Heegaard Floer Knot Homology

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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)

The Heegaard-Floer Knot Homology is a categorification of the Alexander polynomial. Let us test that for the knot 8_19:

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: