Statistics on some classes of knot shadows

The present paper is concerned with the enumeration of the state diagrams for some classes of knot shadows endowed with the usual connected sum operation. We focus on shadows that are recursively generated by knot shadows with up to 3 crossings, and for which the enumeration problem is solved with the help of generating polynomials.Comment: 37 pages, 27 figures, 20 tables; added new OEIS entries, closed form for the generating polynomial

Structure and enumeration of K4-minor-free links and link diagrams

We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with respect to the minimum number of crossings or edges in a projection of L' in L. Further, we obtain counting formulas and asymptotic estimates for the connected K4-minor-free link-diagrams, minimal K4-minor-free link-diagrams, and K4-minor-free diagrams of the unknot.Peer ReviewedPostprint (author's final draft

On the number of unknot diagrams

International audienceLet D be a knot diagram, and let D denote the set of diagrams that can be obtained from D by crossing exchanges. If D has n crossings, then D consists of 2 n diagrams. A folklore argument shows that at least one of these 2 n diagrams is unknot, from which it follows that every diagram has finite unknotting number. It is easy to see that this argument can be used to show that actually D has more than one unknot diagram, but it cannot yield more than 4n unknot diagrams. We improve this linear bound to a superpolynomial bound, by showing that at least 2 3 √ n of the diagrams in D are unknot. We also show that either all the diagrams in D are unknot, or there is a diagram in D that is a diagram of the trefoil knot

On the number of unknot diagrams

Non UBCUnreviewedAuthor affiliation: Universidad Autónoma de San Luis PotosíFacult