Two applications of elementary knot theory to Lie algebras and Vassiliev invariants

Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [Bar-Natan, Garoufalidis, Rozansky and Thurston, arXiv:q-alg/9703025] and [Deligne, letter to Bar-Natan, January 1996, http://www.ma.huji.ac.il/~drorbn/Deligne/], which give, respectively, the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its intertwining property is analogous to the computation of 1+1=2 on an abacus. The Wheels conjecture is proved from the fact that the k-fold connected cover of the unknot is the unknot for all k. Along the way, we find a formula for the invariant of the general (k,l) cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo-Kirillov map S(g)-->U(g) for metrized Lie (super-)algebras g.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper1.abs.htm

Antimagic Labelings of Weighted and Oriented Graphs

A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ for all $e\in E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $\lfloor{\frac{4n}{3}}\rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into $\{1,\dotsc,|E(G)|+k\}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $\lfloor{\frac{2n}{3}}\rfloor$-oriented-antimagic.Comment: 10 pages, 1 figur

Words on Wheels

A bicycle could help end book deserts in the Miami Valley. The Words on Wheels program will use a bicycle customized by University of Dayton engineering students to distribute reading materials to local children in areas where access to a library is limited. “We want to create and sustain a literary oasis,” said Karlos Marshall, a University of Dayton alumnus who founded the nonprofit The Conscious Connect in 2015 to end book deserts in Ohio’s urban cities by 2021

Exploring the magnetic properties of the largest single molecule magnets

The giant {Mn₇₀} and {Mn₈₄} wheels are the largest nuclearity single-molecule magnets synthesized to date, and understanding their magnetic properties poses a challenge to theory. Starting from first-principles calculations, we explore the magnetic properties and excitations in these wheels using effective spin Hamiltonians. We find that the unusual geometry of the superexchange pathways leads to weakly coupled {Mn₇} subunits carrying an effective S = 2 spin. The spectrum exhibits a hierarchy of energy scales and massive degeneracies, with the lowest-energy excitations arising from Heisenberg-ring-like excitations of the {Mn₇} subunits around the wheel. We further describe how weak longer-range couplings can select the precise spin ground-state of the Mn wheels out of the nearly degenerate ground-state band

Wandering Wheels Newsletter, December 2004

https://pillars.taylor.edu/wanderingwheels-newsletter/1014/thumbnail.jp

Design of an add-on device for transform a standard wheelchair on an affordable and motorized

[EN] In the following article a design to adapt a low-cost power unit to a conventional wheelchair is presented. The adaptation is carried out looking for a simple, low cost solution that does not involve any modification or structural alteration of the wheelchair itself. In this work we present a proposal of a mechanical adaptation that will allow connecting a self-balancing scooter to a wheelchair and be controlled by the user as well as by another person. So that the pushing effort is eliminated.Prusas, B.; Mansoor, K.; Engelhardt, L.; Walgers, B.; Pirtilä, SI.; Lukoschek, L.; Defez Garcia, B.... (2020). Design of an add-on device for transform a standard wheelchair on an affordable and motorized. Editorial Universitat Politècnica de València. 371-380. https://doi.org/10.4995/INN2019.2019.10226OCS37138

Uniform cost accounting system

An examination of the cost accounting systems of the members of the Association shows that a variety of widely different methods of cost accounting are used to ascertain the cost of producing car wheels. This is especially true with respect to methods of handling old wheels received through exchange contracts; to methods of distributing general plant expense and administrative and general overhead expense where products in addition to car wheels are manufactured; to methods of handling depreciation; failed wheels; replacements of pattern and flask equipment; and to methods of classifying cost information

Wheels on Wheels on Wheels-Surprising Symmetry

While designing a computer laboratory exercise for my calculus students, I happened to sketch the curve defined by this vector equation: (x, y) = (cos(t), sin(t)) + 1/2(cos(7t), sin(7t)) + 1/3(sin(17t), cos(17t)). I was thinking of the curve traced by a particle on a wheel mounted on a wheel mounted on a wheel, each turning at a different rate. The first term represents the largest wheel, of radius 1, turning counter-clockwise at one radian per second. The second term represents a smaller wheel centered at the edge of the first, turning 7 times as fast. The third term is for the smallest wheel centered at the edge of the second, turning 17 times as fast as the first, clockwise and out of phase