# A GENERALIZED BURAU REPRESENTATION FOR STRING LINKS

@article{Silver2001AGB, title={A GENERALIZED BURAU REPRESENTATION FOR STRING LINKS}, author={Daniel S. Silver and Susan G. Williams}, journal={Pacific Journal of Mathematics}, year={2001}, volume={197}, pages={241-255} }

A 2-variable matrix B ∈ GLn(Z[u±1, v±1]) is defined for any n-string link, generalizing the Burau matrix of an nbraid. The specialization u = 1, v = t−1 recovers the generalized Burau matrix recently defined by X. S. Lin, F. Tian and Z. Wang using probabilistic methods. The specialization u = t−1, v = 1 results in a matrix with a natural algebraic interpretation, and one that yields homological information about the complement of the closed string link.

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## References

SHOWING 1-10 OF 22 REFERENCES

BURAU REPRESENTATION AND RANDOM WALKS ON STRING LINKS

- Mathematics
- 1996

Using a probabilistic interpretation of the Burau representation of the braid group oered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string…

THE GASSNER REPRESENTATION FOR STRING LINKS

- Mathematics
- 1998

The Gassner representation of the pure braid group to $GL_n(Z[Z^n])$ can be extended to give a representation of the concordance group of $n$-strand string links to $GL_n(F)$, where $F$ is the field…

Entropie topologique et représentation de Burau

- Mathematics
- 1989

Let f be an orientation-preserving homeomorphism of the disk D, P a finite invariant subset and [f] the isotopy class of f in D\P. We give a non trivial lower bound of the topological entropy for…

Knots and Links

- Mathematics
- 2003

Introduction Codimension one and other matters The fundamental group Three-dimensional PL geometry Seifert surfaces Finite cyclic coverings and the torsion invariants Infinite cyclic coverings and…

Knots And Physics

- Mathematics
- 1991

Physical Knots States and the Bracket Polynomial The Jones Polynominal and Its Generalizations Braids and Polynomials: Formal Feynman Diagrams, Bracket as Vacuum-Vacmum expectation and the Quantum…

Braids, Links, and Mapping Class Groups.

- Mathematics
- 1975

The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with…

A Course in Group Theory

- Mathematics
- 1996

1. Definitions and examples 2. Maps and relations on sets 3. Elementary consequences of the definitions 4. Subgroups 5. Cosets and Lagrange's Theorem 6. Error-correcting codes 7. Normal subgroups and…

An application of Jensen's formula to polynomials

- Mathematics
- 1960

In this note new proofs will be given for two inequalities on polynomials due to N. I. Feldman [1] and A. 0. Gelfond [2], respectively; these inequalities are of importance in the theory of…