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.

We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/\pi, under any equitable partition \pi, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs \Box_{k} G_{k}/\pi_{k} is isomorphic to the quotient graph \Box_{k} G_{k}/\pi, for some equitable partition \pi. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.

Let M be a closed hyperbolic 3-manifold with a fibered face σ of the unit ball of the Thurston norm on H 2 (M ). If M satisfies a certain condition related to Agol's veering triangulations, we construct a taut branched surface in M spanning σ. This partially answers a 1986 question of Oertel, and extends an earlier partial answer due to Mosher.

We introduce a polynomial invariant V_\tau \in \mathbb{Z}[H_1(M)/\text{torsion}] associated to a veering triangulation \tau of a 3 -manifold M . In the special case where the triangulation is layered, i.e. comes from a fibration, V_\tau recovers the Teichmüller polynomial of the fibered face canonically associated to \tau . Via Dehn filling, this gives a combinatorial description of the Teichmüller polynomial for any hyperbolic fibered 3 -manifold. For a general veering triangulation \tau , we show that the surfaces carried by \tau determine a cone in homology that is dual to its cone of positive closed transversals. Moreover, we prove that this is equal to the cone over a (generally non-fibered) face of the Thurston norm ball, and that \tau computes the norm on this cone in a precise sense. We also give a combinatorial description of V_\tau in terms of the flow graph for \tau and its Perron polynomial. This perspective allows us to characterize when a veering triangulation comes from a fibration, and more generally to compute the face of the Thurston norm ball determined by \tau .

We show that a veering triangulation τ specifies a face σ of the Thurston norm ball of a closed three-manifold, and computes the Thurston norm in the cone over σ. Further, we show that τ collates exactly the taut surfaces representing classes in the cone over σ up to isotopy. The analysis includes nonlayered veering triangulations and nonfibered faces.