Heegaard Floer Knot Homology: Difference between revisions
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In 2007, [http://www.math.unizh.ch/user/jdroz/ Jean-Marie Droz] of the University of Zurich (working along with [http://www.math.unizh.ch/index.php?id=1819&no_cache=1&key1=578&no_cache=1 Anna Beliakova]) wrote a Python program to compute the (hat-version) Heegaard-Floer Knot Homology <math>\widehat{\operatorname{HFK}}(K)</math> of a knot <math>K</math>. His program is integrated into <code>KnotTheory`</code>, though to run it, you must have [http://python.org/ Python] as well as the Python library [http://psyco.sourceforge.net/ Psycho] installed on your system. |
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{{In| |
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in = <nowiki>ToExpression[Import["http://katlas.org/wiki/InterfaceToDroz.m&action=raw"]]</nowiki>}} |
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The Heegaard-Floer Knot Homology is a categorification of the [[The Alexander-Conway Polynomial|Alexander polynomial]]. Let us test that for the knot [[8_19]]: |
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{{HelpAndAbout| |
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n = 2 | |
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n1 = 3 | |
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in = <nowiki>HFK</nowiki> | |
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out= <nowiki>HFK[K][q,t] returns the Poincare polynomial of the Heegaard-Floer Knot Homology of the knot K, in the Alexander variable q and the Maslov variable t.</nowiki> | |
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about= <nowiki>The Heegaard-Floer Knot Homology program was written by Jean-Marie Droz in 2007 at the University of Zurich.</nowiki>}} |
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{{InOut| |
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n = 4 | |
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out= <nowiki> 1 5 2 2 |
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9 + ----- + --- + 5 q t + q t |
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2 2 q t |
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q t</nowiki>}} |
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The knot [[8_19]] is the first knot in the [[The Rolfsen Knot Table|Rolfsen Knot Table]] whose Heegaard-Floer Knot Homology is not "diagonal". Let us test that. The homology <math>\widehat{\operatorname{HFK}}(K)</math> is "on diagonal", iff its Poincare polynomial, evaluated at <math>m=1/t</math>, is a monomial: |
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<!--$$Select[AllKnots[{3, 8}], (Head[HFKHat[#][t, 1/t]] == Plus) &]$$--> |
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{{InOut| |
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n = 5 | |
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in = <nowiki>{hfk /. t -> -1, Alexander[K][q]}</nowiki> | |
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out= <nowiki> -2 5 2 -2 5 2 |
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{9 + q - - - 5 q + q , 9 + q - - - 5 q + q } |
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q q</nowiki>}} |
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Revision as of 11:12, 3 December 2007
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 of a knot . 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 is "on diagonal", iff its Poincare polynomial, evaluated at , is a monomial:
