Homfly Polynomials of Generalized Hopf Links

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

THE ALEXANDER POLYNOMIAL OF A TORUS KNOT WITH TWISTS

This note gives an explicit calculation of the doubly infinite sequence Δ(p, q, 2m), m ∈ Z of Alexander polynomials of the (p, q) torus knot with m extra full twists on two adjacent strings, where p and q are both positive. The knots can be presented as the closure of the p-string braids [Formula: see text], where δp = σp-1σp-2 · σ2σ1, or equally of the q-string braids [Formula: see text]. As an application we give conditions on (p, q) which ensure that all the polynomials Δ(p, q, 2m) with |m| ≥ 2 have at least one coefficient a with |a| &gt; 1. A theorem of Ozsvath and Szabo then ensures that no lens space can arise by Dehn surgery on any of these knots. The calculations depend on finding a formula for the multivariable Alexander polynomial of the 3-component link consisting of the torus knot with twists and the two core curves of the complementary solid tori. </jats:p

Power sums and Homfly skein theory

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements, whose symmetric functions are central in H_n. In [Skein theory and the Murphy operators, J. Knot Theory Ramif. 11 (2002), 475-492] I defined geometrically a homomorphism from the Homfly skein C of the annulus to the centre of each algebra H_n, and found an element P_m in C, independent of n, whose image, up to an explicit linear combination with the identity of H_n, is the m-th power sum of the Murphy operators. The aim of this paper is to give simple geometric representatives for the elements P_m, and to discuss their role in a similar construction for central elements of an extended family of algebras H_{n,p}.Comment: Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper15.abs.htm

DISTINGUISHING MUTANTS BY KNOT POLYNOMIALS

We consider the problem of distinguishing mutant knots using invariants of their satellites. We show, by explicit calculation, that the Homfly polynomial of the 3-parallel (and hence the related quantum invariants) will distinguish some mutant pairs. Having established a condition on the colouring module which forces a quantum invariant to agree on mutants, we explain several features of the difference between the Homfly polynomials of satellites constructed from mutants using more general patterns. We illustrate this by our calculations; from these we isolate some simple quantum invariants, and a framed Vassiliev invariant of type 11, which distinguish certain mutants, including the Conway and Kinoshita-Teresaka pair. </jats:p

Is there an optimal whole-body vibration exposure ‘dosage’ for performance improvement?

International Journal of Exercise Science 7(3) : 169-178, 2014. Whole-body vibration exposure has been shown to improve performance in vertical jumping and knee extensions. Some studies have addressed the question of dose optimality, but are inconclusive and inappropriately designed. Our purpose was to more thoroughly seek an optimum combination of duration, amplitude and frequency of exposure to side-alternating whole-body vibration. We used experimental designs constructed for response surface fitting and optimisation, using both blocked and unblocked second order central composite designs with 12 participants. Immediately after each exposure a discomfort index was recorded, then peak and average torque, peak and average jump height, together with peak and average jump power were recorded over three trials both pre- and post-exposure at each treatment combination. ANOVA revealed that all performance measures improved after vibration exposure. However, no successful response surface fits could be achieved for any of the performance measures, except weakly for average jump height and average jump power for a single subject. Conversely, the discomfort index increased linearly with both vibration amplitude and frequency, more steeply as exposure duration increased. We conclude that although vibration exposure has a significant positive effect on performance, its effect is so variable both between and within individuals that no real optimum can be discerned; and that high amplitudes, frequencies and durations lead to excessive discomfort

Conjugacy for positive permutation braids

Positive permutation braids on n strings, which are defined to be positive n-braids where each pair of strings crosses at most once, form the elementary but non-trivial building blocks in many studies of conjugacy in the braid groups. We consider conjugacy among these elementary braids which close to knots, and show that those which close to the trivial knot or to the trefoil are all conjugate. All such n-braids with the maximum possible crossing number are also shown to be conjugate. We note that conjugacy of these braids for n<6 depends only on the crossing number. In contrast, we exhibit two such braids on 6 strings with 9 crossings which are not conjugate but whose closures are each isotopic to the (2,5) torus knot

The critical power concept and bench press: Modeling 1RM and repetitions to failure

International Journal of Exercise Science 7(2) : 152-160, 2014. Introduction: We demonstrate application of the 3-parameter critical power (CP) model derived for cycling and running, to performance at bench press exercise. We apply the model to both performance of a single repetition maximum (1RM) and multiple repetitions (reps) to failure at different sub-maximal weights. Methods: Sixteen weight-trained young adult male participants each performed a modified YMCA 1RM test and four sets of fixed cadence reps to failure at different sub-maximal weights. The CP model equation takes the form: n = ALC/(m – CL) + ALC/(CL – Lmax,), where n is the number of reps to failure and m is the sub-maximal weight lifted (kg). ALC is the anaerobic lift capacity (kg), CL is the critical lift (the maximal continuous aerobic ability at bench pressing, kg), and Lmax is the maximal ‘instantaneous’ lift (kg). Results: The 3-parameter critical power model fits recorded reps to failure very well in almost all subjects (0.9556 \u3c R2 \u3c 0.9999), and provides estimates of the three model parameters for each individual. CL was not significantly different from zero, suggesting that the aerobic energy contribution to short duration bench press sessions is negligible. When used to estimate 1RM for each subject, the CP model produces estimates significantly greater (p \u3c 0.05) than those obtained using the YMCA procedure. Conclusion: The CP concept can be used to accurately model bench press reps to failure at different submaximal weights in a homogeneous group of individuals. Prediction of 1RM is possible, but caution should be exercised in interpreting and using the prediction