Fingerloop Braids and System of Yang-Baxter Type Equations

1. Fingerloop Braids and System of Yang-Baxter Type Equations Xiao-Song Lin University of California, Riverside Preliminary Report

2. Fingerloop braiding is a weaving technique from Medieval Europe and England.

3. Fingerloop Braids as Mathematical Objects Fingerloop braids form a subgroup FBn of the braid group B2n . We have the following set of generators for this subgroup FBn : Ri = σ2i σ2i-1 σ2i+1 σ2i Xi= σ2i-1 σ2i-1-1 σ2i+1 σ2i Yi= σ2i σ2i-1 σ2i+1-1 σ2i-1 for i= 1,2,…,n-1.

4. Theorem. The fingerloop braid group FBn generated by Ri , Xi ,and Yi has the following defining relations: (1) Ri Xi = Yi Ri , for all i. (2) For P,Q∈ {R, X, Y}, Pi Qj = Qj Pi when |i – j|>1. (3) For P,Q∈ {R, X, Y}, we have (i) Pi Qi+1 Qi = Qi+1 Qi Pi+1 , and (ii) Pi Pi+1 Qi = Qi+1 Pi Pi+1 , with the exceptions of P≠Y but Q = Y in (i) and Q≠ X and P = X in (ii).

5. Fingerloop Braids and Loop Braids The motion group Mn of the oriented unlink in R3 was first studied by Dahm and Goldsmith in the 1960-70’s. They gave a finite set of generators for this group. A finite complete set of relations was given by McCool in 1986. We may visualize a motion of the unlink in R3 by a loop braid in R3×[0,1].

6. Loop braid relations:

7. Another form of loop braid relations:

8. Theorem. The quotient group of the fingerloop braid group FBn by the relations Ri2= 1, for all iis isomorphic to the motion group Mnof the oriented unlink of n components in R3.Note that Mn is isomorphic to the basis conjugatingautomorphism subgroup of Aut(Fn), where Fn is the free group of rank n. So we may have many matrix representations of FBnnot necessarilycoming frommatrix representations of B2n. Such representations map Ri2 to the identity matrix.

9. How to Find Other Representations ofFBn Note that the subgroup of FBn generated by each of the following sets {R1,R2, …, Rn-1}, {X1,X2, …, Xn-1}, and {Y1,Y2, …, Yn-1} is isomorphic to the braid group Bn. Given a representation ρ and 1-dimensional representations λ and μ of Bn. Proposition. The matrices ρ(Ri), λ(Xi) ρ(Xi), μ(Yi) ρ(Yi) give rise to a representation of FBn..

10. TheMain Question of Interest In general, how to find irreducible representations of the braid group Bn, say ρ0,, ρ1, and ρ2of the same dimension, not only differing from each other like in the previous proposition, such that the matrices ρ0(Ri), ρ1(Xi), and ρ2(Yi)give rise to a representation of FBn? Answer to this question involves solving a system of Yang-Baxter type equations.

11. Some Patterns of Fingerloop Braids