1,994 research outputs found

    Analogies between the crossing number and the tangle crossing number

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    Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum crossing number over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts. Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with nn leaves decreases the tangle crossing number by at most n3n-3, and this is sharp. Additionally, if γ(n)\gamma(n) is the maximum tangle crossing number of a tanglegram with nn leaves, we prove 12(n2)(1o(1))γ(n)<12(n2)\frac{1}{2}\binom{n}{2}(1-o(1))\le\gamma(n)<\frac{1}{2}\binom{n}{2}. Further, we provide an algorithm for computing non-trivial lower bounds on the tangle crossing number in O(n4)O(n^4) time. This lower bound may be tight, even for tanglegrams with tangle crossing number Θ(n2)\Theta(n^2).Comment: 13 pages, 6 figure

    The Jones polynomial of ribbon links

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    For every n-component ribbon link L we prove that the Jones polynomial V(L) is divisible by the polynomial V(O^n) of the trivial link. This integrality property allows us to define a generalized determinant det V(L) := [V(L)/V(O^n)]_(t=-1), for which we derive congruences reminiscent of the Arf invariant: every ribbon link L = (K_1,...,K_n) satisfies det V(L) = det(K_1) >... det(K_n) modulo 32, whence in particular det V(L) = 1 modulo 8. These results motivate to study the power series expansion V(L) = \sum_{k=0}^\infty d_k(L) h^k at t=-1, instead of t=1 as usual. We obtain a family of link invariants d_k(L), starting with the link determinant d_0(L) = det(L) obtained from a Seifert surface S spanning L. The invariants d_k(L) are not of finite type with respect to crossing changes of L, but they turn out to be of finite type with respect to band crossing changes of S. This discovery is the starting point of a theory of surface invariants of finite type, which promises to reconcile quantum invariants with the theory of Seifert surfaces, or more generally ribbon surfaces.Comment: 38 pages, reformatted in G&T style; minor changes suggested by the refere

    Linking number and Milnor invariants

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    This is a concise overview of the definitions and properties of the linking number and its higher-order generalization, Milnor invariants.Comment: 7 pages. Expository article written for an upcoming concise encyclopaedia of knot theor

    Spectral geometry, link complements and surgery diagrams

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    We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require 'complicated' surgery diagrams, thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the bridge number of a knot that is not tangle composite, in terms of the twist number of any diagram of the knot. The proofs rely on a theorem of Lipton and Tarjan on planar graphs, and also the relationship between many different notions of width for knots and 3-manifolds.Comment: 20 pages, 11 figure

    Driven Tunneling: Chaos and Decoherence

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    Chaotic tunneling in a driven double-well system is investigated in absence as well as in the presence of dissipation. As the constitutive mechanism of chaos-assisted tunneling, we focus on the dynamics in the vicinity of three-level crossings in the quasienergy spectrum. The coherent quantum dynamics near the crossing is described satisfactorily by a three-state model. It fails, however, for the corresponding dissipative dynamics, because incoherent transitions due to the interaction with the environment indirectly couple the three states in the crossing to the remaining quasienergy states. The asymptotic state of the driven dissipative quantum dynamics partially resembles the, possibly strange, attractor of the corresponding damped driven classical dynamics, but also exhibits characteristic quantum effects.Comment: 32 pages, 35 figures, lamuphys.st

    Higher-dimensional Algebra and Topological Quantum Field Theory

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    The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a "suspension" operation on n-categories, and hypothesize that the k-fold suspension of a weak n-category stabilizes for k >= n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n-dimensional unitary extended TQFTs as weak n-functors from the "free stable weak n-category with duals on one object" to the n-category of "n-Hilbert spaces". We conclude by describing n-categorical generalizations of deformation quantization and the quantum double construction.Comment: 36 pages, LaTeX; this version includes all 36 figure

    The largest crossing number of tanglegrams

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    A tanglegram T\cal T consists of two rooted binary trees with the same number of leaves, and a perfect matching between the two leaf sets. In a layout, the tanglegrams is drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines are drawn as plane trees, and the perfect matching is drawn in straight line segments inside the strip. The tanglegram crossing number cr(T){\rm cr}({\cal T}) of T\cal T is the smallest number of crossings of pairs of matching edges, over all possible layouts of T\cal T. The size of the tanglegram is the number of matching edges, say nn. An earlier paper showed that the maximum of the tanglegram crossing number of size nn tanglegrams is <12(n2)<\frac{1}{2}\binom{n}{2}; but is at least 12(n2)n3/2n2\frac{1}{2}\binom{n}{2}-\frac{n^{3/2}-n}{2} for infinitely many nn. Now we make better bounds: the maximum crossing number of a size nn tanglegram is at most 12(n2)n4 \frac{1}{2}\binom{n}{2}-\frac{n}{4}, but for infinitely many nn, at least 12(n2)nlog2n4\frac{1}{2}\binom{n}{2}-\frac{n\log_2 n}{4}. The problem shows analogy with the Unbalancing Lights Problem of Gale and Berlekamp

    Deformation of Crystals: Connections with Statistical Physics

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    We give a bird's-eye view of the plastic deformation of crystals aimed at the statistical physics community, as well as a broad introduction to the statistical theories of forced rigid systems aimed at the plasticity community. Memory effects in magnets, spin glasses, charge density waves, and dilute colloidal suspensions are discussed in relation to the onset of plastic yielding in crystals. Dislocation avalanches and complex dislocation tangles are discussed via a brief introduction to the renormalization group and scaling. Analogies to emergent scale invariance in fracture, jamming, coarsening, and a variety of depinning transitions are explored. Dislocation dynamics in crystals challenge nonequilibrium statistical physics. Statistical physics provides both cautionary tales of subtle memory effects in nonequilibrium systems and systematic tools designed to address complex scale-invariant behavior on multiple length scales and timescales
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