Related papers: On positive and almost alternating links
This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory.
We discuss relations among various positivities of knots and links, such as strong quasipositivity and quasipositivity. We give several pieces of supporting evidence for conjectural statements concerning these positivities and the defect of…
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to…
A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives…
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…
A link diagram can be considered as a $4$-valent graph embedded in the $2$-sphere and divides the sphere into complementary regions. In this paper, we show that any link has a diagram with only triangles and quadrilaterals. This extends…
We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.
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…
It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due…
Menasco showed that a closed surface in the complement of a non-split prime alternating link in $S^3$ contains a circle isotopic in the link complement to a meridian of the links. This result is known as the meridian lemma for alternating…
We say that a link $L_1$ is an s-major of a link $L_2$ if any diagram of $L_1$ can be transformed into a diagram of $L_2$ by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime…
We show that one can interweave an unknot into any non-alternating connected projection of a link so that the resulting augmented projection is alternating.
The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram…
Generalizing Howie and Greene's characterization of alternating knots, we give a topological characterization of almost alternating knots.
Every link diagram can be represented as a signed ribbon graph. However, different link diagrams can be represented by the same ribbon graphs. We determine how checkerboard colourable diagrams of links in real projective space, and virtual…
We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot.
A plane graph $H$ is a {\em plane minor} of a plane graph $G$ if there is a sequence of vertex and edge deletions, and edge contractions performed on the plane, that takes $G$ to $H$. Motivated by knot theory problems, it has been asked if…
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.
Qazaqzeh and Chbili showed that for any quasi-alternating link, the degree of $Q$-polynomial is less than its determinant. We give a refinement of their evaluation.
Given a prime, alternating link diagram, we build a special cover of the link complement whose degree is bounded by a factorial function of the crossing number. It follows that a subgroup of the link group of that index embeds into…