HEEGAARD FLOER  KNOT HOMOLOGY

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The latest version of our program is available below. Feel free to read our brief article as well. Download:  [hfknew.cpp] Compile:  g++ -o hfk hfknew.cpp Run:  ./hfk

Here is a perfect list of diagrams for 11 crossing non-alternating knots: 11cknots2.cpp. It is possible to show that all of these are minimal. Lenny Ng managed to draw minimal diagrams for a couple of knots where our December 1st diagrams (derived from gridlink) were not perfect.

A version from December 10, 2006: hfkbatch11.cpp. This includes the above list of perfect diagrams for 11-crossing non-alternating knots. These are mostly derived from Marc Culler's "gridlink" program, with a few problematic knots drawn by Lenny Ng. Warning: Some of these diagrams differ by taking the mirror image from the diagrams we used in our original computation and our article (the latter have ancestry tracable to a program of Dror Bar-Natan). This program simply computes the knot Floer homology for all 11-crossing non-alternating knots and outputs the results in latex compatible format.

This is an older version of our program (updated Nov. 26, 2006) which might be more appropriate for the casual user: hfk-mc.cpp. The changes in this version from the previous version (Oct. 6, 2006) are due to Marc Culler. This version is much faster than previous versions, though it differs little in substance. This should work with most any arc-index 13 presentation (if it fails, you might get it to work by exchanging the black and white dots or trying another presentation of the same knot). We ran all our computations on the Columbia Math Department server video.math.columbia.edu). On this computer an arc-index 13 computation takes less than 10 minutes; arc-index 11 computations require just a few seconds. This is an old list of diagrams which isn't so great: diagrams.txt.

TAU

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In principle it is also possible to compute the Ozsv\'ath-Szab\'o $\tau$ invariant algorithmically. Of course, this is trivial to do for knots whose Heegaard Floer homology in Maslov grading $0$ is supported only in one Alexander grading, but we managed to compute $\tau$ in some non-trivial examples. We have an implementation that works for knots with arc-index at most 11 (see below), and sometimes for knots with arc-index 12. Furthermore we managed to produce grid diagrams of arc-index at most 12 for all 11 crossing knots whose $\tau$ invariant cannot simply be read off from the shape of the Heegaard-Floer homology. Thus we expect to produce a table of $\tau$ invariants for 11 crossing knots in the near future.

We can see immediately from our tables that the Heegaard Floer homology of any knot with at most 11 crossings is small enough (i.e. is supported only on two diagonals) that the $\tau$ invariant is determined by the ranks of the $d^1$ differentials. Now recall that whenever we have a (bounded) filtered chain complex

$$ \cdots F^i \subset F^{i+1} \cdots \subset C $$

then this induces a spectral sequence with $E^1_{p,q} = H_{p+q}(F^p / F^{p-1})$ where the $E^1$ differential

$$d^1 : E^1_{p,q} = H_{p+q}(F^p / F^{p-1}) \to E^1_{p-1,q} = H_{p+q-1}(F^{p-1} / F^{p-2})$$

is given by the connecting homomorphism in the LES in homology associated to the SES of complexes

$$ 0 \to \frac{F^{p-1}}{F^{p-2}} \to \frac{F^p}{F^{p-2}} \to \frac{F^p}{F^{p-1}} \to 0.$$

This is exactly the situation that arises in our Heegaard Floer Knot Homology computations, where the filtration is induced by Alexander grading. Given a grid presentation $\Gamma$ it suffices to compute using the chain complex $C(\Gamma)$ and its filtration, keeping in mind that this complex is actually filtered chain homotopic to the filtered complex $\widehat{CFK}(S^3,K) \otimes K^{\otimes n-1}$.

For example, we can compute $\tau(K=11n_{31})=2$ as follows. We first produce an arc-index 11 grid diagram $\Gamma$ for $K$:

int white[11] = {9,6,4,5,7,8,3,2,1,10,0};
int black[11] = {2,3,1,8,10,6,7,0,9,4,5};

Next, from the shape of the Heegaard-Floer homology, to finish the computation it suffices to show that the $d^1$ differential $d^1 : HFK_0(K,1) \cong \ZZ_2^4 \to HFK_{-1}(K,0) \cong \ZZ_2^6$ has rank 3. This is equivalent to showing that the differential

$$ d^1 : H_0( \frac{C(\Gamma,1)}{C(\Gamma,0)}) \cong \ZZ_2^{14} \to H_{-1}(\frac{C(\Gamma,0)}{C(\Gamma,-1)}) \cong \ZZ_2^{91}$$

has rank 13. We wrote two programs to accomplish this:

hfktaugraph.cpp
hfktaucomp.cpp

We next run the following series of commands (running on video.math.columbia.edu):

g++ -O3 -o hfkgraph hfktaugraph.cpp
./hfkgraph > 11n31.txt
g++ -O3 -o hfkcomp hfktaucomp.cpp
./hfkcomp < 11n31.txt

This completes the computation. The above programs can be modified to work for other knots.


Total Visits:

Relevant Links: My homepage, Bar Natan's Knot Atlas

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W. D. Gillam