an important question whether a knot is the same as its reverse or not! One of the earliest things noticed about knots (and links) is that they can always be made to be the boundary of a surface. More importantly, it can always be made into the boundary of an orientable surface, one whose orientation matches that of the knot or link, in fact. Here's one way to do it, called the Seifert circle method. At each (oriented!) crossing, replace the crossing with a non-crossing, so that the incoming underpass connects with the outgoing overpass, and vice versa. See the picture: Now the diagram has no actual crossings left in it, so consists of a bunch of disjoint loops. Put a 2-cell on each loop. Then attach these together with little half-twisted bands. The resulting surface is orientable because to go from loop to loop never requires an odd number of twisted bands, so you always end up on the same side of the surface. Now this Seifert surface is not a knot invariant, because different drawings of the same knot give different surfaces. However, it is a knot invariant to ask for the lowest possible genus such a surface could have. This is, in general, difficult to compute. But we can do something amazing that does turn out to be a knot invariant. First, now that we have a surface, find its genus (which may or may not be the genus of the knot itself|the surface for a particular variation of the diagram might have a higher genus than that of the knot). If the genus is g then one can find 2g curves on the surface from which any other curve can be made. These should also be oriented. These are called generators for the homotopy of the surface. We're not going to worry about homotopy right now! 1 For each of these curves, in turn, push it slightly off the surface in the direction in which the surface is oriented. This push-off \links" with each of the generators (including itself!). Count the linking number of each, and make a matrix. This is called a Seifert matrix. In the picture above, note that when we push the left curve off the surface it links with itself once, and −1 times with the right curve. When we push the right curve off the surface, it links once with itself, but not at all with the left. So the Seifert matrix we find is 1 −1 . 0 1 Oddly, it turns out many things about this matrix S are invariants of the knot, even though different surfaces or choices of base curves will lead to different matrices. For instance, det(S+ST ) is called the determinant of the knot and is an invariant. A more interesting thing to look at is the polynomial defined by det(S −tST ). This is called the Alexander polynomial of the knot, denoted ∆K (t) and is a powerful invariant. Note that the determinant of the knot is ∆)K (−1). From the above Seifert matrix for the trefoil, we learn that its Alexander polynomial is it t2 − t + 1. Warning: multiplying or dividing by t is considered the same 4 3 2 −1 Alexander polynomial, so the trefoil could also have ∆K (t) = t −t +t or even 1−t +t−2. Oftentimes the power is set so that the powers match for positives and negatives, so we'd −1 have ∆K (t) = t − 1 + t . How can we show that the Alexander polynomial is immune to Reidemeister moves? The twist is pretty easy, since it does nothing to the Seifert matrix. The poke is trickier, depending on whether the two strands are going the same direction or not. If they are not, then the poke does nothing to the matrix. If they are, though, the Seifert matrix must change as below: 0 ∗ 0 1 0 1 B M ∗ 0 C B C @ M A −! B ∗ 0 C B C @ ∗ ∗ ∗ ∗ 0 A 0 0 0 1 0 Computing det(S −tST ) for the new matrix just multilplies the polynomial by t, which as we noted above really doesn't mean anything as far as the Alexander polynomial is concerned. The slide move is a fairly tricky to deal with, because, as we have seen in class, it changes the layout of the Seifert surface. While it can be done this way, it's actually easier to prove that the Alexander polynomial is a knot invariant by other means. 2 When Alexander first figured out this polynomial, he determined it a completely different way. Starting with a knot diagram, he noted there would be n crossings and n + 2 regions (including the outside). The n + 2 is because of Euler!. So he made an n × (n + 2) matrix. The entry in a particular spot is determined by how the crossing and the region interact, as seen by a traveler moving along the strand that will become the undercrossing: • 0 if the crossing is not adjacent to the region • −t if the region is left of the crossing, before the undercrossing • t if the region is left of the crossing, after the undercrossing • 1 if the region is right of the crossing, before the undercrossing • −1 if the region is right of the crossing, after the undercrossing Now this matrix contains redundant information, because you can reconstruct what hap- pens in two (touching) regions by knowing what happens going around them. So delete any two columns corresponding to touching regions. Take the determinant of what's left. And you get the Alexander polynomial again! Showng this requires knowing some algebraic topology, so we won't do it. There's yet another way to figure out the Alexander polynomial! This was figured out by Conway in the 1960's. It uses a skein relation which is a relationship between the polynomials that would exist if a crossing were replaced by other variations. For Conway's polynomial, let's say you have a knot or link, and there is a right-handed crossing, and we call that knot or link L+. Now reverse the crossing to a left-handed crossing, and call the resulting knot or link L−. Finally, call the variation with the uncrossed strands L0. If r(K) is the Conway polynomial of a link, you can define the Conway polynomial by the two relationships r(unknot) = 1 and r(L+) − r(L−) = zr(L0). From these two relationships, you can work out the Conway polynomial of any knot or link. For example, we compute the Conway polynomial of the trefoil to be 1 + z2 as follows. First, apply the skein relation to any of the three crossings, and we find that r(trefoil) = zr(Hopf link) + r(unknot). We can apply the skein relation again to the Hopf link, to obtain rHopflink = zr(unknot) + r(2-component unlink). A sneaky trick involving the unknot drawn as a double-circle with one crossing shows that the 2-component unlink has r = 0. So the Hopf link has r = z and then the trefoil has r = z(z + 0) + 1 = z2 + 1. Well, amazingly, the Alexander polynomial and Conway polynomial are related|for any knot, ∆(t) = r(t1=2 − t−1=2). The really interesting questions start with: why do these totally different-looking opera- tions all produce the same polynomial? Something deep must be going on, and that's what mathematicians really want to understand. 3.
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