UMCP  November 22, 1997

  The Fifth Conference on Knot Theory and its Ramifications
  "Knots in Washington" will be held Saturday, November 22, 1997 at
  the University of Maryland (College Park).

  This regional miniconference is held every semester, in various locations
  in the Washington area. (The Sixth "Knots in Washington" conference:
  "Knot Theory Days," will take place Feb. 7-9, 1998 at U.S. Naval Academy,
  Annapolis; M.Kidwell, local organizer).

  You are cordially invited to participate in this and future meetings.

  All talks will be in the Colloquium Room in the Mathematics Department
  of the University of Maryland (Room number 3206).


  10:00 - 10:30   Refreshments

  10:30 - 11:30   Charles Frohman, University of Iowa;
                   "Skeins and Characters"

        The lecture will begin with a general method for
        producing 3-manifold invariants from a compact group.
        Out of this we will establish integral formulas for
        the Turaev-Viro invariant.

        More provocatively, we will show that
        the Kauffman bracket skein module at a root of
        unity of a 3-manifold can be viewed as functions
        on a cartesian product of copies of SU(2). The
        mode of evaluation depends on a Heegaard diagram
        of the 3-manifold. This gives rise to invariants
        of Heegaard diagrams coming from the Kauffman bracket
        skein module. It also gives a path to a rigorous
        analysis of the asymptotic behavior of Turaev-Viro
        invariants in terms of the representation theory
        of the fundamental group of the manifold.

  11:45 - 12:05   Jozef Przytycki, GWU;   "Torsion in skein modules:
                        Theorems, Conjectures and Speculations."

        We discuss  torsion in skein modules of 3-manifolds.
        1. A nonseparating 2-sphere or 2-torus in a manifold
           yields a torsion in most of the skein modules.
        2. A separating incompressible 2-sphere or 2-torus is
        often yielding a torsion (e.g. for Kaufman bracket,
        Homflypt and Kauffman skein modules).
        3. A nonseparating  surface (of any genus) is a cause
        of torsion in the second skein module (related to L_+
        -q L_0 skein relation).

        We discuss, with more details, torsion in the Kauffman
        bracket skein module. In particular we show that
        1. If M is a connected sum of M_1 and M_2 then KBSM of
        M has a torsion provided that $M_1$ and $M_2$ have
        first homology groups that are not 2-torsion groups.
        2. If M is the double of a hyperbolic manifold with
        boundary torus then the Kauffman bracket skein module
        of M has a torsion.

  12:10 -  2:10   Lunch

   2:15  - 3:15   Ted Stanford,  Naval Academy; "Vassiliev invariants and the
                  lower central series of the pure braid group."

        Vassiliev defined a new set of knot invariants around
        1990 using singularity theory.  Birman and Lin showed
        that the Jones polynomial and its generalizations
        can be reparametrized to fit into the Vassiliev
        framework.  Finding an interpretation of the Jones
        polynomial and its generalizations in terms of classical
        topology has been notoriously difficult, but Vassiliev
        invariants have proved a little more amenable in this

        We will prove the following theorem:  Let K1 and K2
        be knots.  Then v(K1) = v(K2) for every Vassiliev
        invariant of order less than n if and only if there
        exists a positive integer m and a braid b in Bm
        and a pure braid p in the nth group of the lower
        central series of Pm, such that K1 is the closure
        of b and K2 is the closure of pb.  Bm is the
        braid group on m strands, and Pm is the pure braid
        group on m strands.

        Thus we obtain an interpretation of what it means for
        two knots to have matching invariants up to order n
        in terms of classical group theory and topology.

        The proof of the theorem was inspired by a recent
        result of Habiro, which gives a characterization
        of knots with matching invariants up to order n
        in terms of "claspers", which are curves and
        handlebodies in a knot complement on which surgery
        is performed to modify the knot.

   3:30  - 3:50   Adam Sikora, UMCP;
                  "A topological approach to Sl_n character

        We will present a theorem proved jointly with C. Frohman
        which gives a purely topological description of the
        $SL_n$-character variety of any fundamental group of a
        manifold. We will discuss applications of this
        theorem to the representation theory of groups, the
        theory of character varieties and knot theory.

   4:00  - 4:50   Open problem session

  Local organizer: William Goldman (
  Organizing Committee:  John Millson (
                         Jozef Przytycki (
                         Yongwu Rong  (
                         Sergey Novikov (
                         Adam Sikora (