Linking Numbers in Three-Manifolds

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of $$S^3$$ S 3 branched along a knot $$\alpha \subset S^3$$ α ⊂ S 3 . Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot $$\alpha $$ α can be derived from dihedral covers of $$\alpha $$ α . The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications.

On Frink's Type Metrization of Weighted Graphs

Using the technique of the metrization theorem of uniformities with countable bases, in this note we provide, test and compare an explicit algorithm to produce a metric d(x, y) between the vertices x and y of an anity weighted undirected graph.

Discrimination of POVMs with rank-one effects

The main goal of this work is to provide an insight into the problem of discrimination of positive operator-valued measures with rank-one effects. It is our intention to study multiple-shot discrimination of such measurements, that is the case when we are able to use to unknown measurement a given number of times. Furthermore, we are interested in comparing two possible discrimination schemes: the parallel and adaptive ones. To this end, we construct a pair of symmetric informationally complete positive operator-valued measures which can be perfectly discriminated in a two-shot adaptive scheme but cannot be distinguished in the parallel scheme. On top of this, we provide an explicit algorithm which allows us to find this adaptive scheme.

The Unspoken Shaken Baby Lie Detector Algorithm—An Analysis of Diagnostic Procedures in Cases of Allegedly Abusive Head Trauma without External Signs of Trauma

Various algorithms have been developed for diagnosis of Abusive Head Trauma (AHT); however, there is no explicit algorithm for the 1/3 of alleged AHT cases which present with findings restricted to subdural and retinal hemorrhages, with or without encephalopathy—i.e., isolated triad cases. Moreover, such cases have been lumped together with AHT cases with external sign of trauma and all diagnosed as AHT despite those we are focusing on lack signs of trauma. It is therefore relevant to analyze how this diagnosis can be reached in the absence of unequivocal trauma. Our analysis indicates that with respect to isolated triad cases, a two-step algorithm seems to be in use—but unspoken. The first step concerns a ruling out diagnostic process and the second step addresses the veracity of the caregivers’ narrative. Both steps are based on the traditional AHT mechanism theories and ten tacitly applied auxiliary hypotheses which are, however, questionable or incorrect. If the traditional AHT mechanism theories and necessary auxiliary hypotheses are not valid, then neither is the two-step algorithm. Accordingly, we propose that in alleged AHT cases with no external signs of trauma, the two-step algorithm—here referred to as the unspoken shaken baby lie detector algorithm—should be abandoned.