17 pages, 22 figures. Compared to the previous file, some remarks and a figure have been added. To appear in the Bulletin of the Hellenic Mathematical Society. arXiv admin note: substantial text overlap with arXiv:2101.03538
In this paper we introduce and study the theory of pseudo links in the genus $g$ handlebody, $H_g$. Pseudo links are links with some missing crossing information that naturally generalize the notion of knot diagrams. The motivation for studying these relatively new knotted objects lies in the fact that pseudo links may be used to model DNA knots, since it is not uncommon for biologists to obtain DNA knots for which it is not possible to tell a positive from a negative crossing. We consider pseudo links in $H_g$ as mixed pseudo links in $S^3$ and we generalize the Kauffman bracket polynomial for the category of pseudo links in $H_g$. We then pass on the category of mixed pseudo braids, that is, pseudo braids whose closures are pseudo links in $H_g$, and we formulate the analogue of the Alexander theorem. It is worth mentioning that the theory of pseudo links is close related to the theory of singular links and that all results in this paper may be used for studying singular links in $H_g$.
We consider knots and links in handlebodies that have hyperbolic complements and operations akin to composition. Cutting the complements of two such open along separating twice-punctured disks such that each of the four resulting handlebodies has positive genus, and gluing a pair of pieces together along the twice-punctured disks in their boundaries, we show the result is also hyperbolic. This should be contrasted with composition of any pair of knots in the 3-sphere, which is never hyperbolic. Similar results are obtained when both twice-punctured disks are in the same handlebody and we glue …
We consider oriented knots and links in a handlebody of genus $g$ through appropriate braid representatives in $S^3$, which are elements of the braid groups $B_{g,n}$. We prove a geometric version of the Markov theorem for braid equivalence in the handlebody, which is based on the $L$-moves. Using this we then prove two algebraic versions of the Markov theorem. The first one uses the $L$-moves. The second one uses the Markov moves and conjugation in the groups $B_{g,n}$. We show that not all conjugations correspond to isotopies.
A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalizes the notion of links in classical knot theory. In this paper, we consider handlebody links with one genus 2 handlebody and $n-1$ solid tori, $n>1$. Our main result is the complete classification of such handlebody links with six crossings or less, up to ambient isotopy.
In this paper we study the theory of {\it pseudo knots}, which are knots with some missing crossing information, and we introduce and study the theory of {\it pseudo tied links} and the theory of {\it pseudo knotoids}. In particular, we first present a braiding algorithm for pseudo knots and we then introduce the $L$-moves in that setting, with the use of which we formulate a sharpened version of the analogue of the Markov theorem for pseudo braids. Then we introduce and study the theory of {\it tied pseudo links}, that generalize the notion of tied links, and we exploit the relation between t…
Pseudo links have two crossing types: classical crossings and indeterminate crossings. They were first introduced by Ryo Hanaki as a possible tool for analyzing images produced by electron microscopy of DNA. A normalized bracket polynomial is defined for pseudo links and then used to construct and obstruction to cosmetic crossings in classical links.
We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to the boundary, but only in a concave fashion, and homotopy should preserve tangency type; (2) framed links L in M, up to framed isotopy; (3) triples (M,v,L), with (M,v) as above and L transversal to v, up to pseudo-Legendrian isotopy (transversality-preserving simultaneous homotopy…
We generalize the notion of biquandles to psyquandles and use these to define invariants of oriented singular links and pseudolinks. In addition to psyquandle counting invariants, we introduce Alexander psyquandles and corresponding invariants such as Alexander psyquandle polynomials and Alexander-Gröbner psyquandle invariants of oriented singular knots and links. We consider the relationship between Alexander psyquandle colorings of pseudolinks and p-colorings of pseudolinks. As a special case we define a generalization of the Alexander polynomial for oriented singular links and pseudolinks w…
We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescribed cone angles at all edges is (if non-empty) a smooth complex manifold of dimension the sum of the genera of the vertex links. Moreover, we show that the complex lengths of a collection of peripheral elements give a local holomorphic parameterisation of this manifold.
Pseudodiagrams are diagrams of knots where some information about which strand goes over/under at certain crossings may be missing. Pseudoknots are equivalence classes of pseudodiagrams, with equivalence defined by a class of Reidemeister-type moves. In this paper, we introduce two natural extensions of classical knot colorability to this broader class of knot-like objects. We use these definitions to define the determinant of a pseudoknot (i.e. the pseudodeterminant) that agrees with the classical determinant for classical knots. Moreover, we extend Conway notation to pseudoknots to facilitat…
Let $δ_g$ be the minimal dilatation for pseudo-Anosovs on a closed surface $Σ_g$ of genus $g$ and let $δ_g^+$ be the minimal dilatation for pseudo-Anosovs on $Σ_g$ with orientable invariant foliations. This paper concerns the pseudo-Anosovs which occur as the monodromies on closed fibers for Dehn fillings of $N(r)$ for each $r \in \{-3/2, -1/2, 2\}$ of the magic manifold $N$. The manifold $N(-3/2)$ is homeomorphic to the Whitehead sister link exterior. We consider the set $Λ_g(r)$ (resp. $Λ_g^+(r)$) which consists of the dilatations of all monodromies (resp. monodromies having orientable invar…
A two-component link produces a torus as the product of the component knots in a two-point configuration space of a three-sphere. This space can be identified with a cotangent bundle and also with an indefinite Grassmannian. We show that the integration of the absolute value of the canonical symplectic form is equal to the area of the torus with respect to the pseudo-Riemannian structure, and that it attains the minimum only at the "best" Hopf links.
This paper introduces even triangulations of n-dimensional pseudo-manifolds and links their combinatorics to the topology of the pseudo-manifolds. This is done via normal hypersurface theory and the study of certain symmetric representation. In dimension 3, necessary and sufficient conditions for the existence of even triangulations having one or two vertices are given. For Haken n-manifolds, an interesting connection between very short hierarchies and even triangulations is observed.
For n >1, if the Seifert form of a knotted 2n-1 sphere K in S^{2n+1} has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three (n = 1). However, in the three dimensional case it is true that if the metabolizer has a basis represented by a strongly slice link then K is slice. The question has been asked as to whether it is sufficient that each basis element is represented by a slice knot to assure that K is slice. For genus one knots this is of course true; here we present a genus two counterexample.
By use of a variety of techniques (most based on constructions of quasipositive knots and links, some old and others new), many smooth 3-manifolds are realized as transverse intersections of complex surfaces in complex 3-space with strictly pseudoconvex 5-spheres. These manifolds not only inherit interesting intrinsic structures (eg, they have canonical Stein-fillable contact structures), they also have extrinsic structures of a knot-theoretical nature (eq, the 3-sphere arises in infinitely many distinct ways). This survey is not comprehensive; a number of questions are left open for future wo…
