Fingerloop Braids and System of Yang-Baxter Type Equations

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Fingerloop Braids and System of Yang-Baxter Type Equations. Xiao-Song Lin University of California, Riverside Preliminary Report. Fingerloop braiding is a weaving technique from Medieval Europe and England. Fingerloop Braids as Mathematical Objects.

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## Fingerloop Braids and System of Yang-Baxter Type Equations

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### Fingerloop Braids and System of Yang-Baxter Type Equations

Xiao-Song Lin

University of California, Riverside

Preliminary Report

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.

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).

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].

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.

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..

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.