Topology and Geometry in Polymer Science pp 101-133 | Cite as

# Knots in Graphs in Subsets of *Z*^{3}

## Abstract

The probability that an embedding of a graph in *Z* ^{3} is knotted is investigated. For any given graph (embeddable in *Z* ^{3}) without cut edges, it is shown that this probability approaches 1 at an exponential rate as the number of edges in the embedding goes to infinity. Furthermore, at least for a subset of these graphs, the rate at which the probability approaches 1 does not depend on the particular graph being embedded. Results analogous to these are proved to be true for embeddings of graphs in a subset of *Z* ^{3} bounded by two parallel planes (a slab).

In order to investigate the knotting probability of embeddings of graphs in a rectangular prism (an infinitely long rectangular tube in *Z* ^{3}), a pattern theorem for self-avoiding polygons in a prism is proved. From this it is possible to prove that for any given eulerian graph, the probability that an embedding of the graph in a prism is knotted goes to 1 as the number of edges in the embedding goes to infinity. Then, just as for *Z* ^{3}, there is at least a subset of these graphs for which the rate that this probability approaches 1 does not depend on the particular graph.

Similar results are shown to hold in cases where restrictions are placed on the number of edges per branch in a graph embedding.

## Keywords

knots graph embeddings branched polymer simple cubic lattice## Preview

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