Related papers: A new proof that alternating links are non-trivial

In this paper, we introduce a novel generalization of the classical property of algebras known as "being alternative," which we term "partially alternative." This new concept broadens the scope of alternative algebras, offering a fresh…

We present a concise proof for the supporting hyperplane theorem. We then observe that the proof not only establishes the supporting hyperplane theorem but also extends it to a hyperplane separation theorem for certain non-convex sets. The…

The group of piecewise projective homeomorphisms of the line provides straightforward counter-examples to the so-called von Neumann conjecture. The examples are so simple that many additional properties can be established.

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining…

In an attempt to get some information on the multiplicative structure of the Green ring we study algebraic modules for simple groups, and associated groups such as quasisimple and almost-simple groups. We prove that, for almost all groups…

The core group is an invariant of unoriented virtual links. We introduce a peripheral structure for the core group, in which the longitudes are sensitive to orientations. We show that the combination of the core group and its peripheral…

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.…

This is an expository paper in which we explain how basic, standard, results about simple Lie algebras can be obtained by geometric arguments, following ideas of Cartan, Richardson and others.

We revisit Akbulut and King's first example of a compact semialgebraic set which satisfies Sullivan's local Euler characteristic condition, but which is not homeomorphic to an algebraic set. A nontrivial obstruction is computed using the…

A theorem of alternatives provides a reduction of validity in a substructural logic to validity in its multiplicative fragment. Notable examples include a theorem of Arnon Avron that reduces the validity of a disjunction of multiplicative…

A new classification theorem for links by the authors and Roger Fenn leads to computable link invariants. As an illustration we distinguish the left and right trefoils and recover the result of Carter et al that the 2-twist-spun trefoil is…

The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with…

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the…

It is known that alternative links are pseudoalternating. In 1983 Louis Kauffman conjectured that both classes are identical. In this paper we prove that Kauffman Conjecture holds for those links whose first Betti number is at most 2.…

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…

A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus,…

An isomorphism between two hermitian unitals is proved, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).

Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in $\reals^3$, bringing techniques from topological knot theory to bear on their transversal…

We prove that the vector bundles at the core of the Knizhnik-Zamolodchikov and quantum constructions of braid groups representations are topologically trivial bundles. We provide partial generalizations of this result to generalized braid…

While paradoxical linkages famously violate the Chebyshev-Grubler-Kutzbach criterion by exhibiting unexpected mobility, we identify an opposing phenomenon: a class of linkages that appear mobile according to the same criterion, yet are in…