Related papers: Generalized augmented alternating links and hyperb…
Menasco proved that nontrivial links in the 3-sphere with connected prime alternating non-2-braid projections are hyperbolic. This was further extended to augmented alternating links wherein non-isotopic trivial components bounding disks…
A generalized augmented link of a knot $K$ is a link obtained by adding trivial components to $K$ that bound $n$-punctured disks. In this paper we consider that $K$ is given by a positive braid with at least one full twist. We characterize…
To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these…
If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume…
We give a refined upper bound for the hyperbolic volume of an alternating link in terms of the first three and the last three coefficients of its colored Jones polynomial.
The hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalize this to the simplicial volume of link…
An alternative method is described for determining the hyperbolic structure on a link complement, and some of its elementary consequences are examined. The method is particularly suited to alternating links.
Classical fully augmented links have explicit hyperbolic geometry, and have diagrams on the 2-sphere in the 3-sphere. We generalise to construct fully augmented links projected to the reflection surface of any 3-manifold obtained by…
It has been known for several decades that classical alternating links in the 3-sphere have nice hyperbolic geometric properties. Recent work generalises such results to give hyperbolic geometry of links with alternating projections onto…
We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial.
We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces.…
In this paper, we study the generalized volume conjecture for the colored Jones polynomials of links with complements containing more than one hyperbolic piece. First of all, we construct an infinite family of prime links by considering the…
In Guts, Volume and Skein Modules of 3-Manifolds (arXiv:2010.06559), we showed that the twist number of certain hyperbolic weakly generalized alternating links can be recovered from a Jones-like polynomial, and offers a lower bound for the…
By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges…
We present explicit geometric decompositions of the hyperbolic complements of alternating $k$-uniform tiling links, which are alternating links whose projection graphs are $k$-uniform tilings of $S^2$, $\mathbb{E}^2$, or $\mathbb{H}^2$. A…
We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman…
We show that the volumes of certain hyperbolic A-adequate links can be bounded (above and) below in terms of two diagrammatic quantities: the twist number and the number of certain alternating tangles in an A-adequate diagram. We then…
We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating…
A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a…
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.