Framed Knotoids and Their Quantum Invariants

We modify the definition of spherical knotoids to include a framing, in analogy to framed knots, and define a further modification that includes a secondary 'coframing' to obtain 'biframed' knotoids. We exhibit topological spaces whose ambient isotopy classes are in one-to-one correspondence with framed and biframed knotoids respectively. We then show how framed and biframed knotoids allow us to generalize quantum knot invariants to a knotoid setting, leading to the construction of general Reshetikhin-Turaev type biframed knotoid invariants.