In 2007, Jean-Marie Droz of the University of Zurich (working along with Anna Beliakova) wrote a Python program to compute the (hat-version) Heegaard-Floer Knot Homology [math]\displaystyle{ \widehat{\operatorname{HFK}}(K) }[/math] of a knot [math]\displaystyle{ K }[/math]. His program is integrated into KnotTheory`, though to run it, you must have Python as well as the Python library Psycho installed on your system.
(For In[1] see Setup)
| In[1]:=
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?HFKHat
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| HFKHat[K][t,m] returns the Poincare polynomial of the Heegaard-Floer Knot Homology (hat version) of the knot K, in the Alexander variable t and the Maslov variable m.
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| In[2]:=
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HFKHat::about
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| The Heegaard-Floer Knot Homology program was written by Jean-Marie Droz in 2007 at the University of Zurich, based on methods of Anna Beliakova's arXiv:07050669.
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The Heegaard-Floer Knot Homology is a categorification of the Alexander polynomial. Let us test that for the knot 8_19:
In[3]:=
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hfk = HFKHat[K = Knot[8, 19]][t, m]
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Out[3]=
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2 -3 m 5 2 6 3
m + t + -- + m t + m t
2
t
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In[4]:=
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{hfk /. m -> -1, Alexander[K][t]}
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Out[4]=
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-3 -2 2 3 -3 -2 2 3
{1 + t - t - t + t , 1 + t - t - t + t }
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The knot 8_19 is the first knot in the Rolfsen Knot Table whose Heegaard-Floer Knot Homology is not "diagonal". Let us test that. The homology [math]\displaystyle{ \widehat{\operatorname{HFK}}(K) }[/math] is "on diagonal", iff its Poincare polynomial, evaluated at [math]\displaystyle{ m=1/t }[/math], is a monomial:
In[5]:=
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Select[AllKnots[{3, 8}], (Head[HFKHat[#][t, 1/t]] == Plus) &]
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Out[5]=
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{Knot[8, 19]}
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In[6]:=
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hfk /. m -> 1/t
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Out[6]=
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4 -2
-- + t
3
t
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