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Burau Representation

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Gives a matrix representation b_i of a braid group in terms of (n-1)×(n-1) matrices. A -t always appears in the (i,i) position.

b_1=[-t 0 0 ... 0; -1 1 0 ... 0; 0 0 1 ... 0; | | | ... |; 0 0 0 ... 1]
(1)
b_i=[1 ... 0 0 ... 0; | ... | | ... |; 0 ... -t 0 ... 0; 0 ... -t 0 ... 0; 0 ... -1 1 ... 0; 0 ... 0 0 ... |; 0 ... 0 0 ... 1]
(2)
b_(n-1)=[1 0 ... 0 0; 0 1 ... 0 0; | | ... | |; 0 0 ... 1 -t; 0 0 ... 0 -t].
(3)

Let Psi be the matrix product of braid words, then

(det(I-Psi))/(1+t+...+t^(n-1))=Delta_L,
(4)

where Delta_L is the Alexander polynomial and det is the determinant.

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References

Burau, W. "Über Zopfgruppen und gleichsinnig verdrilte Verkettungen." Abh. Math. Sem. Hanischen Univ. 11, 171-178, 1936.Jones, V. "Hecke Algebra Representation of Braid Groups and Link Polynomials." Ann. Math. 126, 335-388, 1987.

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Burau Representation

Cite this as:

Weisstein, Eric W. "Burau Representation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BurauRepresentation.html

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