The extension of the theory of links in \(\mathbb{S}^3\) to links embedded in closed, compact, oriented 3-manifolds \(\Sigma\) has been a topic of interest for knot theorists over the past few decades. To facilitate this, the concept of a mixed link has been created. A mixed link \(L\) consists of a fixed part representing the complement of \(\Sigma\) in \(\mathbb{S}^3\), and a moving part that represents \(L\). On the other hand, scientists from various fields have shown interest in generalizing the topological study of classical knot diagrams to open knotted curves, namely knotoids in \(\mathbb{S}^2\), as first introduced by V. Turaev in [Osaka J. Math. 49, No. 1, 195–223 (2012; Zbl 1271.57030)].
Motivated by the study of knotoids on the torus, the author combines these concepts and introduces the notion of mixed knotoids in \(\mathbb{S}^2\) in this paper. Specifically, a mixed knotoid is defined as a multi-knotoid, which is the union of a finite number of knot diagrams (here, the unknot represents the fixed part of the mixed knotoid, representing the complementary solid torus in \(\mathbb{S}^3\)) and the knotoid diagram (the moving part of the mixed knotoid). Moreover, as in the case of classical knots and knotoids, the author also defines the notion of mixed pseudo knotoids, which are mixed knotoids with missing crossing information. This concept follows the definitions of pseudo knots introduced by R. Hanaki [ibid. 47, No. 3, 863–883 (2010; Zbl 1219.57006)] and of pseudo knotoids introduced by the author in a previous paper [I. Diamantis, Mediterr. J. Math. 18, No. 5, Paper No. 201, 31 p. (2021; Zbl 1480.57004)], allowing for the study of pseudo knotoids on the torus.
Generalizing the classical Reidemeister theorem to mixed knotoids and mixed pseudo knotoids, the author presents the notion of isotopy for these two classes of knotoids on the torus. This allows an extension of the Kauffman bracket polynomial, particularly by viewing the torus as a punctured disk. Furthermore, this polynomial is proven to be an invariant for both cases, and an equivalent definition of the bracket polynomial is also presented in the form of a state-sum formula. Here, a state of a mixed (pseudo) knotoid is defined as a corresponding diagram where each crossing has been smoothed, resulting in a diagram with no crossings. The bracket polynomial of a mixed (pseudo) knotoid can thus be seen as a linear sum of corresponding trivial diagrams. It should be noted that for a mixed pseudo knotoid, orienting the diagram is necessary to define the Kauffman bracket polynomial.
As classical knots and links can be studied from the perspective of braid theory, the concept of braidoids has been developed by N. Gügümcü and S. Lambropoulou [Isr. J. Math. 242, No. 2, 955–995 (2021; Zbl 1470.57011)] to study knotoids. The author thus generalizes these concepts and presents the notions of mixed braidoids and mixed pseudo braidoids as counterparts to the theories of mixed knotoids and mixed pseudo knotoids, respectively. A mixed braidoid diagram on \(n\) strands is defined as a braidoid diagram formed by two disjoint sets of strands: a single fixed strand representing the complementary solid torus, and a set of moving strands representing the knotoid on the torus. This latter set consists of usual braid strands along with two free strands, as defined for classical braidoids. For mixed pseudo knotoids, the definition is similar, although some crossing information is missing. Then, moves that translate isotopy of mixed (pseudo) knotoids are introduced at the level of mixed (pseudo) braidoids. These moves enable the author to state and prove the analogue of the Alexander theorem, relating mixed (pseudo) braidoids to mixed (pseudo) knotoids. For both cases, this theorem can be proved using a similar strategy to that used for classical braidoids. A well-defined closure operation on mixed (pseudo) braidoids is also presented, which is akin to the one for mixed braids in handlebodies but differs in that forbidden moves (introduced to define isotopy of mixed (pseudo) knotoids) must be considered. In addition, specific attention is given to the pseudo case, where the missing crossings, termed pre-crossings, are defined up to isotopy such that they contain only down arcs, as done in the literature for virtual knots. Indeed, using this approach, the braidoid algorithm does not affect the pre-crossings. Finally, following the natural strategy adopted in braid theory, the author extends the notion of L-moves to mixed (pseudo) knotoids and demonstrates a geometric analogue of the Markov theorem for mixed braidoids.
Note that the author also highlights potential applications of the results of this paper that could be considered in the study of DNA and proteins in molecular biology, for example.