Related papers: Grid Diagrams, Braids, and Contact Geometry
We compute the treewidth of a family of graphs we refer to as the glued grids, consisting of the stacked prism graphs and the toroidal grids. Our main technique is constructing strict brambles of large orders. We discuss connections to…
We define a new kind of Gauss diagrams to describe knots in the solid torus with projections in the annulus. We see that it provides an efficient tool for showing that a knot diagram can be fully recovered from its decorated Gauss diagram,…
By defining grids as graphs, geometric graphs can be represented in a very concise way.
In this series we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids. In…
We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot detection. We also explain how grid diagrams can be…
We partially determine grid homology (combinatorial knot Floer homology) of diagonal knots, which are conjectured to be equivalent to positive braid knots, by exploiting nice grid diagrams. Its next-to-top term detects the number of prime…
We prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all with the same self-linking number. In some cases, we can even classify all such knots. We also show…
We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We…
In this paper we study Legendrian knots in the knot types of satellite knots. In particular, we classify Legendrian Whitehead patterns and learn a great deal about Legendrian braided patterns. We also show how the classification of…
We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present an explicit combinatorial formulation for these invariants. The algebras conjecturally give the relative…
We define a graph encoding the structure of contact surgery on contact 3-manifolds and analyze its basic properties and some of its interesting subgraphs.
We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up…
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…
We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coefficients for moving…
We construct a map from knots to (abstract) 2-knots which can be extended to higher dimensions; this map is the natural "knot" counterpart for "braid" theory of groups $G_{n}^{k}$.
Links in $S^3$ can be encoded by grid diagrams; a grid diagram is a collection of points on a toroidal grid such that each row and column of the grid contains exactly two points. Grid diagrams can be reinterpreted as front projections of…
Chord diagrams and related enlacement graphs of alternating knots are enhanced to obtain complete invariant graphs including chirality detection. Moreover, the equivalence by common enlacement graph is specified and the neighborhood graph…
In this short note we highlight some of the differences between cube diagrams and grid diagrams. We also list examples of small cube diagrams for all knots up to 7 crossings and give some examples of links.
Directed graphs are widely used in modelling of nonsymmetric relations in various sciences and engineering disciplines. We discuss invariants of strongly connected directed graphs - minimal number of vertices or edges necessary to remove to…
This paper explores the problem of unknotting closed braids and classical knots in mathematical knot theory. We apply evolutionary computation methods to learn sequences of moves that simplify knot diagrams, and show that this can be…