In 2007, JeanMarie Droz of the University of Zurich (working along with Anna Beliakova) wrote a Python program to compute the (hatversion) HeegaardFloer Knot Homology ${\widehat {\operatorname {HFK} }}(K)$ of a knot $K$ (see arXiv:0803.2379). His program is integrated into KnotTheory`
, though to run it, you must have Python as well as the Python library Psyco installed on your system.
(For In[1] see Setup)
In[1]:=

?HFKHat

HFKHat[K][t,m] returns the Poincare polynomial of the HeegaardFloer Knot Homology (hat version) of the knot K, in the Alexander variable t and the Maslov variable m.


In[2]:=

HFKHat::about

The HeegaardFloer Knot Homology program was written by JeanMarie Droz in 2007 at the University of Zurich, based on methods of Anna Beliakova's arXiv:07050669.


The HeegaardFloer Knot Homology is a categorification of the Alexander polynomial. Let us test that for the knot 8_19:
In[3]:=

hfk = HFKHat[K = Knot[8, 19]][t, m]

Out[3]=

2 3 m 5 2 6 3
m + t +  + m t + m t
2
t

In[4]:=

{hfk /. m > 1, Alexander[K][t]}

Out[4]=

3 2 2 3 3 2 2 3
{1 + t  t  t + t , 1 + t  t  t + t }

The knot 8_19 is the first knot in the Rolfsen Knot Table whose HeegaardFloer Knot Homology is not "diagonal". Let us test that. The homology ${\widehat {\operatorname {HFK} }}(K)$ is "on diagonal", iff its Poincare polynomial, evaluated at $m=1/t$, is a monomial:
In[5]:=

Select[AllKnots[{3, 8}], (Head[HFKHat[#][t, 1/t]] == Plus) &]

Out[5]=

{Knot[8, 19]}

In[6]:=

hfk /. m > 1/t

Out[6]=

4 2
 + t
3
t

The (mirrored) Conway knot K11n34 and the (mirrored) KinoshitaTerasaka knot K11n42 are a mutant pair, and are notoriously difficult to tell apart. Let us check that an array of standard knot polynomials fails to separate them, yet ${\widehat {\operatorname {HFK} }}$ succeeds:
In[7]:=

K1 = Knot["K11n34"]; K2 = Knot["K11n42"];
test[invt_] := (invt[K1] =!= invt[K2]);
test /@ {
Alexander, MultivariableAlexander, Jones, HOMFLYPT, Kauffman, Kh, HFKHat
}

Out[7]=

{False, False, False, False, False, False, True}

Indeed,
In[8]:=

{HFKHat[K1][t, m], HFKHat[K2][t, m]}

Out[8]=

2 1 1 3 3 3 3
{3 +  +  +  +  +  +  +  + 3 t + 3 m t +
m 4 3 3 3 3 2 2 2 2 m t
m t m t m t m t m t
2 2 2 2 3 3 3
3 m t + 3 m t + m t + m t ,
6 1 1 4 4 2 2 2
7 +  +  +  +  +  + 4 t + 4 m t + m t + m t }
m 3 2 2 2 2 m t
m t m t m t

On July 6, 2006, User:AnonMoos asked User:Drorbn if he could identify the knot in the left hand side picture below. At the time it was impossible using the tools available with KnotTheory`
 using any of many invariants, the answer can be found to be either the mirror of K11n34 or the mirror of K11n42, but KnotTheory`
couldn't tell which one it is (though of course, it is possible to do it "by hand"). The 2007 addition ${\widehat {\operatorname {HFK} }}$ does the job, though. Indeed, we first extract the mystery knot's DT (DowkerThistlethwaite) Code using the picture on the right hand side below, then compute ${\widehat {\operatorname {HFK} }}$, and then search for it within the ${\widehat {\operatorname {HFK} }}$'s of all knots with up to 11 crossings:
In[9]:=

K3 = DTCode[6, 8, 14, 12, 4, 18, 2, 20, 22, 10, 16];

In[10]:=

H = HFKHat[Mirror[K3]][t, m]

Out[10]=

2 1 1 3 3 3 3
3 +  +  +  +  +  +  +  + 3 t + 3 m t +
m 4 3 3 3 3 2 2 2 2 m t
m t m t m t m t m t
2 2 2 2 3 3 3
3 m t + 3 m t + m t + m t

In[11]:=

Select[AllKnots[], HFKHat[#][t, m] == H &]

Out[11]=

{Knot[11, NonAlternating, 34]}

And so the mystery knot is the Conway knot, the mirror of K11n34.