Related papers: Mirror-Curves and Knot Mosaics
Mirror symmetry has proven to be a powerful tool to study several properties of higher dimensional superconformal field theories upon compactification to three dimensions. We propose a quiver description for the mirror theories of the…
In this paper, we research the grid homology for spatial graphs with cut edges. We show that the grid homology for spatial graph $f$ is trivial if $f$ has sinks, sources, or cut edges. As an application, we give purely combinatorial proofs…
We propose a mirror derivation of the quiver description of open topological strings known as the knots-quivers correspondence, based on enumerative invariants of augmentation curves encoded by exponential networks. Quivers are obtained by…
We develop techniques for computing the equivariant local mirror symmetry of curves, i.e. mirror symmetry for O(k)+O(-2-k) over P^1 for k greater than 0. We also describe related methods for dealing with mirror symmetry of non-nef toric…
Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to…
Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual…
Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot $n$-mosaic is an $n \times n$ matrix of 11 kinds of specific…
Virtual knots are defined diagrammatically as a collection of figures, called virtual knot diagrams, that are considered equivalent up to finite sequences of extended Reidemeister moves. By contrast, knots in $\mathbb{R}^3$ can be defined…
We study in detail mirror symmetry for the quartic K3 surface in P3 and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison, mirror symmetry for K3 surfaces can be entirely described in terms of…
Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole…
This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a…
Grid diagrams with their relatively simple mathematical formalism provide a convenient way to generate and model projections of various knots. It has been an open question whether these 2D diagrams can be used to model a complex 3D process…
A virtual knot is an equivalence class of embeddings of $ S^1 $ into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined…
Borcea-Voisin threefolds provided some of the first examples of mirror pairs in the Hodge-theoretic sense, but their mirror symmetry at the quantum level have not previously been shown. We prove a Givental-style quantum mirror theorem for…
To every knot (or link) diagram K, we associate a cluster algebra A that contains a cluster x with the property that every cluster variable in x specializes to the Alexander polynomial of K. We call x the knot cluster of A. Furthermore,…
Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly…
We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary…
We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each…
To a nullhomologous knot $K$ in a 3-manifold $Y$, knot Floer homology associates a bigraded chain complex over $\mathbb{F}[U,V]$ as well as a collection of flip maps; we show that this data can be interpretted as a collection of decorated…
The concept of a knot mosaic was introduced by Lomonaco and Kauffman as a means to construct a quantum knot system. The mosaic number of a given knot $K$ is defined as the minimum integer $n$ that allows the representation of $K$ on an $n…