Zoltan Szabo discussed knot Floer homology and Bordered Algebras on Friday. Knot Floer homology is an invariant of a knot in a 3-manifold, which is a bigraded homology theory. The two gradings are known as the Maslov grading and the Alexander grading. The chain complex has essentially the same generators as the chain complex for the ambient 3-manifold, but there is an additional filtration keeping track of the knot data, obtained by adding an additional basepoint to the Heegaard diagram. If is a doubly pointed Heegaard diagram for a 3-manifold Y, then the knot K associated to the diagram is the knot obtained by connecting the two basepoints, z and w, first by an arc in in the complement of the circles and pushing this slightly into the handlebody, and then connecting the basepoints by an arc in in the complement of the circles, and pushing this slightly into the handlebody. An example of a Heegaard diagram for the trefoil knot in is shown in the picture below.
Here are some facts about knot Floer homology:
1. The Euler characteristic of the knot Floer homology for a knot in the 3-sphere is the Alexander polynomial, i.e. if denotes the knot Floer homology in the jth Alexander grading and the ith Maslov grading then the Alexander polynomial is given by .
2. Conway mutation is an operation on knots, that is not detected by the Alexander polynomial, but is detected by the bigraded knot Floer homology (i.e. the Alexander polynomial of two Conway mutants is the same, but there are examples of Conway mutants whose knot Floer homology is not the same). The Kinoshita-Terasaka and Conway mutant knots provide an example.
On the other hand, the total rank of knot Floer homology appears to be unchanged by Conway mutation. Levine and Baldwin conjecture that -graded knot Floer homology (singly graded by the difference of the Alexander and Maslov gradings) is unchanged under Conway mutation. Allison Moore and I gave some evidence recently that the rank of knot Floer homology may also be unchanged by more general genus 2 mutation.
3. Knot Floer homology detects the unknot (no other knot in has the same knot Floer homology). This follows from the theorem that knot Floer homology detects the Seifert genus of a knot.
Another conjecture being considered is that there should be some relationship between Khovanov homology and knot Floer homology. Rasmussen conjectures that there is a spectral sequence starting at reduced Khovanov homology and ending at knot Floer homology.
After discussing these properties, Szabo went on to discuss computability of knot Floer homology. There is a combinatorial definition of knot Floer homology for knots in due to Manolescu, Ozsvath, and Sarkar, involving grid diagrams. The disadvantage of this technique is that there are a huge number of generators of the chain complex, as a tradeoff for making the differential easily computable.
Szabo described a method of generating a Heegaard diagram for a knot, using a knot projection. The generators of the chain complex will be determined by the knot projection, but the differential is not always straightforward to compute. The way to get the Heegaard diagram is to take the boundary of a thickened neighborhood of the graph coming from a marked knot projection, and place curves at each crossing, as shown in the diagram, with an additional circle a meridian near the marked point. The circles surround all but one of the regions in the complement of the graph, where the excluded region is unbounded and touches the marked edge of the knot. See this picture for the trefoil.
It is clear that all the intersections between and curves occur locally near the crossings except for the unique intersection corresponding to the marked point. Therefore generators of the chain complex correspond to certain choices at each crossing. The generators given by all possible choices correspond precisely to Kauffman states. The Alexander and Maslov gradings can also be computed by a sum over local contributions associated to each crossing.
Ozsvath and Szabo used these Kauffman state chain complexes to compute the knot Floer homology of alternating knots. In this case the differential is simple to compute. There are some non-alternating cases which can also be computed by this method, but in general, it is not possible to compute the entire differential.
This provides motivation to create a theory to compute knot Floer homology without introducing a huge excess of generators of the chain complex, but also keeping the differential computable. The idea is to find a theory that allows one to cut up a knot into simpler pieces and glue the pieces together to get the knot Floer homology of the whole knot, as bordered Heegaard Floer homology does for 3-manifolds. To do this, they start with a projection of the knot in bridge form, and cut at a horizontal level. To the piece of the knot above the horizontal level, they associate a D-module (to the bottom presumably they associate an A-module). Generators and relations can be determined based on how strands come in or out of a horizontal level, and there is a complicated algebra involved which can be represented combinatorially by choices of dots between the strands of the knot.
Unfortunately, my notes have degenerated into lots of pictures of lines, arrows, and dots at this point, so we’ll need a bordered Floer expert to fill in the details. The takeaway message should be that this will give a computable, more intuitive cut-and-paste way to determine knot Floer homology, but the cost is some complicated algebraic structure.