28,945 research outputs found

    A basis for the Birman-Wenzl algebra

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    An explicit isomorphism is constructed between the Birman-Wenzl algebra, defined algebraically by J. Birman and H. Wenzl using generators and relations, and the Kauffman algebra, constructed geometrically by H. R. Morton and P. Traczyk in terms of tangles. The isomorphism is obtained by constructing an explicit basis in the Birman-Wenzl algebra, analogous to a basis previously constructed for the Kauffman algebra using 'Brauer connectors'. The geometric isotopy arguments for the Kauffman algebra are systematically replaced by algebraic versions using the Birman-Wenzl relations.Comment: This is a lightly edited version of an article written in 1989 but never fully completed. It was originally intended as a joint paper with A. J.Wassermann. 33 pages, 20 figure

    Magnetohydrodynamic waves in a compressible magnetic flux tube with elliptical cross-section

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    Aims. The propagation of magnetohydrodynamic (MHD) waves in a finite, compressible magnetic flux tube with an elliptical cross-section embedded in a magnetic environment is investigated. Methods. We present the derivation of the general dispersion relation of linear magneto-acoustic wave propagation for a compressible magnetic flux tube with elliptical cross-section in a plasma with finite beta. The wave modes of propagation for the n = 0 (symmetric) sausage and n = 1 (anti-symmetric) kink oscillations are then examined within the limit of the thin flux tube approximation. Results. It is shown that a compressible magnetic tube with elliptical cross-section supports slow and fast magneto-acoustic waves. In the thin tube approximation, the slow sausage mode and the slow and fast kink modes are found in analogue to a circular cross-section. However, the kink modes propagate with different phase speeds depending on whether the axial displacement takes place along the major or minor axis of the ellipse. This feature is present in both the slow and the fast bands, providing two infinite sets of slow kink modes and two infinite sets of fast kink modes, i.e. each corresponding cylindrical mode splits into two sets of modes due to the ellipticity. The difference between the phase speeds along the different axis is dependent on the ratio of the lengths of the two axes. Analytical expressions for the phase speeds are found. We show that the sausage modes do not split due to the introduced ellipticity and only the phase speed is modified when compared to the appropriate cylindrical counterpart. The percentage difference between the periods of the circular and elliptical cross-sections is also calculated, which reaches up to 21% for oscillations along the major axis. The level of difference in period could be very important in magneto-seismological applications, when observed periods are inverted into diagnostic properties (e. g. magnetic field strength, gravitational scale height, tube expansion parameter). Also shown is the perturbation of focal points of the elliptical cross-section for different modes. It is found that the focal points are unperturbed for the sausage mode, but are perturbed for all higher modes

    Homfly Polynomials of Generalized Hopf Links

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    Following the recent work by T.-H. Chan in [HOMFLY polynomial of some generalized Hopf links, J. Knot Theory Ramif. 9 (2000) 865--883] on reverse string parallels of the Hopf link we give an alternative approach to finding the Homfly polynomials of these links, based on the Homfly skein of the annulus. We establish that two natural skein maps have distinct eigenvalues, answering a question raised by Chan, and use this result to calculate the Homfly polynomial of some more general reverse string satellites of the Hopf link.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-2.abs.htm

    Mutants and SU(3)_q invariants

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    Details of quantum knot invariant calculations using a specific SU(3)_q-module are given which distinguish the Conway and Kinoshita-Teresaka pair of mutant knots. Features of Kuperberg's skein-theoretic techniques for SU(3)_q invariants in the context of mutant knots are also discussed.Comment: 17 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper18.abs.htm

    Mutant knots with symmetry

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    Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m>2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial for m<6 in general, and for all choices of m when the satellite is based on a cable knot pattern. We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantum sl(3) invariants.Comment: 15 page
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