For every natural number n, we exhibit a graph with the property that every embedding of it in M 3 contains a non-split n-component link. Furthermore, we prove that our graph is minor minimal in the sense that every minor of it has an embedding in M 3 that contains no non-split n-component link. 1.1991 Mathematics Subject Classification. 57M25, 57M15.

The applications of topological techniques for understanding molecular structures have become increasingly important over the past thirty years. In this topology text, the reader will learn about knot theory, 3-dimensional manifolds, and the topology of embedded graphs, while learning the role these play in understanding molecular structures. Most of the results that are described in the text are motivated by questions asked by chemists or molecular biologists, though the results themselves often go beyond answering the original question asked. There is no specific mathematical or chemical prerequisite; all the relevant background is provided. The text is enhanced by nearly 200 illustrations and more than 100 exercises. Reading this fascinating book, undergraduate mathematics students can escape the world of pure abstract theory and enter that of real molecules, while chemists and biologists will find simple, clear but rigorous definitions of mathematical concepts they handle intuitively in their work.

We show that for every m ∈ N, there exists an n ∈ N such that every embedding of the complete graph K n in R 3 contains a link of two components whose linking number is at least m. Furthermore, there exists an r ∈ N such that every embedding of K r in R 3 contains a knot Q with |a 2 (Q)| ≥ m, where a 2 (Q) denotes the second coefficient of the Conway polynomial of Q.

How knotted proteins fold has remained controversial since the identification of deeply knotted proteins nearly two decades ago. Both computational and experimental approaches have been used to investigate protein knot formation. Motivated by the computer simulations of Bölinger et al. [Bölinger D, et al. (2010) PLoS Comput Biol 6:e1000731] for the folding of the 61-knotted α-haloacid dehalogenase (DehI) protein, we introduce a topological description of knot folding that could describe pathways for the formation of all currently known protein knot types and predicts knot types that might be identified in the future. We analyze fingerprint data from crystal structures of protein knots as evidence that particular protein knots may fold according to specific pathways from our theory. Our results confirm Taylor’s twisted hairpin theory of knot folding for the 31-knotted proteins and the 41-knotted ketol-acid reductoisomerases and present alternative folding mechanisms for the 41-knotted phytochromes and the 52- and 61-knotted proteins.

We prove that a graph is intrinsically linked in an arbitrary 3-manifold M if and only if it is intrinsically linked in S 3 . Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S 3 .05C10, 57M25