arXiv research

Constructs links that are both quasi-alternating and almost alternating.

Paper explores conditions for constructing new quasi-alternating links.

An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain alternating distances is arbitrarily large. We also show that two alternating distances, t…

New quasi-alternating links created from existing ones.

Classifies involutions on alternating prime non-split links.

A link is almost alternating if it is non-alternating and has a diagram that can be transformed into an alternating diagram via one crossing change. We give formulas for the first two and last two potential coefficients of the Jones polynomial of an almost alternating link. Using these formulas, we show that the Jones …

Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alter…

Odd primes act on alternating knots via flypes.

In this paper, we show that a link which has a positive and almost alternating diagram is alternating, besides that a positive and non-alternating Montesinos link has an almost positive-alternating diagram.

We construct an infinite family of quasi-alternating links from a given quasi-alternating link by replacing a crossing by a product of rational tangles each of which extends that crossing. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family contains all the classes of quasi-…

Study shows bounds on volumes of weakly generalised alternating knots.

Study of alternating links on nonorientable surfaces, extending results to nonorientable projections.

The study calculates and analyzes alternating surgeries for various knots.

Decomposes prime alternating links into prime tangles.

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as …

Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with an alternating rational tangle. Consequently, we show that many pretzel links are…

Extends classical results to virtual links, proving new properties of alternating and semi-alternating virtual links.

We prove that twisting any quasi-alternating link $L$ with no gaps in its Jones polynomial $V_L(t)$ at the crossing where it is quasi-alternating produces a link $L^{*}$ with no gaps in its Jones polynomial $V_{L^*}(t)$. This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other th…

Proves special alternating knots cannot be decomposed as non-trivial band sums.

Proves Girth Alternative for some HNN extensions, finds counterexamples.

The study examines quasi-alternating surgeries on knots and their properties.

New proof limits Jones polynomial values for quasi-alternating links.

Characterizes strongly quasipositive quasi-alternating and Montesinos links.

Explains alternating knots for a concise encyclopedia.

Introduces alternators for modeling sequences, outperforming baselines.

We extend Howie's characterization of alternating knots to give a topological characterization of toroidally alternating knots, which were defined by Adams. We provide necessary and sufficient conditions for a knot to be toroidally alternating. We also give a topological characterization of almost-alternating knots whi…

We prove that the degree of the Brandt-Lickorish-Millet polynomial of any quasi-alternating link is less than its determinant. Therefore, we obtain a new and a simple obstruction criterion for quasi-alternateness. As an application, we identify some knots of 12 crossings or less and some links of 9 crossings or less th…

It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module, we describe an alt…

We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alte…

We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagra…

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 alternating knots of n crossings, the set of all prime alternating knots of n+1 crossing…

Meridian lemma extended to fully alternating links in thickened surfaces.

This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (T …

Extended Thistlethwaite's result on Jones polynomials of quasi-alternating links.

Proves strong Tits alternative for 3D automorphisms over zero char fields.

Study bounds on cusp volumes of alternating knots on surfaces.

Improved clustering algorithm for large datasets.

New invariants derived from Seifert graphs help distinguish alternating links.

Generalizing Howie and Greene's characterization of alternating knots, we give a topological characterization of almost alternating knots.

Study sharpens unlinking number bounds for special alternating links.

The paper proves periodic projections of alternating knots using Flyping theorem.

The paper tabulates and computes the number of alternating pretzel links up to a given crossing number.

New findings on hyperbolicity of augmented links in thickened surfaces.

Algorithm finds ribbon disks for alternating knots, resolving sliceness for most prime knots.

We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert surfaces for…

Automatic computation speeds up crosscap number calculation for alternating knots.

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.

New measure of knot complexity tied to twist number.