Oriented Quantum Algebras and Coalgebras, Invariants of Oriented 1-1 Tangles, Knots and Links

TL;DR

This paper introduces oriented quantum coalgebras and demonstrates their use in constructing invariants for oriented tangles, knots, and links, highlighting parallels with quantum coalgebra theory.

Contribution

It establishes the relationship between oriented quantum coalgebras and other algebraic structures and develops new isotopy invariants for knots and links based on these coalgebras.

Findings

Regular isotopy invariants for oriented tangles and knots are constructed.

Oriented quantum coalgebras are closely related to oriented quantum algebras.

The theory of oriented quantum coalgebras parallels that of quantum coalgebras.

Abstract

In this paper we study oriented quantum coalgebras which are structures closely related to oriented quantum algebras. We study the relationship between oriented quantum coalgebras and oriented quantum algebras and the relationship between oriented quantum coalgebras and quantum coalgebras. We show that there are regular isotopy invariants of oriented 1-1 tangles and of oriented knots and links associated to oriented and twist oriented quantum coalgebras respectively. There are many parallels between the theory of oriented quantum coalgebras and the theory of quantum coalgebras