As is well known, the closing operation connects braids and links. The Alexander theorem states that any link can be obtained by closing a braid. The Markov theorem describes the equivalence relations of braids and as a consequence each link is considered as Markov class of some braid. For a braid \(b\in B_n\) and its closing \(K_b,\) Artin’s theorem gives the presentation of the fundamental group of link complement: \(\pi_1(S^3\smallsetminus K_b) = \langle x_1,\dots ,x_n\mid x_1=\rho_n(b)(x_1),\dots , x_n=\rho_n(b)(x_n)\rangle,\) where \(\rho_n:B_n\to Aut(F^n)\) is the Artin representation of braid group \(B_n\) into the automorphism group of the free group \(F^n\) on \(n\) generators \(x_1,\dots ,x_n.\) Burau’s theorem gives the formula for the Alexander polynomial of \(K_b:\) \({\mathcal A}(K_b)=(-1)^{n-1}\frac {det(\hat\beta_n (b))-Id}{x^{n-1}+X^{n-2}+\dots +1},\) where \(\widehat\beta_n :B_n\to Gl_{n-1}({\mathbb{Z}}[x^{\pm}])\) are the reduced Burau representations of \(B_n.\) Singular braids and singular links (in particular, singular knots) are defined similarly to ordinary braids and links but they are allowed to have a finite number of transversal self-intersections.
The main goal of the article is the extension of the theorems listed above to singular braids and singular links. The set of singular braids on \(n\) strings does not form a group. However, it still forms the \(SB_n\) monoid which has the presentation: \[ \begin{aligned} SB_n=\langle &s_i,s_i^{-1},t_i, i=1,\ldots ,n-1|s_is_i^{-1}=1, i<n; \\ &s_is_j=s_js_i, s_it_j=t_js_i, t_it_j=t_jt_i |i-j|>1; \\ & s_it_i=t_is_i, i<n; s_is_{i+1}s_i=s_{i+1}s_is_{i+1}, \\&t_is_{i+1}s_i=s_{i+1}s_it_{i+1}, s_is_{i+1}t_i=t_{i+1}s_is_{i+1}\rangle\end{aligned} \] Here \(s_i, i=1,...,n-1\) are standard braid generators and \(t_i\) are singular analogues of \(s_i\) with one singular point each. The braid group \(B_n\) is the group of all invertible elements in \(SB_n.\) It is proved that there exist extensions \(\widetilde\rho_n^{\omega};SB_n\to End(F^n)\) and \(\tilde\beta_n^p:SB_n\to M_n({\mathbb Z}[x^{\pm 1}])\) of the Artin representation \(\rho_n\) and the Burau representation \(\beta_n\) of \(B_n.\) These extensions possess some local properties and they correspond to an element \(\omega\) of the free group \(F^2\) and a Laurent polynomial \(p\in {\mathbb Z}[x^{\pm 1}].\) It is noted that any extended Burau representation \(\widetilde\beta_n^p\) can be used for the definiton of the Alexander polynomial of singular knots. Using the representation \(\tilde\rho_n^{\omega}\) and a singular braid \(b\in SB_n\) the group \(G_{\tilde\rho_n^{\omega}(b)}\) is defined which is a group invariant of singular links \(K_b.\) It is proved that the group \(G_{\tilde\rho_n^{\omega}(b)}\) is a quotient group depending on \(\omega\) of fundamental group \(\pi_1(S^3\smallsetminus K_b)\) of singular links \(K_b.\) In the end of the article a list of non-equivalent singular knots and their group invariants is presented.