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
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
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
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
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
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.
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
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
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
...
1
2
3
...