Lagrangian tangles in Fox coloring spaces and their t-deformations
by
Jozef H. Przytycki
George Washington University, Washington, D.C.

Y.Nakanishi conjectured twenty years ago that every link is 3-move equivalent to a trivial link. The natural extension of the problem is to find all 3-move equivalence classes of n-tangles. In particular I conjectured that the number of such classes (up to trivial components) is finite, for any n (let denote it by f_n). I conjectured that f_2=4 and f_3=40. The calculation of Tsukamoto suggested that f_4 is about 1120. My approach to the conjecture was the observation that the space of n-colorings is unchanged by n-moves. This led to a natural question: which subspaces of Fox p-colorings of boundary points are realized by n-tangles (denote that number by f_n(p))? I new that f_2(p) =p+1 and that f_3(3)=40 and f_4(3) is about 1120. I asked the question T.Januszkiewicz, after his talk on Tits buildings (May 2, 2000, Warsaw), and he answered that the sequence 4, 40, 1120 suggested that I am counting Lagrangians in a symplectic space (then f_n(p) should be equal to (p+1)(p^2+1)...(p^n-1+1)). Soon after I constructed an appropriate symplectic form (as did Januszkiewicz's student J.Dymara) and the theory of Lagrangians tangles was born. In this talk I will discuss the grow of the theory, its generalizations and applications.

Date received: April 3, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahe-09.