Pseudo links and singular links in the Solid Torus

In this paper we introduce and study the theories of pseudo links singular in Solid Torus, ST. Pseudo are with some missing crossing information that naturally generalize notion knot diagrams, have potential use molecular biology, while contain a finite number self-intersections. We consider ST set up appropriate topological theory order to construct invariants for these types particular, formulate prove analogue Alexander theorem then mixed braid monoid monoid, which, Markov \smallbreak Moreover, Hecke algebra type A, $P\mathcal{H}_n$, cyclotomic generalized algebras B, $P\mathcal{H}_{1, n}$, discuss how (cor. monoid) can be represented by $P\mathcal{H}_{n}$ n}$). This is first step toward construction HOMFLYPT-type $S^3$ also $S\mathcal{H}_{1, present two sets conjecture they form linear bases n}$. Finally, bracket polynomial