arXiv subjects

Ryan Maguire

Publications and source records attributed to Ryan Maguire.

3 recordsLinked to original sources

arXiv2024

Affine linking number estimates for the number of times an observer sees a star

Affine linking numbers are the generalization of linking numbers to the case of nonzero homologous linked submanifolds. They were introduced by Rudyak and the first author who used them to study causality in globally hyperbolic spacetimes. In this paper we use affine linking numbers to estimate the number of times an observer sees light from a star, that is how many copies of the star do they see on the sky due to gravitational lensing.

gr-qc↗

arXiv2022

Conjectures on the Khovanov Homology of Torus Knots, Twist Knots, and Legendrian Simple Knots

A theorem of Kronheimer and Mrowka states that Khovanov homology is able to detect the unknot. That is, if a knot has the Khovanov homology of the unknot, then it is equivalent to it. Similar results hold for the trefoils and the figure-eight knot. We conjecture that Khovanov homology is able to distinguish all torus and twist knots. Numerical evidence has been gathered by examining all prime knots with 20 or fewer crossings, a total of 2,199,471,680 knots (not including mirrors). We found that all knots with the same Khovanov polynomial (the Poincar\'{e} polynomial of Khovanov homology) as a torus or twist knot are indeed torus or twist knots themselves. Since torus knots are known to be Legendrian simple, and since all twist knots $K_{m}$ with $m\geq{-3}$ are Legendrian simple, this provides evidence for the claim that Khovanov homology and Legendrian simplicity may be connected. We conjecture that indeed Khovanov homology is able to distinguish Legendrian simple knots and use the (conjectured) Legendrian simple knots from the Legendrian knot atlas to test this claim. A similar observation was made, and no knots with 20 or fewer crossing share their Khovanov polynomial with the knots in the Legendrian knot atlas (except for the knots that are a part of this atlas).

math.GT↗

arXiv2019

Properties of an infinite dimensional Banach space over the field with two elements

A banach space X is a normed vector space, which is complete with respect to the metric induced by the norm. Given a bounded linear operator T acting on a banach space X, T is said to attain its norm if there is a unit vector z in X, such that the norm of Tz equals the norm of T. The existence of an infinite dimensional banach space X, in which each bounded linear operator acting on X attains its norm, is still undetermined. This question was posed by M.I. Ostrovskii at St. John's University. In this paper we show that if an infinite dimensional banach space is considered over GF(2), then it is possible for every bounded linear operator to attain its norm.

math.FA↗