PrefaceAround 1980, W. Thurston proved that every knot complement satisfies the geometrization conjecture: it decomposes into pieces that admit locally homogeneous geometric structures. In addition, he proved that the complement of any non-torus, non-satellite knot admits a complete hyperbolic metric which, by the Mostow-Prasad rigidity theorem, is is necessarily unique up to isometry. As a result, geometric information about a knot complement, such as its volume, gives topological invariants of the knot.Since the mid-1980's, knot theory has also been invigorated by ideas from quantum physics, which have led to powerful and subtle knot invariants, including the Jones polynomial and its relatives, the colored Jones polynomials. Topological quantum field theory predicts that these quantum invariants are very closely connected to geometric structures on knot complements, and particularly to hyperbolic geometry. The volume conjecture of R. Kashaev, H. Murakami, and J. Murakami, which asserts that the volume of a hyperbolic knot is determined by certain asymptotics of colored Jones polynomials, fits into the context of these predictions. Despite compelling experimental evidence, these conjectures are currently verified for only a few examples of hyperbolic knots.This monograph initiates a systematic study of relations between quantum and geometric knot invariants. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A-or B-adequacy), we derive direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. We prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement, and that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement. In particular, the surface is a fiber if and only if a certain coefficient vanishes.Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our methods here provide a deeper and more intrinsic explanation for similar connections that have been previously observed.Our approach is to generalize the checkerboard decompositions of alternating knots and links. For A-or B-adequate diagrams, we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to establish a dictionary between the pieces of the JSJ decomposition of the surface complement and the combinatorial structure of certain spines of the checkerboard surface (state graphs). In particular, we give a combinatorial formula 4 for the complexity of the hyperbolic part of the JSJ decomposition (the guts) of the surface complement in terms of the diagram of the knot, and use this to give lower bounds on volumes of several ...

SummaryBackgroundResults of small trials indicate that fluoxetine might improve functional outcomes after stroke. The FOCUS trial aimed to provide a precise estimate of these effects.MethodsFOCUS was a pragmatic, multicentre, parallel group, double-blind, randomised, placebo-controlled trial done at 103 hospitals in the UK. Patients were eligible if they were aged 18 years or older, had a clinical stroke diagnosis, were enrolled and randomly assigned between 2 days and 15 days after onset, and had focal neurological deficits. Patients were randomly allocated fluoxetine 20 mg or matching placebo orally once daily for 6 months via a web-based system by use of a minimisation algorithm. The primary outcome was functional status, measured with the modified Rankin Scale (mRS), at 6 months. Patients, carers, health-care staff, and the trial team were masked to treatment allocation. Functional status was assessed at 6 months and 12 months after randomisation. Patients were analysed according to their treatment allocation. This trial is registered with the ISRCTN registry, number ISRCTN83290762.FindingsBetween Sept 10, 2012, and March 31, 2017, 3127 patients were recruited. 1564 patients were allocated fluoxetine and 1563 allocated placebo. mRS data at 6 months were available for 1553 (99·3%) patients in each treatment group. The distribution across mRS categories at 6 months was similar in the fluoxetine and placebo groups (common odds ratio adjusted for minimisation variables 0·951 [95% CI 0·839–1·079]; p=0·439). Patients allocated fluoxetine were less likely than those allocated placebo to develop new depression by 6 months (210 [13·43%] patients vs 269 [17·21%]; difference 3·78% [95% CI 1·26–6·30]; p=0·0033), but they had more bone fractures (45 [2·88%] vs 23 [1·47%]; difference 1·41% [95% CI 0·38–2·43]; p=0·0070). There were no significant differences in any other event at 6 or 12 months.InterpretationFluoxetine 20 mg given daily for 6 months after acute stroke does not seem to improve functional outcomes. Although the treatment reduced the occurrence of depression, it increased the frequency of bone fractures. These results do not support the routine use of fluoxetine either for the prevention of post-stroke depression or to promote recovery of function.FundingUK Stroke Association and NIHR Health Technology Assessment Programme.

Abstract. Given a hyperbolic 3-manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones polynomials.

Abstract. We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link. Mathematics Subject Classification (2000). 57M25, 57M5.

We consider links that are alternating on surfaces embedded in a compact 3manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometry of generalisations of alternating links, including weakly generalised alternating links described by the first author. We give diagrammatical properties that determine when such links are hyperbolic, find the geometry of their checkerboard surfaces, bound volume, and exclude exceptional Dehn fillings.