Framed Knotoids and Their Quantum Invariants

TL;DR

This paper introduces framed and biframed knotoids, extending the concept of knotoids with additional structure, and develops quantum invariants for these new objects, generalizing existing knot invariants to a broader setting.

Contribution

It defines framed and biframed knotoids, establishes their topological classification, and constructs quantum invariants analogous to Reshetikhin-Turaev invariants for these structures.

Findings

Topological spaces classified by framed and biframed knotoids

Construction of Reshetikhin-Turaev type invariants for biframed knotoids

Extension of quantum knot invariants to knotoid setting

Abstract

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.