The Fundamental Crossed Module of the Complement of a Knotted Surface

We prove that if $M$ is a CW-complex and $M^1$ is its 1-skeleton then the crossed module $\Pi_2(M,M^1)$ depends only on the homotopy type of $M$ as a space, up to free products, in the category of crossed modules, with $\Pi_2(D^2,S^1)$. From this it follows that, if $G$ is a finite crossed module and $M$ is finite, then the number of crossed module morphisms $\Pi_2(M,M^1) \to G$ can be re-scaled to a homotopy invariant $I_G(M)$, depending only on the homotopy 2-type of $M$. We describe an algorithm for calculating $\pi_2(M,M^{(1)})$ as a crossed module over $\pi_1(M^{(1)})$, in the case when $M$ is the complement of a knotted surface $\Sigma$ in $S^4$ and $M^{(1)}$ is the handlebody made from the 0- and 1-handles of a handle decomposition of $M$. Here $\Sigma$ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant $I_G$ yields a non-trivial invariant of knotted surfaces in $S^4$ with good properties with regards to explicit calculations.Comment: A perfected version will appear in Transactions of the American Mathematical Societ

On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups

We give an interpretation of Yetter's Invariant of manifolds $M$ in terms of the homotopy type of the function space $TOP(M,B(G))$, where $G$ is a crossed module and $B(G)$ is its classifying space. From this formulation, there follows that Yetter's invariant depends only on the homotopy type of $M$, and the weak homotopy type of the crossed module $G$. We use this interpretation to define a twisting of Yetter's Invariant by cohomology classes of crossed modules, defined as cohomology classes of their classifying spaces, in the form of a state sum invariant. In particular, we obtain an extension of the Dijkgraaf-Witten Invariant of manifolds to categorical groups. The straightforward extension to crossed complexes is also considered.Comment: 45 pages. Several improvement

Infinitesimal 2-braidings and differential crossed modules

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of $n$ particles in the complex plane, hence to a categorification of the Knizhnik-Zamolodchikov connection. We discuss infinitesimal 2-braidings in a 2-category naturally assigned to every differential crossed module, leading to the notion of a quasi-invariant tensor in a differential crossed module. Finally we prove that quasi-invariant tensors exist in the differential crossed module associated to the String Lie-2-algebra.Comment: v3 - the introduction has been expanded, overall improvements in the presentation. Final version, to appear in Adv. Mat

Categorifying the sl(2,C)sl(2,C) Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra

We construct a flat (and fake-flat) 2-connection in the configuration space of $n$ indistinguishable particles in the complex plane, which categorifies the $sl(2,C)$-Knizhnik-Zamolodchikov connection obtained from the adjoint representation of $sl(2,C)$. This will be done by considering the adjoint categorical representation of the string Lie 2-algebra and the notion of an infinitesimal 2-Yang-Baxter operator in a differential crossed module. Specifically, we find an infinitesimal 2-Yang-Baxter operator in the string Lie 2-algebra, proving that any (strict) categorical representation of the string Lie-2-algebra, in a chain-complex of vector spaces, yields a flat and (fake flat) 2-connection in the configuration space, categorifying the $sl(2,C)$-Knizhnik-Zamolodchikov connection. We will give very detailed explanation of all concepts involved, in particular discussing the relevant theory of 2-connections and their two dimensional holonomy, in the specific case of 2-groups derived from chain complexes of vector spaces.Comment: The main result was considerably sharpened. Title, abstract and introduction updated. 50 page

Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces

We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way.Comment: New results. A perfected version will appear in Compositio Mathematic

Decision Support System for facility location problems in fleet management

Businesses that are growing by providing more services, reaching more customers or improving their business strategy, might need to create or relocate a facility location to expand the geographical coverage and improve their services. This decision is complex, and it is crucial to analyse their client locations, journeys and be aware of the factors that may affect their geographical decision. These organisations must weigh all these factors, such as security levels, taxes or costs due to the importance and impact that they can have in the short and long term business strategy. Therefore, the decision-maker needs to ensure that the location is the most profitable site according to the business scope and future perspectives. To help the businesses on this complex decision, this dissertation details the development of a Decision Support System (DSS) capable of providing facility location suggestions based on the existing journeys and the factors that the decision-maker considers more relevant to the company. The developed DSS has three main components: (1) Decision Support System; (2) Geospatial Analysis System; and (3) Facility Location Factors System. The Decision Support System is responsible for producing ordered facility location suggestions by performing the multicriteria decision analysis (MCDA) that implements the TOPSIS algorithm. The Geospatial Analysis System, through the use of the DBSCAN algorithm, is responsible for retrieving the alternatives by identifying the geospatial clusters based on the existing journeys. Lastly, the Facility Location Factors System is responsible for retrieving the criterion by gathering the data from external sources according to the chosen factors. The evaluation analysis shows that the perspective of the users about the assistance of the system by helping them choose appropriate facility location is favourable. This analysis showed that the users agree about the accuracy and the value of the facility location suggestions. The output helps the business managers to make better decisions by returning facility locations that have potential to maximise the company’s profit by reducing transportation and fuel costs and maximise the number of covered customers by expanding their territorial coverage. This project handles data provided by Fonix Telematics from their United Kingdom clients that have relevance to the study, such as a high number of assets, journeys and geographical coverage.As empresas em crescimento por via da disponibilização de mais serviços, do aumento do seu leque de clientes ou da melhoria da sua estratégia, podem pretender criar ou realocar um centro de operações de modo a expandir a sua cobertura geográfica e, consequentemente, melhorar os seus serviços. Esta decisão é complexa e é fundamental analisar vários aspetos, assim como, a localização dos seus clientes, as viagens recorrentes e, acima de tudo, estar consciente dos fatores que podem afetar a sua decisão geográfica. As organizações devem pesar todos esses fatores, assim como níveis de segurança, impostos ou custos, devido à importância e ao impacto que podem ter na estratégia da empresa a curto e a longo prazo. Portanto, o decisor necessita de garantir que o local é rentável e que capta o âmbito do negócio e as perspetivas futuras. De modo a auxiliar as empresas nesta complexa decisão, esta dissertação detalha o processo de desenvolvimento de um Sistema de Apoio à Decisão (SAD) capaz de fornecer um conjunto de sugestões com os locais mais indicados para a criação de um centro de operações com base nas viagens efetuadas e nos fatores que o decisor considera mais relevantes para a organização. O SAD desenvolvido possui três componentes: (1) Decision Support System; (2) Geospatial Analysis System; e (3) Facility Location Factors System. O Decision Support System é responsável por produzir as sugestões geoespaciais, através da Análise de Decisão Multicritério (MCDA) que por sua vez implementa o algoritmo TOPSIS. O Geospatial Analysis System, através da utilização do algoritmo DBSCAN, é responsável por retornar as alternativas através da identificação dos clusters geográficos com base nas viagens existentes. Por último, o Facility Location Factors System é responsável por retornar os critérios, que são compostos por dados recolhidos através de fontes externas de acordo com os fatores previamente selecionados. A avaliação da solução demonstra que a perspetiva dos utilizadores sobre o sistema é positiva e que, de facto, os auxilia na decisão do local mais indicado para as suas instalações. A análise indica ainda que os utilizadores estão de acordo com a precisão e com locais sugeridos para os centros de operações. Estas sugestões auxiliam os decisores a tomarem decisões mais sustentadas, visto que os locais sugeridos possuem potencial para maximizar a rentabilidade da empresa, reduzir os custos de transporte e combustível, assim como maximizar a cobertura de clientes através do posicionamento geográfico. Este trabalho utiliza dados de clientes da Fonix Telematics que atuam no Reino Unido e que possuem relevância para o estudo, como um número significativo de veículos, viagens e cobertura geográfica