Related papers: Alternating links and definite surfaces

We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. The main theorem gives us a way to see if a given almost alternating link diagram represents a splittable link without increasing…

Traditionally, alternating links are studied with alternating diagrams on $S^2$ in $S^3$. In this paper, we consider links which are alternating on higher genus surfaces $S_g$ in $S_g \times I$. We define what it means for such a link to be…

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. The converse statement is easy…

Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a…

New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used…

We investigate the coefficients of the highest and lowest terms (also called the head and the tail) of the colored Jones polynomial and show that they stabilize for alternating links and for adequate links. To do this we apply techniques…

Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is…

We look at complete minimal surfaces of finite total curvature in $\mathbb{R}^4$. Similarly to the case of complex curves in $\mathbb{C}^2$ we introduce their {\it link at infinity}; we derive the {\it writhe number at infinity} which gives…

We study invariant Seifert surfaces for strongly invertible knots, and prove that the gap between the equivariant genus (the minimum of the genera of invariant Seifert surfaces) of a strongly invertible knot and the (usual) genus of the…

Tait's flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S^3. The proof of this version…

There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While…

A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain…

This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime…

Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper we introduce twisted intersection colorings of a diagram and construct two invariants of a…

Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. Permutation triangles are the alternating sign triangles whose entries are…

We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather…

We extend the Wirtinger number of links, an invariant originally defined by Blair, Kjuchukova, Velazquez, and Villanueva in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove…

In this chapter (Chapter V) we present several results which demonstrate a close connection and useful exchange of ideas between graph theory and knot theory. These disciplines were shown to be related from the time of Tait (if not Listing)…

We generalize Milnor link invariants to all types of surface-links in $4$--space (possibly with boundary). This is achieved by using the notion of cut-diagram, which is a 2-dimensional generalization of Gauss diagrams, associated to…

We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number.