Invariants for Braids

Section 1.3 Invariants for Braids

Objectives

Make connections, informally and formally by homomorphism, between (finite) groups of permutations and (infinite) braid groups.
Use a presentation of the braid group on \(n\) strands to simplify braid expressions and solve equations.
Explore the extent to which the Burau matrix of a braid is a (complete?) invariant for braids, and use technology to compute its Alexander polynomial.

The permutation group on \(n\) symbols, and the braid group on \(n\) strands, have very similar presentations.

Theorem 1.3.1.

Let \(G\) be a finitely presented group on \((n-1)\) generators, called

\begin{equation*} \sigma_1,\sigma_2,\ldots,\sigma_{n-1}. \end{equation*}

We define three sets of relations:

Order: For all \(1\leq i \leq n-1\text{,}\) we have

\begin{align} \sigma_i^2 &= e\label{eq_orderrelation}\tag{1.3.1} \end{align}
Disjoint pairs commute: For all \(i,j\) satisfying \(|i-j|\geq 2\text{,}\) we have

\begin{align} \sigma_i\sigma_j &= \sigma_j\sigma_i.\label{eq_commuterelation}\tag{1.3.2} \end{align}
Skein relations: For all \(1\leq i \leq n-2\text{,}\) we have

\begin{align} \sigma_i\sigma_{i+1}\sigma_i &= \sigma_{i+1}\sigma_i\sigma_{i+1}.\label{eq_skeinrelation}\tag{1.3.3} \end{align}

If a presentation of \(G\) consists of all three sets of relations (1.3.1), (1.3.2), and (1.3.3), then \(G\) is isomorphic to the permutation group on \(n\) strands:

\begin{equation*} G \cong S_n. \end{equation*}

If a presentation of \(G\) consists of only the second and third sets of relations (1.3.2) and (1.3.3), then \(G\) is isomorphic to the braid group on \(n\) strands:

\begin{equation*} G\cong B_n. \end{equation*}

Our goal in this section will be to discover a collection of square matrices whose properties -- to the extent possible -- mirror the properties of the generators of these groups

Worksheet 1.3.1 Matrices for Braids: The Burau Representation

We begin by defining a pair of \(2\times 2\) matrices that we would like to “represent” the exchange of two positions in a permutation and/or in a braid, as follows.

\begin{equation*} X = \left[\begin{array}{rr} 0&1\\ 1&0 \end{array}\right] \quad \text{and} \quad X(t) = \left[\begin{array}{cc} 1-t & t \\ 1& 0 \end{array}\right]. \end{equation*}

1.

Check directly that, for any \(x,y\text{,}\) we have

\begin{equation*} X \cdot \left[\begin{array}{r}x\\y\end{array}\right] = \left[\begin{array}{r} y\\x\end{array}\right]. \end{equation*}

Where and why do you think this matrix might be helpful in representing permutations and/or braids?

2.

Compute the matrix product \(X(t) \cdot \left[\begin{array}{rr} x\\y\end{array}\right]\text{.}\) What value of \(t\) could be substituted to make this answer agree with the previous problem?

3.

Check directly that \(X^2 = X\cdot X\) is the identity matrix \(\left[\begin{array}{rr}1&0\\0&1\end{array}\right]\text{,}\) but that \(X(t)^2 = X(t)\cdot X(t)\) is not. What might this mean about the possbility that these matrices can help us represent group elements that satisfy the relations (1.3.1), (1.3.2), and (1.3.3)?

To help us with our computations, we'll use the open-source computer algebra system Sage (http://sagecell.sagemath.org) from here on.

4.

Consider the \(3\times 3\) matrices

\begin{equation*} B_1 = \left[\begin{array}{ccc}1-t&t&0\\1&0&0\\0&0&1\end{array}\right] \quad \text{and} \quad B_2 = \left[\begin{array}{ccc}1&0&0\\0&1-t&t\\0&1&0\end{array}\right]. \end{equation*}

These two matrices are defined in the Sage code block below. Use it to verify that the matrices satisfy the skein relation (1.3.3).

5.

Describe how you think these matrices were created out of the matrices \(X\) and/or \(X(t)\text{,}\) above. What might these matrices represent?

6.

Use Sage below to determine the inverses of the matrices \(B_1\) and \(B_2\text{.}\)

7.

Compute the matrix product \(B_2B_1B_2B_1^{-1}B_2^{-1}B_1^{-1}\) using Sage.

8.

Now sketch the braid in the braid group on 3 strands represented by the word

\begin{equation*} \sigma_2\sigma_1\sigma_2\sigma_1^{-1}\sigma_2^{-1}\sigma_1^{-1}. \end{equation*}

The Sage code below will do this for you. What do you notice about this braid that is less surprising given your result in the previous problem?

Hint

Remember, what you're seeing is a diagram of a braid... so our diagram rules can help us out!

9.

Every braid can be “closed” to form a knot. Consult the KnotInfo database at http://knotinfo.math.indiana.edu and select a knot with 8 or fewer crossings, whose braid notation consists of only \(\pm 1\) and \(\pm 2\)'s. (This indicates the knot can be formed from a braid of three strands.)

Find and paste an image of your knot's diagram from Rolfsen's knot table.

Use Sage to determine the following for your braid:

A braid diagram (plot) for your braid, and
A \(3\times 3\) matrix representing your braid word.

Finally, use your result from #2 to determine a permutation matrix associated with your braid. What permutation of the word \(CAT\) does your matrix achieve, and how do you connect this information with your selected braid and/or knot?