Lune -- Free Knot Graphs

Details Lune -- Free Knot Graphs Lune -- Free Knot Graphs

2005-06-18

arxiv.org/abs/math/0506371v5

Mathematics

Combinatorics

24 pages, 27 figures, LaTeX document

Scientific paper

This paper is an exploration of simple four-regular graphs in the plane (i.e. loopless and with no more than one edge between any two nodes). Such graphs are fundamental to the theory of knots and links in three dimensional space, and their planar diagrams. We dedicate this paper to Frank Harary (1921 -- 2005) whose fascination with graphs of knots inspired this work and with whom we had the pleasure of developing this paper. We prove that for v (the number of nodes) greater than or equal to 8 there always exist such knot-graphs. Our proof gives rise to a class of "tight" graphs that do not submit to the recursive procedure that generates graphs with v+2 nodes from "admissible" graphs with v nodes. We then classify such tight graphs.

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