Matrix presentations of braids and applications

Lee, Sang

doi:10.1090/S0002-9939-99-04948-5

Matrix presentations of braids and applications
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by Sang Youl Lee PDF

Proc. Amer. Math. Soc. 127 (1999), 3403-3412 Request permission

Abstract:

We show that there exists a one-to-one correspondence between the class of certain block tridiagonal matrices with the entries $-1, 0,$ or $1$ and the free monoid generated by $2n$ generators $\sigma _{1}, \cdots ,\sigma _{n}, \sigma _{1}^{-1},\cdots , \sigma _{n}^{-1}$ and relation $\sigma _{i}^{\pm 1}\sigma _{j}^{\pm 1} = \sigma _{j}^{\pm 1}\sigma _{i}^{\pm 1}~ (|i-j| \geq 2)$ and give some applications for braids. In particular, we give new formulation of the reduced Alexander matrices for closed braids.

References

J. W. Alexander, A lemma on systems of knotted curves., Proc. Nat. Acad. Sci. USA 9 (1923), 93–95.
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
C. McA. Gordon and R. A. Litherland, On the signature of a link, Invent. Math. 47 (1978), no. 1, 53–69. MR 500905, DOI 10.1007/BF01609479
Patrick M. Fitzpatrick and Maria Testa, The parity of paths of closed Fredholm operators of index zero, Differential Integral Equations 7 (1994), no. 3-4, 823–846. MR 1270106
Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422. MR 171275, DOI 10.1090/S0002-9947-1965-0171275-5
Lorenzo Traldi, On the Goeritz matrix of a link, Math. Z. 188 (1985), no. 2, 203–213. MR 772349, DOI 10.1007/BF01304208