Knot at Lunch on June 16-22

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The Plan

See The Proposed Thomas Fiedler Marathon.

Fiedler's Abstract

 Title : A candidate for a calculable complete invariant for classical knots

  Abstract :
   To each oriented classical knot K and each natural number n one can
associate an isotopy class of a (n,n)-tangle which is an isotopy
invariant of K.
We construct two combinatorial relative 1-cocycles, called Y and Sing,
for spaces of tangels. The cocycle Y takes values in a Hecke algebra
H_n+1 with coefficients in a polynomial ring of three variables. The
cocycle Sing takes values in a module
(over some polynomial ring) freely generated by all 1-singular tangels.
For each 1-singular tangle we can consider its two non-singular
resolutions and we can apply the cocycle Y to these resolutions.
Iterating this proces, with starting point the above (n,n)-tangle,
creates a "wave" in Hecke algebras of increasing dimension. We show that
this wave is indeed "expanding" and it is a good candidate for a
complete knot invariant.

The cours will be structured as follows:
-basic notions from singularity theory and a higher order Reidemeister
-construction of polynomial valued 1-cocycles for knot spaces. The
tetrahedron and the cube equations. Calculations
-integer-valued 1-cocycles for closed braids and a new filtration on the
space of all finite type invariants for closed braids
-essential homotopies of knots and their 1-cocycle. Specific invariants
for knots of unknotting number one.


Blackboard shots are at BBS/Fiedler-080616-084319.jpg. The programs written and a very condensed summary of our results are at Odds, Ends, Unfinished: Some HOMFLY-PT One Parameter Knot Theory Computations.