Knot at Lunch on June 16-22

Knot at Lunch/Navigation  Date(s) Link(s) 2010/02/22 ??? 2010/01/20 Formal integration 2010/01/13 Combing wB 2010/01/06 Exponentiation in tder 2009/09/22 descending v-knots 2009/08/26 red over green v-tangles 2009/08/19 Polyak Algebra 2009/07/08 Immanants 2009/07/01 Alexander modules 2009/06/24 Alexander modules 2009/06/10 Alexander, PBW for A^w, class videos 2009/06/03 Low key 2009/05/06 Low key 2009/04/29 Winter on Ribbons 2009/04/22 Misc 2009/04/15 KV 2009/03/25 KV 2009/03/18 Peter Lee 2009/03/04 Kirby calculus 2009/02/25 Karene on Reidemeister-Schreier 2009/02/11 Dror on Trotter, Jana on Alexander 2009/02/04 Bracelets 2009/01/28 gl(N) chickens 2009/01/15 2D Gauss Diagrams, FiC 2009/01/08 S&G update and more 2008/12/11 Chu on Garside, II 2008/12/04 wZ is 1-1 2008/11/27 The Wen 2008/11/20 The Zoom Space 2008/11/13 Chu on Garside 2008/11/06 Z and GPV 2008/10/30 Peter Lee on EHKR 2008/10/23 Map of the Field 2008/09/25 Hirasawa on Open Books 2008/09/18 Odd Khovanov 2008/09/17 Categorification.m 2008/09/11 More wAlex 2008/09/03 ? 2008/08/27 Dexp and BCH 2008/08/06 Z, A, det, tr, log 2008/07/30 Alexander Relations Marathon 2008/07/02 Peter Lee on horizontal Aw 2008/06/25 w-Alexander 2008/06 16-22 Thomas Fiedler Marathon 2008/06/11 ? 2008/06/04 Dylan Thurston 2008/05/28 Welded Tangles 2008/05/21 Bruce, Lucy 2008/04/23 Welded Knots 2008/04/16 Quandles and Lie algebras 2008/04/09 No-div Alekseev-Torossian 2008/04/02 Knotted Kung Fu Pandas 2008/03/26 Homotopy invariants 2008/03/19 Infinitesimal Artin 2008/03/12 Infinitesimalization of Artin 2008/03/05 Krzysztof Putyra on Odd Khovanov Homology 2008/02/27 Karene Chu on Proof of Artin 2008/02/20 Organizational, Hecke algebras 2008/02/13 Exponential and Magnus expansions 2008/02/06 cancelled 2008/01/30 Artin's theorem 2008/01/16 Hutchings' work, 2 2008/01/09 Hutchings' work, 1 2007/12/12 Bone soup 2007/12/05 Expansions 2007/11/28 Quantum groups 2007/11/21 Surfaces and gl(N)/so(N) 2007/11/07 Expansions for Groups 2007/10/31 Louis Leung on bialgebra weight systems 2007/10/24 Zsuzsi Dancso, continued 2007/10/17 Jana Archibald on the multivariable Alexander 2007/10/10 Zsuzsi Dancso on diagrammatic su(2) 2007/10/03 Hernando Burgos on alternating tangles 2007/09/26 Peter Lee on homology 2007/09/06 Garoufalidis' visit 2007/08/30 Art and enumeration 2007/08/23 My Hanoi talk? 2007/08/16 Lie bialgebra weight systems and more 2007/07/19 Subdiagram formulas 2007/07/12 Playing with Brunnians 2007/07/05 Virtualization 2007/06/28 Virtual braids 2007/06/07 Virtual knots 2007/05/31 Social gathering 2007/05/24 Lee on Frozen Feet

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
theorem
-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.

Content

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.