How can I tell if two braids are equal?

Question

I was reading this wikipedia article. In it they claim that

\begin{equation}\sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2\end{equation}

Here $\sigma_1, \sigma_2, \sigma_3$ represent the 4 generators of the braid group $B_4$

I went verify this on paper and I was having a hard time, so I went and got out a piece of string and tried it. Sure enough it is in fact true, if you apply $\sigma_1^-$ to each side you get:

\begin{equation}\sigma_1\sigma_2 = \sigma_2\sigma_1\sigma_2\sigma_1^-\end{equation}

And I was able to convince myself, via physical manipulation that

\begin{equation}\sigma_2\sigma_1\sigma_2\sigma_1^-=\sigma_1\sigma_2 \end{equation}

However this doesn't sit well with me. Having to verify if two braids are equal by playing around with string each time seems quite cumbersome. I tried the other example on the wiki as well as a few ones I made up (which all turned out to be unequal) but I wasn't seeing any pattern.

What ways are there for determining the equality of two braids? I don't necessarily want a closed form, an algorithm or a rule of thumb will suffice as far as I am concerned.

Answer 1

The braid groups are known to be linear, so you can work with a faithful matrix representation such as the Lawrence-Krammer representation. Then you can just compute some matrices and check whether they are equal.