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I consider the group $B_{n,n}$, the braids, colored in two colors, say all odd strings are black and all even strings are white. It is easy to find a set of generators for $B_{n,n}$: $$ \begin{cases} \def\s{\sigma} a)&M_i=\s_{2i-1} \s_{2i}\s_{2i-1} = \s_{2i}\s_{2i-1}\s_{2i}, i=1,\dots,n-1 \\ b)&N_i=\s_{2i}\s_{2i+1} \s_{2i} = \s_{2i+1} \s_{2i} \s_{2i+1}, i=1,\dots,n-1\\ c)&P_i=\s_{2i}^2, i=1,\dots,n-1 \\ d)&Q_i=\s_{2i-1}^2, i=1,\dots,n \\ \end{cases} $$ and relations $(M_iP_i^{-1}N_i)Q_i(M_iP_i^{-1}N_i)^{-1} = Q_{i+1}, (N_iQ_{i+1}^{-1}M_{i+1})P_i(N_iQ_{i+1}^{-1}M_{i+1})^{-1} = P_{i+1} $

But I can not find any source where it is proven. Could you give me a reference (paper, textbook)?

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А. А. Марков, Основы алгебраической теории кос, Тр. Матем. ин-та им. В. А. Стеклова, 1945, том 16, страницы 3–53 (Mi tm911), a link is available here

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  • $\begingroup$ I found there nothing about $B_{n,n}$. Which page did you refer to? As far as I see, the article concerns the group $R_n$ which is the groop of pure braids($n$ colours), not coloured in two colors. $\endgroup$
    – Nikita Kalinin
    Commented Apr 24, 2015 at 8:25

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