In this paper, we extend the definition of a knotoid to multi-linkoids that consist of a finite number of knot and knotoid components. We study invariants of multi-linkoids, such as the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the T-invariant in relation with generalized $$\Theta $$Θ -graphs.

In this paper, we study knotoids with extra graphical structure (bonded knotoids) in the settings of rigid vertex and topological vertex graphs. We construct bonded knotoid invariants by applying tangle insertion and unplugging at bonding sites of a bonded knotoid. We demonstrate that our invariants can be used for the analysis of the topological structure of proteins.

In this paper, we extend the definition of a knotoid that was introduced by Turaev, to multi-linkoids that consist of a number of knot and knotoid components. We study invariants of multi-linkoids that lie in a closed orientable surface, namely the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the $T$-invariant in relation with generalized $\Theta$-graphs.

In this paper, we introduce biquandle power brackets, an infinite family of invariants of oriented links containing the classical skein invariants and the quandle and biquandle 2-cocycle invariants as special cases. Biquandle power brackets are generalizations of biquandle brackets in which the values of Kauffman states also depend on the biquandle colors they admit. We provide example computations and discuss the relationship between these new invariants and the previous cases.