Braid Word

Any $n$-braid is expressed as a braid word, e.g., $\sigma_1\sigma_2\sigma_3\sigma_2^{-1}\sigma_1$ is a braid word for the Braid Group $B_3$. By Alexander's Theorem, any Link is representable by a closed braid, but there is no general procedure for reducing a braid word to its simplest form. However, Markov's Theorem gives a procedure for identifying different braid words which represent the same Link.

Let $b_+$ be the sum of Positive exponents, and $b_-$ the sum of Negative exponents in the Braid Group $B_n$. If

$$b_+-3b_-\geq n,$$

then the closed braid $b$ is not Amphichiral (Jones 1985).

See also Braid Group

References

Jones, V. F. R. ``A Polynomial Invariant for Knots via von Neumann Algebras.'' Bull. Amer. Math. Soc. 12, 103-111, 1985.

Jones, V. F. R. ``Hecke Algebra Representations of Braid Groups and Link Polynomials.'' Ann. Math. 126, 335-388, 1987.