View source for 07-1352/Class Notes for February 6

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{{In Preparation}}

==Today's Agenda==
* A bit about genus, finite type invariants and the Alexander polynomial.

==Genus and the Alexander Polynomial==

{{In|n=1|in=<< KnotTheory`}}

{{Print|Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.}}

{{In|n=2|in=Ks = Select[AllKnots[], ThreeGenus[#] == 1 &]}}

{{Message|KnotTheory::credits: The 3-genus data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.}}

{{Message|KnotTheory::loading: Loading precomputed data in IndianaData`.}}

{{Out|n=2|out=<nowiki>{Knot[3, 1], Knot[4, 1], Knot[5, 2], Knot[6, 1], Knot[7, 2],
 Knot[7, 4], Knot[8, 1], Knot[8, 3], Knot[9, 2], Knot[9, 5], Knot[9, 35],
 Knot[9, 46],Knot[10, 1], Knot[10, 3], Knot[11, Alternating, 247],
 Knot[11, Alternating, 343], Knot[11, Alternating, 362], Knot[11, Alternating, 363],
 Knot[11, NonAlternating, 139], Knot[11, NonAlternating, 141]}</nowiki>}}

{{In|n=3|in=Conway[#][z] & /@ Ks}}

{{Message|KnotTheory::loading: Loading precomputed data in PD4Knots`.}}

{{Message|KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.}}

{{Message|KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.}}

{{Out|n=3|out=<nowiki>{1+z^2, 1-z^2, 1+2z^2, 1-2z^2, 1+3z^2, 1+4z^2, 1-3z^2, 1-4z^2,
 1+4z^2, 1+6z^2, 1+7z^2, 1-2z^2, 1-4z^2, 1-6z^2, 1+5z^2, 1+8z^2, 1+10z^2, 1+9z^2,
 1-2z^2, 1-5z^2}</nowiki>}}