Related papers: Strongly-cyclic branched coverings of knots via $(…
We show that complex symplectic structures need not be preserved under small deformations, and we find sufficient conditions for this to happen. We study various cohomologies of compact complex symplectic manifolds, obtaining some…
We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored Jones polynomials to the topology of…
We discuss geometric properties of covers of closed hyperbolic manifolds of dimension $n\geq 3$, branched along a totally geodesic codimension two submanifold $\Sigma$. The results are mostly known to the experts but hard to find in the…
For a closed 3-manifold $M$ in a certain class, we give a presentation of the cellular chain complex of the universal cover of $M$. The class includes all surface bundles, some surgeries of knots in $S^3$, some cyclic branched cover of…
We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural…
We present a systematic classification of uncolored bonded knots with singularity number at most seven. Bonded knots provide a topological model for closed protein chains with intramolecular bridges, such as disulfide bonds. Following the…
Optical knots and links have attracted great attention because of their exotic topological characteristics. Recent investigations have shown that the information encoding based on optical knots could possess robust features against external…
There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every…
We explore the possibility of applying the framework of frequent pattern mining to a class of continuous objects appearing in nature, namely knots. We introduce the frequent knot mining problem and present a solution. The key observation is…
We describe the genus two knots which admit a genus one, one bridge position. These are divided into several families, one consists of vertical bandings of two genus one $(1,1)$-knots, other consists of vertical bandings of two cross cap…
This work is concerned with the calculation of the fundamental group of torus knots. Torus knots are special types of knots which wind around a torus a number of times in the longitudinal and meridional directions. We compute and describe…
A knot k is called ``strongly (n-1)-trivial.'' if there exists a projection of k, such that one can choose n crossings of the projection with the property that making the crossing changes corresponding to any of the $2^{n}-1$ nontrivial…
We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly…
We describe a new method of producing equations for the canonical component of representation variety of a knot group into $PSL_2(\mathbb{C})$. Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of…
Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have…
We consider collections of disjoint simple closed curves in a compact orientable surface which decompose the surface into pairs of pants. The isotopy classes of such curve systems form the vertices of a 2-complex, whose edges correspond to…
Knotted line defects in continuous fields entrain a complex arrangement of the material sur- rounding them. Recent experimental realisations in optics, fluids and nematic liquid crystals make it important to fully characterise these…
Whitehead doubles provide a plethora of examples of knots that are topologically slice but not smoothly slice. We discuss the problem of the Whitehead double of the Figure 8 knot and survey commonly used techniques to obstructing sliceness.…
Scaling arguments are used to analyze the size of topologically constrained closed ring polymer with excluded volume. It is found that there exists a finite range of polymer thickness (excluded volume) in which self-avoidance is unimportant…
In this paper, we study a special family of $(1,1)$ knots called constrained knots, which includes 2-bridge knots in the 3-sphere $S^3$ and simple knots in lens spaces. Constrained knots are parameterized by five integers and characterized…