Braid structures in knot complements, handlebodies and 3-manifolds (Q2725552)

The author gives one form of the presentation of braid groups in handlebodies. Other forms of the presentations were studied in the 90's by A. Leeves, A. Sossinsky and V. Vershinin. Some special cases were well-known earlier in physical papers. Let \(B_{m+n}\) be Artin braid group on \(m+n\) strands and \(P_{m+n}\subset B_{m+n}\) be a pure braid subgroup. Let \(a_{ij}\), \( 1\leq i<j\leq m+n\) and \(\sigma_k\), \(k=1,\ldots,m+n-1\) be the standard generators of the \(P_{m+n}\) and \(B_{m+n}.\) The subgroup \(B_{m,n}\subset B_{m+n}\) is generated by braids with the first \(m\) strands trivially fixed (that is, if we remove the last \(n\) strands we obtain the trivial braid with \(m\) strands). By definition, \(P_{m,n} =P_{m+n}\cap B_{m,n}.\) The group \(P_{m,n}\) is generated by elements \(a_{ij}, i<j, i=1,\ldots, m+n-1, j=m+1,\ldots ,m+n\) with relations NEWLINE\[NEWLINEa_{ij}^{-1}a_{rs}a_{ij}=a_{rs}\text{ if }i<j<r<s\text{ or }r<i<j<s,\tag{i}NEWLINE\]NEWLINE NEWLINE\[NEWLINEa_{ij}^{-1}a_{js}a_{ij}=a_{is}a_{js}a_{is}^{-1}\text{ if }i<j<s,\tag{ii}NEWLINE\]NEWLINE NEWLINE\[NEWLINEa_{ij}^{-1}a_{is}a_{ij}=a_{is}a_{js}a_{is}a_{js}^{-1}a_{is}^{-1}\text{ if }i<j<s,\tag{iii}NEWLINE\]NEWLINE NEWLINE\[NEWLINEa_{ij}^{-1}a_{rs}a_{ij}=a_{is}a_{js}a_{is}^{-1}a_{js}^{-1}a_{rs}a_{js}a_{is}a_{js}^{-1}a_{is}^{-1}\text{ if }i<r<j<s.\tag{iv}NEWLINE\]NEWLINE Denote \(a_i=a_{i,m+1}, i=1,\ldots ,m\) and \(\sigma_i =\sigma_{m+i}, i=1,\ldots ,m+n-1\) For the group \(B_{m,n}\) the following presentation is given: NEWLINE\[NEWLINE B_{m,n}=\langle a_1,\dots ,a_m,\sigma_1,\ldots ,\sigma_{n-1}\mid \sigma_i\sigma_j=\sigma_j\sigma_i,\;|i-j|>1; NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sigma_k\sigma_{k+1}\sigma_k= \sigma_{k+1}\sigma_k\sigma_{k+1},\;k=1,\ldots,n-1; NEWLINE\]NEWLINE NEWLINE\[NEWLINE a_i\sigma_k=\sigma_ka_i,\;k\leq 2,i=1,\ldots m; NEWLINE\]NEWLINE NEWLINE\[NEWLINE a_i\sigma_1a_i\sigma_1=\sigma_1a_i\sigma_1a_i,\;i=1,\ldots,m; NEWLINE\]NEWLINE NEWLINE\[NEWLINE a_i(\sigma_1a_r\sigma_1^{-1})=(\sigma_1a_r\sigma_1^{-1})a_i,\;r<i\rangle. NEWLINE\]NEWLINE The defined braids can be used for the construction of knots and links in different 3-manifolds and for proving analogues of the Markov theorem.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00034].