Grid diagram for singular links

In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on $\mathcal{SG}$ which induce the bijection onto each singular object. This is an extension of the known result of Ng-Thurston for non-singular links and braids.Comment: 33 pages, 34 figure

Bennequin type inequalities in lens spaces

We give criteria for an invariant of lens space links to bound the maximal self-linking number in certain tight contact lens spaces. As a corollary we extend the Franks-Williams-Morton inequality to the setting of lens spaces.Comment: 21 pages, 13 figures; International Mathematics Research Notices 201

Braids: A Survey

This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literature is extensive, and for reasons of space our coverage necessarily omits many very interesting developments. Open problems are noted and so-labelled, as we encounter them.Comment: Final version, revised to take account of the comments of readers. A review article, to appear in the Handbook of Knot Theory, edited by W. Menasco and M. Thistlethwaite. 91 pages, 24 figure

HOMFLY-PT polynomial and normal rulings of Legendrian solid torus links

We show that for any Legendrian link $L$ in the $1$-jet space of $S^1$ the $2$-graded ruling polynomial, $R^2_L(z)$, is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover $R^2_L(z)$ as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, we show that this specialization may be interpreted as the standard inner product on the algebra of symmetric functions that is often identified with a certain subalgebra of the HOMFLY-PT skein module of the solid torus. In contrast to the $2$-graded case, we are able to use $0$-graded ruling polynomials to distinguish many homotopically non-trivial Legendrian links with identical classical invariants.Comment: 30 pages, 9 figure

A family of transversely nonsimple knots

We apply knot Floer homology to exhibit an infinite family of transversely nonsimple prime knots starting with $10_{132}$. We also discuss the combinatorial relationship between grid diagrams, braids, and Legendrian and transverse knots in standard contact $\mathbb{R}^3$.Comment: 19 pages, v2: minor corrections to statements in section 2.

Taut foliations, braid positivity, and unknot detection

We study positive braid knots (the knots in the three-sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if $K$ is a non-trivial positive braid knot, then for all $r < 2g(K)-1$, the 3-manifold obtained via $r$-framed Dehn surgery along $K$ admits a taut foliation. Our main result provides some positive evidence towards this conjecture: we construct taut foliations in such manifolds whenever $r<g(K)+1$. As an application, we produce a novel braid positivity obstruction for cable knots by proving that the $(n,\pm 1)$-cable of a knot $K$ is braid positive if and only if $K$ is the unknot. We also present some curious examples demonstrating the limitations of our construction; these examples can also be viewed as providing some negative evidence towards the L-space conjecture. Finally, we apply our main result to produce taut foliations in some splicings of knot exteriors.Comment: 91 pages, 49 figures, 5 tables, 1 flowchart, 1 appendi

Dynamics and steady-state properties of adaptive networks

Tese de doutoramento, Física, Universidade de Lisboa, Faculdade de Ciências, 2013Collective phenomena often arise through structured interactions among a system's constituents. In the subclass of adaptive networks, the interaction structure coevolves with the dynamics it supports, yielding a feedback loop that is common in a variety of complex systems. To understand and steer such systems, modeling their asymptotic regimes is an essential prerequisite. In the particular case of a dynamic equilibrium, each node in the adaptive network experiences a perpetual change in connections and state, while a comprehensive set of measures characterizing the node ensemble are stationary. Furthermore, the dynamic equilibria of a wide class of adaptive networks appear to be unique, as their characteristic measures are insensitive to initial conditions in both state and topology. This work focuses on dynamic equilibria in adaptive networks, and while it does so in the context of two paradigmatic coevolutionary processes, obtained results easily generalize to other dynamics. In the rst part, a low-dimensional framework is elaborated on using the adaptive contact process. A tentative description of the phase diagram and the steady state is obtained, and a parameter region identi ed where asymmetric microscopic dynamics yield a symmetry between node subensembles. This symmetry is accounted for by novel recurrence relations, which predict it for a wide range of adaptive networks. Furthermore, stationary nodeensemble distributions are analytically generated by these relations from one free parameter. Secondly, another analytic framework is put forward that detects and describes dynamic equilibria, while assigning to them general properties that must hold for a variety of adaptive networks. Modeling a single node's evolution in state and connections as a random walk, the ergodic properties of the network process are used to extract node-ensemble statistics from the node's long-term behavior. These statistical measures are composed of a variety of stationary distributions that are related to one another through simple transformations. Applying this fully self-su cient framework, the dynamic equilibria of three di erent avors of the adaptive contact process are subsequently described and compared. Lastly, an asymmetric variant of the coevolutionary voter model is motivated and proposed, and as for the adaptive contact process, a low-dimensional description is given. In a parameter region where a dynamic equilibrium lets the in nite system display perpetual dynamics, this description can be further reduced to a one-dimensional random walk. For nite system sizes, this allows to analytically characterize longevity of the dynamic equilibrium, with results being compared to the symmetric variant of the process. A nontrivial parameter combination is identi ed for which, in the low-dimensional description of the process, the asymmetric coevolutionary model emulates symmetric voter dynamics without topological coevolution. This emerging symmetry is partially con rmed for the full system and subsequently elaborated on. Slightly varying the original asymmetric model, an additional asymptotic regime is shown to occur that coexists with all others and complicates system description.A estrutura das interacções entre os constituintes elementares de um sistema está frequentemente na origem de comportamentos colectivos não triviais. Em redes adaptativas, esta estrutura de interacção evolui a par com a dinâaica que nela assenta, traduzindo uma retroacção que de comum encontrar em vários sistemas complexos. Resultados analíticos sobre os estados assimptóticos destes sistemas são uma peça essencial para a sua compreensão e controlo. O equilíbrio dinâmico de um caso particular de estado assimptótico em que cada nodo da rede adaptativa vai sempre mudando o seu estado e as suas ligações a outros nodos, enquanto que um conjunto de medidas que caracterizam estatisticamente o ensemble dos nodos mantêm valores fixos. Alémm disso, uma classe muito geral de redes adaptativas apresenta equilíbrios dinâmicos que parecem ser únicos, no sentido em que aqueles valores estacionários não dependem das condições iniciais, quer em termos do estados dos nodos quer em termos da topologia da rede.Este trabalho incide no estudo do equilíbrio dinâmiico de redes adaptativas no contexto particular de dois modelos paradigmáticos de coevolação, mas os principais resultados podem ser facilmente generalizados a outros processos. Na primeira parte, revisita-se e desenvolve-se uma abordagem da variante adaptativa do processo de contacto baseada num modelo de baixa dimensão. Obtem-se uma descrição aproximada do diagrama de fases do sistema e do equilíbrio dinâmico, e identifica-se nessa fase uma combinação de parâmetros para a qual a dinâmica microscópica, que de assimétrica nos estados dos nodos, da origem a uma simetria entre os dois subconjuntos de nodos. Esta simetria é explicada através da derivação de relações de recorrência para as distribuições de grau, que a preveêm para uma ampla classe de redes adaptativas. Estas relações permitem também gerar analiticamente as distribuições de grau estacionárias de cada subconjunto de nodos a partir de um parâmetro livre.Na segunda parte, desenvolve-se uma outra abordagem analítica que permite detectar e descrever o equilíbrio dinâmico, a partir de propriedades gerais que se têm que verificar em muitas redes adaptativas. Na base desta abordagem está a descrição do processo estocástico associado à evolução do estado e das ligações de cada nó, e as propriedades ergódicas que permitem obter as estatísticas de ensemble na rede a partir do comportamento a longo termo de um nó. Estas medidas estatísticas podem ser calculadas a partir de várias distribuições estacionárias que se relacionam umas com as outras através de transformações simples. Como aplicação desta abordagem completa, os equilíbrios dinâmicos de três diferentes variantes do processo de contacto adaptativo são descritos e comparados. Finalmente, motiva-se e propõe-se uma variante assimétrica do voter model coevolutivo. A fase activa metastável é tentativamente descrita como uma random walk ao longo de uma variedade lenta, à semelhan ca do que foi feito na literatura para o modelo simétrico, e os resultados para os dois casos são comparados.É identicada uma combinação de parâmetros particular para a qual este modelo assim etrico emula o modelo simétrico em rede fixa, o que é mais um exemplo da simetria emergente prevista pelas relações de recorrência estabelecidas na primeira parte. Considera-se ainda uma outra variante assimétrica, mais complexa, do voter model co-evolutivo, que apresenta um diagrama de fases essencialmente diferente, e cuja descrição se mostra requerer novas abordagens.Fundação para a Ciência e a Tecnologia (FCT, SFRH/BD/45179/2008