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In this paper, we extend the theory of planar pseudo knots to the theories of annular and toroidal pseudo knots. Pseudo knots are defined as equivalence classes under Reidemeister-like moves of knot diagrams characterized by crossings with undefined over/under information. In the theories of annular and toroidal pseudo knots, we introduce their respective lifts to the solid and the thickened torus. Then, we interlink these theories by representing annular and toroidal pseudo knots as planar O -mixed and H -mixed pseudo links. We also explore the inclusion relations between planar, annular and toroidal pseudo knots, as well as of O -mixed and H -mixed pseudo links. Finally, we extend the planar weighted resolution set to annular and toroidal pseudo knots, defining new invariants for classifying pseudo knots and links in the solid and in the thickened torus. [ABSTRACT FROM AUTHOR]

In this paper, we present two different ways for computing the Kauffman bracket skein module of S 1 × S 2 , KBSM S 1 × S 2 , via braids. We first extend the universal Kauffman bracket type invariant V for knots and links in the Solid Torus ST, which is obtained via a unique Markov trace constructed on the generalized Temperley–Lieb algebra of type B, to an invariant for knots and links in S 1 × S 2 . We do that by imposing on V relations coming from the braid band moves. These moves reflect isotopy in S 1 × S 2 and they are similar to the second Kirby move. We obtain an infinite system of equations, a solution of which is equivalent to computing KBSM S 1 × S 2 . We show that KBSM S 1 × S 2 is not torsion free and that its free part is generated by the unknot (or the empty knot). We then present a diagrammatic method for computing KBSM S 1 × S 2 via braids. Using this diagrammatic method, we also obtain a closed formula for the torsion part of KBSM S 1 × S 2 . [ABSTRACT FROM AUTHOR]

Let F be a Morse–Bott foliation on the solid torus T = S 1 × D 2 into 2-tori parallel to the boundary and one singular central circle. Gluing two copies of T by some diffeomorphism between their boundaries, one gets a lens space L p , q with a Morse–Bott foliation F p , q obtained from F on each copy of T and thus consisting of two singular circles and parallel 2-tori. In the previous paper Khokliuk and Maksymenko (J Homotopy Relat Struct 18:313–356. https://doi.org/10.1007/s40062-023-00328-z, 2024) there were computed weak homotopy types of the groups D lp (F p , q) of leaf preserving (i.e. leaving invariant each leaf) diffeomorphisms of such foliations. In the present paper it is shown that the inclusion of these groups into the corresponding group D + fol (F p , q) of foliated (i.e. sending leaves to leaves) diffeomorphisms which do not interchange singular circles are homotopy equivalences. [ABSTRACT FROM AUTHOR]

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces L (p , q) , KBSM(L (p , q)), for q ≠ 0. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, TL 1 , n , which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, V , for knots and links in ST, via a unique Markov trace constructed on TL 1 , n . The universal invariant V is equivalent to the KBSM(ST). For passing now to the KBSM(L (p , q)), we impose on V relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in L (p , q) but not in ST, and which reflect the surgery description of L (p , q) , obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM(L (p , q)). We first present the solution for the case q = 1 , which corresponds to obtaining a new basis, ℬ p , for KBSM(L (p , 1)) with (⌊ p / 2 ⌋ + 1) elements. We note that the basis ℬ p is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case q > 1 , we first show how the new basis ℬ p of KBSM(L (p , 1)) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case q > 1. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements. [ABSTRACT FROM AUTHOR]

In this paper, we extend the theory of planar pseudo knots to the theories of annular and toroidal pseudo knots. Pseudo knots are defined as equivalence classes under Reidemeister-like moves of knot diagrams characterized by crossings with undefined over/under information. In the theories of annular and toroidal pseudo knots, we introduce their respective lifts to the solid and the thickened torus. Then, we interlink these theories by representing annular and toroidal pseudo knots as planar O-mixed and H-mixed pseudo links. We also explore the inclusion relations between planar, annular and toroidal pseudo knots, as well as of O-mixed and H-mixed pseudo links. Finally, we extend the planar weighted resolution set to annular and toroidal pseudo knots, defining new invariants for classifying pseudo knots and links in the solid and in the thickened torus.

In this paper, we present two different ways for computing the Kauffman bracket skein module of S1×S2, KBSMS1×S2, via braids. We first extend the universal Kauffman bracket type invariant V for knots and links in the Solid Torus ST, which is obtained via a unique Markov trace constructed on the generalized Temperley–Lieb algebra of type B, to an invariant for knots and links in S1×S2. We do that by imposing on V relations coming from the braid band moves. These moves reflect isotopy in S1×S2 and they are similar to the second Kirby move. We obtain an infinite system of equations, a solution of which is equivalent to computing KBSMS1×S2. We show that KBSMS1×S2 is not torsion free and that its free part is generated by the unknot (or the empty knot). We then present a diagrammatic method for computing KBSMS1×S2 via braids. Using this diagrammatic method, we also obtain a closed formula for the torsion part of KBSMS1×S2.

Let T = S 1 × D 2 be the solid torus, F the Morse–Bott foliation on T into 2-tori parallel to the boundary and one singular circle S 1 × 0 , which is the central circle of the torus T, and D (F , ∂ T) the group of diffeomorphisms of T fixed on ∂ T and leaving each leaf of the foliation F invariant. We prove that D (F , ∂ T) is contractible. Gluing two copies of T by some diffeomorphism between their boundaries, we will get a lens space L p , q with a Morse–Bott foliation F p , q obtained from F on each copy of T. We also compute the homotopy type of the group D (F p , q) of diffeomorphisms of L p , q leaving invariant each leaf of F p , q . [ABSTRACT FROM AUTHOR]

In this chapter, the vector analysis on compact Riemannian manifolds with boundary in the presence of symmetries, is presented, with the functions being invariant under the action of a compact subgroup G of the isometry group Is(M, g). In connection with the Sobolev inequalities in these manifolds, the calculation of their best constants is studied. [ABSTRACT FROM AUTHOR]

Tied links in S 3 were introduced by Aicardi and Juyumaya as standard links in S 3 equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces L (p , 1) , in handlebodies of genus g , and in the complement of the g -component unlink. We introduce the tied braid monoids T M g , n by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an L -move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology. [ABSTRACT FROM AUTHOR]

In this paper we present two new bases, B H 2 ′ and ℬ H 2 , for the Kauffman bracket skein module of the handlebody of genus 2 H 2 , KBSM (H 2). We start from the well-known Przytycki-basis of KBSM (H 2) , B H 2 , and using the technique of parting we present elements in B H 2 in open braid form. We define an ordering relation on an augmented set L consisting of monomials of all different "loopings" in H 2 , that contains the sets B H 2 , B H 2 ′ and ℬ H 2 as proper subsets. Using the Kauffman bracket skein relation we relate B H 2 to the sets B H 2 ′ and ℬ H 2 via a lower triangular infinite matrix with invertible elements in the diagonal. The basis B H 2 ′ is an intermediate step in order to reach at elements in ℬ H 2 that have no crossings on the level of braids, and in that sense, ℬ H 2 is a more natural basis of KBSM (H 2). Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of closed–connected–oriented (c.c.o.) 3-manifolds M that are obtained from H 2 by surgery, since isotopy moves in M are naturally described by elements in ℬ H 2 . [ABSTRACT FROM AUTHOR]

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces L (p , 1) from the HOMFLYPT skein module of the solid torus, 𝒮 (ST) , it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant X for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set Λ aug , augmenting the basis Λ of 𝒮 (ST). [ABSTRACT FROM AUTHOR]

The construction of the extensions of a 2-torus diffeomorphism up to volume preserving diffeomorphism of solid tori is proposed. The construction is illustrated on the example of extension of diffeomorphism of action on the torus. These extensions are used for the construction of kinematic dynamo examples onto solid torus. For the case of n -sphere a generation of results obtained before is proposed.

In this paper, we analyze the problem of determining the best constants for the Sobolev inequalities in the limiting case where p=1${p=1}$. Firstly, the special case of the solid torus is studied, whenever it is proved that the solid torus is an extremal domain with respect to the second best constant and totally optimal with respect to the best constants in the trace Sobolev inequality. Secondly, in the spirit of Andreu, Mazon and Rossi [3], a Neumann problem involving the 1-Laplace operator in the solid torus is solved. Finally, the existence of both best constants in the case of a manifold with boundary is studied, when they exist. Further examples are provided where they do not exist. The impact of symmetries which appear in the manifold is also discussed.