Narrow Results

Let M be a closed oriented surface of genus g≥1, let B_n(M) be the braid group of M on n strings, and let SB_n(M) be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map η : SB_n(M)→ℤ[B_n(M)], introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective.

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y_d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SB_n into the algebra Y_d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y_d,n(u).

In this paper, we define the set of singular grid diagrams ${𝒮}{𝒢}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on ${𝒮}{𝒢}$ which induce the bijection onto each singular object. This is an extension of the known result of Ng–Thurston [Grid diagrams, braids, and contact geometry, in Proc. Gökova Geometry-Topology Conf. 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 120–136] for nonsingular links and braids.

In this paper, we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side, we state the relations between classical and singular virtual objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove that there is a bijection between singular abstract braids, singular horizontal Gauss diagrams up to a certain equivalence relation, and singular virtual braids, in particular using singular horizontal Gauss diagrams we obtain a presentation of the singular pure virtual braid monoid.