The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in three-dimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in three-dimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or self-intersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a

The size sz() of an `1-graph = (V; E) is the minimum of n f =tf over all the possible `1-embeddings f into n f -dimensional hypercube with scale t f . The sum of distances between all the pairs of vertices of is at most sz()dv=2ebv=2c (v = jV j). The latter is an equality if and only if is equicut graph, that is, admits an `1 -embedding f that for any 1 i n f satis es x2V f(x) i 2 fdv=2e; bv=2cg. Basic properties of equicut graphs are investigated. A construction of equicut graphs from `1-graphs via a natural doubling construction is given. It generalizes several well-known constructions of polytopes and distance-regular graphs. Finally, large families of examples, mostly related to polytopes and distance-regular graphs, are presented.

The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) with Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semi-regular maps which correspond to the so-called Archimedean solids. R#sum# La notion de fonction de Belyi est un outil technique qui relie la combinatoire des cartes (c'est-#-dire, des graphes plong#s sur des surfaces) avec la th#orie de Galois, la th#orie des nombres alg#briques et la th#orie des surfaces de Riemann. Dans cet article nous calculons les fonctions de Belyi pour une classe des cartes semi-reguli#res, correspondant # ce qu'on appelle les solides d'Archim#de. 1 Introduction The title of the present paper attempts to link together traditional and contemporary mathematics. The name of Archimedes represents tradition; that of Belyi, one of the most recent advances in Galois theory, known (even i...

A fullerene F n is a 3-regular (or cubic) polyhedral carbon molecule for which the n vertices - the carbons atoms - are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onion-like metallic clusters and geodesic domes. Quasi-embeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...

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Abstract. A polytope is called regular faced if every one of its facets is a regular polytope. The 4-dimensional regular-faced polytopes were determined by [2, 3, 1]. The last class of such poytopes are the one obtained by removing a set of non-adjacent vertices of 600-cell. Here we determine all such independent sets up to isomorphism and find 314248344 polytopes. 1.