An n-crossing is a singular point in a projection of a link at which n strands cross such that each strand travels straight through the crossing. We introduce the notion of an ubercrossing projection, a knot projection with a single n-crossing. Such a projection is necessarily composed of a collection of loops emanating from the crossing. We prove the surprising fact that all knots have a special type ofübercrossing projection, which we call a petal projection, in which no loops contain any others. The rigidity of this form allows all the information about the knot to be concentrated in a permutation corresponding to the levels at which the strands lie within the crossing. These ideas give rise to two new invariants for a knot K: theübercrossing numberü(K), and petal number p(K). These are the least number of loops in anyübercrossing or petal projection of K, respectively. We relateü(K) and p(K) to other knot invariants, and compute p(K) for several classes of knots, including all knots of nine or fewer crossings.

The k-gonal models of random groups are defined as the quotients of free groups on n generators by cyclically reduced words of length k. As k tends to infinity, this model approaches the Gromov density model. In this paper we show that for any fixed d0 ∈ (0, 1), if positive k-gonal random groups satisfy Property (T) with overwhelming probability for densities d > d0, then so do nk-gonal random groups, for any n ∈ N. In particular, this shows that for densities above 1/3, groups in 3k-gonal models satisfy Property (T) with probability 1 as n approaches infinity.

For random groups in the Gromov density model at d > 3 / 14 d>3/14 , we construct walls in the Cayley complex X X which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities d > 1 / 5 d>1/5 and d > 5 / 24 d>5/24 , respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density d > 1 / 4 d>1/4 .

No abstract

A graph G with n vertices is Hamiltonian if it admits an embedded cycle containing all vertices of G. In any Hamiltonian graph, each vertex is the starting point of a Hamiltonian path. In this paper we explore the converse. We show that for 2 < n < 9, if G admits Hamiltonian paths starting at every vertex then G is Hamiltonian. We also show that this is not true for n > 9. We then investigate the number of pairs of vertices in a non-Hamiltonian graph G which can be connected by Hamiltonian paths. In particular we construct a family of non-Hamiltonian graphs with approximately 4/5 of the pairs of vertices connected by Hamiltonian paths.