Braid group

review of symmetric groups

• 원소의 개수가 n인 집합의 전단사함수들의 모임
• \(n!\) 개의 원소가 존재함
• 대칭군의 부분군은 치환군(permutation group)이라 불림

presentation of symmetric groups

• \(S_n\)
• generators \(\sigma_1, \ldots, \sigma_{n-1}\)
• relations
  • \({\sigma_i}^2 = 1\)
  • \(\sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1\)
  • \(\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}\)

presentation of braid groups

• \(B_n\)
• generators \(\sigma_1,...,\sigma_{n-1}\)
• relations (known as the braid or Artin relations):
  • \(\sigma_i\sigma_j =\sigma_j \sigma_i\) whenever \(|i-j| \geq 2 \)
  • \(\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i \sigma_{i+1}\) for \(i = 1,..., n-2\)
• Yang-Baxter equation (YBE)
• For a solution of the YBE \(\bar{R}\), we can construct a representation \(\rho\) of the braid group by

\[ \rho : B_n \to \rm{Aut}(V^{\otimes n}) \] where \(\rho(\sigma_i)=\bar{R}_i\)

There is also a natural surjective morphism from \(B_n\) to the symmetric group \(\mathfrak{S}_n\), given on the generators by \(B_n\ni\sigma_i\mapsto s_i\in \mathfrak{S}_n\), \(i=1,\dots,n-1\). For a braid \(\beta\in B_n\), we denote \(p_{\beta}\) its image in \(\mathfrak{S}_n\), and refer to \(p_{\beta}\) as to the underlying permutation of \(\beta\).

examples

• in a braid diagram, read from bottom to top and we number all strands of the braid with the indices it starts at the bottom

• read the braid word from left to right accordingly.
• For instance, the braid word corresponding to the braid above is \(\sigma_1^{-1}\sigma_2\sigma_1^{-1}\sigma_2\sigma_1^{-1}\)

Markov moves

• braid group version of Reidemeister moves

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxZ3NjMGpGUWI0QkE/edit

related items

• Jones polynomials
• Hecke algebra
• Loop braid group

encyclopedia

• http://en.wikipedia.org/wiki/Braid_group

expositions

• Abad, Camilo Arias. 2014. “Introduction to Representations of Braid Groups.” arXiv:1404.0724 [math], April. http://arxiv.org/abs/1404.0724.

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• ID : Q220409