07-1352/Class Notes for February 6

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In Preparation

The information below is preliminary and cannot be trusted! (v)

Today's Agenda

• A bit about genus, finite type invariants and the Alexander polynomial.

Genus and the Alexander Polynomial

In[1]:=

<< KnotTheory`

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

In[2]:=

Ks = Select[AllKnots[], ThreeGenus[#] == 1 &]

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

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[2]=

{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]}

In[3]:=

Conway[#][z] & /@ Ks

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

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

Out[3]=

{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}