An Introduction to Braid Theory

The n-braid group, B_n, is the group generated by n-1 generators, s_1, s_2, ..., s_{n-1} satisfying the following relations:

  1. | i-j | > 1 => s_i s_j = s_j s_i

  2. s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}

These relations resemble the ones satisfied by the generators of the permutation group, with the sole difference that now the square of each generator is not identity. This is the reason why the braid group is relevant in the study of statistics in 2-dimensional manifolds.

In Knot Theory braids are relevant since it has been shown that any knot may be considered as the closure of some braid. In general the closure of a braid is a link; one may find the number of components of the link as follows.

Let a braid b. This braid can be mapped to a permutation p by mapping the generators of the braid group s_i to the exchanges (i,i+1). If p belongs to the conjugacy class a_1,a_2,...,a_k where a_k > 0 , the corresponding link consists of k components.

The closure of equal braids gives ambient isotopic links. Unequal braids may also give isotopic links; this will occur if and only if the braids are connected through a series of Markov moves . There are two kinds of such moves.

  1. b -> xbx^{-1}

  2. b belonging to B_n -> b s_n^{\pm 1} belonging to B_{n+1}

One may thus construct knot invariants by obtaining a function of a representation of the braid group such that for any two braids differing by a Markov move, the function remains invariant. If, for example, one considers a function of the eigenvalues of a representation (i.e. the trace), one need only check its invariance under the second move. It is through appropriate representations of the braid group, such as the Burau, that many new knot invariants have been recently discovered. Such invariants may distinguish among mirror symmetric knots, which is not possible through the color tests. The representations of the braid group have not yet been fully classified, and thus new knot invariants may arise in the future.

In 1993, Sofia Lambropoulou showed that these two moves are just two special cases of a more general move, the L-move. Through this L-move she obtained a new proof of the fact that knots and links are braid closures. To view the relevant paper, which she coauthored with Colin Rourke, click here .

Knot invariants are usually defined through the skein relations. One first sets P(unknot), the usual choice is 1. Then one imposes a linear condition between the three links that are the closure of braids b, b s_i, b s_i^{-1} . The coefficients of the three links do not depend on b or on s_i, but they do depend on the value of one (Jones) or two (HOMFLY) polynomials. Inductively one may always calculate P(any link) as a function of P(unknot).

This is not however the case with the Akutsu-Wadati polynomials which were recently invented; here, the linear relation connects more than three braids (in general); the N-polynomial connects the braids b, b s_i, b s_i^2, ..., b s_i^N . P(unknot) is not sufficient to obtain P(link) when N > 2 . When N = 2 the Akutsu-Wadati is identical to the Jones polynomial. Through the use of the N=3 polynomial Akutsu and Wadati were able to distinguish two knots that had identical Jones polynomials.

A footnote: the symbol s_i means s with index i, the symbol a^b means a to the b power, while \pm 1 means plus or minus 1. This is the standard symbolism for mathematics in TeX.

Charilaos Aneziris,

Copyright 1995


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