Toroidal embeddings of abstractly planar graphs are knotted or linked

Abstract

We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus T 2 ⊂ R 3 and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not embed on the torus. Our results provide general insight into properties of molecules that are synthesized on a torus.

Deque Languages, Automata and Planar Graphs

“…2. Evidence for similar patterns occurs in the linear representation of some models for chemical and biological molecular structures (e.g., [4,13]). It is known that some of those patterns also occur in natural languages and are to some extent captured by existing grammar models, in particular by dependency grammars.…”

“…deque graph, i.e., the vertices of the graph can be processed according to a linear order and the edges correspond to items in the deque inserted and removed at their end vertices". Similar more complex embeddings of planar graphs on a cylinder are considered in natural sciences, e.g., in [4,13] for chemistry and for RNA.…”

On chirality of toroidal embeddings of polyhedral graphs

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We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1], [2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3connected planar graphs in the standard torus are chiral.

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“…For the proof we rely on the fact that all planar toroidal spatial graphs contain a nontrivial knot or a nonsplit link: Theorem 1.3 (Existence of knots and links [1], [2]). Let G be an planar graph and f : G → R 3 be a nontrivial embedding of G with image G. If G is contained in the torus T 2 , it contains a subgraph which is a nontrivial knot or a nonsplit link.…”

Gauss diagram formulas of Vassiliev invariants of spatial 2-bouquet graphs

“…[2,3]). One of his motivation, in practical application of the Jones polynomial, is to solve two problems: (1) Divergence occurs when we evaluate polynomials; (2) Computational time is growing exponentially with respect to the number of crossings of link diagrams.…”