Almost unknotted embeddings of graphs and surfaces(Knots and soft-matter physics: Topology of polymers and related topics in physics, mathematics and biology)

Abstract

We consider the number of embeddings of almost unknotted Θ_k-graphs, 3≤k≤6, in the simple cubic lattice Z^3. We show that to exponential order this number is the same as the number of unknotted Θ_k-graphs. This implies that almost unknotted Θ_k-graphs are exponentially rare in the set of embeddings of Θ_k-graphs. We construct almost unknotted surfaces in Z^4 by spinning and show that to exponential order the numbers of almost unknotted spun Θ_k are equal to the numbers of unknotted spun Θ_k, 4≤k≤6. The case of k=3 is open.