07-1352/Class Notes for February 6: Difference between revisions

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Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.}}
Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.}}


{{In|n=2|Ks = Select[AllKnots[], ThreeGenus[#] == 1 &]}}
{{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::credits: The 3-genus data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.}}
Line 23: Line 23:
Knot[11, Alternating, 343], Knot[11, Alternating, 362], Knot[11, Alternating, 363],
Knot[11, Alternating, 343], Knot[11, Alternating, 362], Knot[11, Alternating, 363],
Knot[11, NonAlternating, 139], Knot[11, NonAlternating, 141]}</nowiki>}}
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>}}

Revision as of 11:05, 6 February 2007

07-1352/Navigation Panel  
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# Week of... Links
1 Jan 9 About, Notes
2 Jan 16 Notes
3 Jan 23 Notes
4 Jan 30 Notes and Plans
5 Feb 6 Notes
6 Feb 13 Notes
R Feb 20 Reading week
7 Feb 27 Notes
8 Mar 6 Notes
9 Mar 13 Notes
10 Mar 20 Notes
11 Mar 27 Notes
12 Apr 3 Map
13 Apr 10 Dancso
SP Apr 17 Chu, Sankaran, Amir-Khosravi
Suggested Student Topics
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}
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