The first set of projects in the topology component revolve around questions on counting simple closed curves and their geometric intersection numbers on punctured discs and surfaces.
The area of classical knot and link theory is attractive to many undergraduates, but it requires limitation in scope as well as substantial simplification in the introductory development in order to be accessible. Our REU philosophy is to chose research directions that take off from linear algebra, elementary group theory and combinatorics. For example, knots are developed from the knot diagram and the relation of Reidemeister moves.
This past year, a participant, Aaron Bronfman, gave an unexpected complete calculation of the word growth function for the positive braid monoid that opens up a whole new set of questions and potential projects directly related to the theme: Growth and Counting functions in Knot and Braid Theory
One of the initial problems posed last year was to find the word length growth function $h_n(t) = \sum_{g} t^{\length(g)}$ of the pure braid monoid. Known to be a rational function, the exact expression had only been computed for n = 3 and 4 by Xu (and known trivially for n=2.) Utilizing insights gained from an attempt to use the Anick resolution to compute the Hilbert series for the monoid ring of the positive braids for a small number of strands, Aaron Bronfman was eventually able to completely solve the problem and compute a closed form expression for all the positive braid monoids. The proof utilizes the structure of the the Garside-Brieskorn-Saito-Deligne solution to the word problem and a clever inclusion-exclusion argument.
In developing the central theme (counting and zeta functions) of our proposed REU, we have the following possible directions for student projects in topology.
The usual generators for the pure braid group involve conjugates and therefore inverses. On the other hand, this intersection is the kernel of the natural morphism from the positive braid monoid to the symmetric group, hence a significant fraction $\frac{1}{n!}$ of positive braids are pure. It would be interesting to find a nice generating set and study the relationship of this submonoid to the positive braid monoid.
Stoltzfus developed an elementary approach to the Gassner representation (as well as others related to the lower central series) using the following two additional ingredients: twisting of the matrix multiplication and the existence of an invariant sesqui-linear form for the ``representation." This gives a purely algebraic and combinatorial approach to certain problems in braid and knot theory. Artin demonstrated that the braid group on $n$ strands, denoted $B_n$, is a subgroup of the automorphism group of the free group, $F_n$. Using the free derivatives $\partial_i$, the Jacobian function $J$ can be defined from the Artin braid group to the set of square matrices over the ring of integral Laurent polynomials in $n$ variables which is a crossed homomorphism satisfying $J(ab) = \mu(b)_*(J(a)) J(b)$. This extends the Gassner representation from the pure braid subgroup with, of course, the loss of the strict homomorphism condition. Next, Artin's necessary conditions for an automorphism of the free group to be in the braid group translate into algebraic conditions reminiscent of orthogonal transformations as follows:
where $A$ is a matrix in the image of the representation $J$, $\Lambda_n $ denotes a fixed matrix depending only on the number of strands $n$ and $\mu(A)$ is the permutation of variables associated to $A$. In this context the following questions are open.
These difficult questions hold the continued interest of mathematicians. It would be unrealistic to routinely expect complete answers within the constraints of a summer REU, but it would be quite appropriate to work on tractable pieces of them.
To help with the computation of examples, the manipulation of elements in a free group, the computation of their free derivatives, twisted matrix multiplication, the Jacobian matrix and the $\Lambda$-form have already been encoded in a Mathematica(TM) notebook by students in previous summers.
Now, with the Bigelow-Krammer proof of the faithfulness of the BMW representation, similar questions and projects can be done for this representation as well.