Related papers: Essential twisted surfaces in alternating link com…
State surfaces are spanning surfaces of links that are obtained from link diagrams guided by the combinatorics underlying Kauffman's construction of the Jones polynomial via state models. Geometric properties of such surfaces are often…
Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications…
We extend the Gordon-Litherland pairing to links in thickened surfaces, and use it to define signature, determinant, and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the…
We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of topologically embedded graphs is non-restrictive…
We interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an…
A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of the boundary of M. We derive a…
Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting…
We present a general method for predicting bond percolation thresholds and critical surfaces for a broad class of two-dimensional periodic lattices, reproducing many known exact results and providing excellent approximations for several…
Those maps of a closed surface to the three-dimensional torus that are homotopic to embeddings are characterized. Particular attention is paid to the somewhat intricate case when the surface is nonorientable.
Using a new tool called lassos, we establish a new correspondence between cellular link {diagrams} on closed surfaces and equivalence classes of virtual link {diagrams}. This is analogous to a well-known correspondence among the links…
We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to…
We study torsion properties of the twisted Alexander modules of the affine complement $M$ of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane…
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study…
Let $L$ be a prime alternating link with $n$ crossings. We show that for each fixed $g$, the number of genus $g$ incompressible surfaces in the complement of $L$ is bounded by a polynomial in $n$. Previous bounds were exponential in $n$.
Laundry surfaces for closed braid diagrams are presented. It is shown that braid diagrams are characterized by linking matrices obtained by lifting cycles from these surfaces. Oriented link types are then characterized by equivalence…
This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of…
Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such…
We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based…
In this paper we review the definitions of homogeneous and alternative links. We also give two new characterizations of an alternative link diagram, one within the context of the enhanced checkerboard graph and another from the labeled…
A cobordism between links in thickened surfaces consists of a surface $ S $ and a $3$-manifold $M $, with $ S $ properly embedded in $ M \times I $. We show that there exist links in thickened surfaces such that if $(S,M) $ is a cobordism…