The Homflypt skein module of a connected sum of 3-manifolds

If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M. We show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensor S(M_2) modulo torsion. In fact, we show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensot S(M_2) if we are working over a certain localized ring. We show the similar result holds for relative skein modules. If M contains a separating 2-sphere, we give conditions under which certain relative skein modules of M vanish over specified localized rings.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-31.abs.htm

Mutual braiding and the band presentation of braid groups

This paper is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The geometric condition for mutual braiding refers to the existence of a close control on the way in which the whole family of fibre surfaces meet the family of discs spanning the braid axis. We show how such a braid can be presented naturally as a word in the `band generators' of the braid group discussed by Birman, Ko and Lee in their recent account of the band presentation of the braid groups. In this context we are able to convert the conditions for mutual braiding into the existence of a suitable sequence of band relations and other moves on the braid word, and derive a combinatorial method for deciding whether a braid is mutually braided.Comment: 13 pages, 10 figures. To appear in the proceedings of 'Knots in Hellas 98

On the accuracy of surface spline interpolation on the unit sphere

This paper considers a novel modification to the surface splines that have previously been used on the unit sphere. The surface splines considered are a natural analogue of surface splines in IRd and possess a unique Fourier expansion in terms of an orthonormal basis of spherical harmonics. Knowing the decay of the associated Fourier coefficients is important because they enable error estimates for spherical interpolation. In this paper we explicitly compute the Fourier coefficients of the surface splines and employ a recent theoretical result [8] to provide a useful error bound. We illuminate our theoretical findings by performing numerical experiments on the sphere and also on the hemisphere

Believing in Others

Suppose some person 'A' sets out to accomplish a difficult, long-term goal such as writing a passable Ph.D. thesis. What should you believe about whether A will succeed? The default answer is that you should believe whatever the total accessible evidence concerning A's abilities, circumstances, capacity for self-discipline, and so forth supports. But could it be that what you should believe depends in part on the relationship you have with A? We argue that it does, in the case where A is yourself. The capacity for "grit" involves a kind of epistemic resilience in the face of evidence suggesting that one might fail, and this makes it rational to respond to the relevant evidence differently when you are the agent in question. We then explore whether similar arguments extend to the case of "believing in" our significant others -- our friends, lovers, family members, colleagues, patients, and students

Maximizing Academic Success for Foster Care Students: A Trauma-Informed Approach

Children in foster care have experienced significant trauma due to the loss of primary attachment figures and the circumstances associated with that loss. Children who have suffered trauma generally present with cognitive, social, physical, and emotional vulnerabilities. These vulnerabilities are often expressed in the P–12 academic setting through difficulties with behavioral and emotional self-regulation, academic functioning, and physical ailments and illness related to chronic stress-induced compromised immune systems. This results in academic failure for half of all children in care. Training in how to respond to children who have suffered trauma is essential to ensure that children are comfortable and feel secure in the classroom so that they can access their education. To that end, a framework to support children in P–12 settings who are particularly vulnerable to academic failure due to trauma is presented

Geometrical relations and plethysms in the Homfly skein of the annulus

The oriented framed Homfly skein C of the annulus provides the natural parameter space for the Homfly satellite invariants of a knot. It contains a submodule C+ isomorphic to the algebra of the symmetric functions. We collect and expand formulae relating elements expressed in terms of symmetric functions to Turaev's geometrical basis of C+. We reformulate the formulae of Rosso and Jones for quantum sl(N) invariants of cables in terms of plethysms of symmetric functions, and use the connection between quantum sl(N) invariants and C+ to give a formula for the satellite of a cable as an element of C+. We then analyse the case where a cable is decorated by the pattern which corresponds to a power sum in the symmetric function interpretation of C+ to get direct relations between the Homfly invariants of some diagrams decorated by power sums.Comment: 28 pages, 15 figure

Non-genomic regulation of intermediate conductance potassium channels by aldosterone in human colonic crypt cells

BACKGROUND: Aldosterone has a rapid, non-genomic, inhibitory effect on macroscopic basolateral K+ conductance in the human colon, reducing its capacity for Cl− secretion. The molecular identity of the K+ channels constituting this aldosterone inhibitable K+ conductance is unclear. AIM: To characterise the K+ channel inhibited by aldosterone present in the basolateral membrane of human colonic crypt cells. METHODS: Crypts were isolated from biopsies of healthy sigmoid colon obtained during colonoscopy. The effect of aldosterone on basolateral K+ channels, and the possible involvement of Na+:H+ exchange, were studied by patch clamp techniques. Total RNA from isolated crypts was subjected to reverse transcriptase-polymerase chain reaction (RT-PCR) using primers specific to intermediate conductance K+ channels (KCNN4) previously identified in other human tissues. RESULTS: In cell attached patches, 1 nmol/l aldosterone significantly decreased the activity of intermediate conductance (27 pS) K+ channels by 31%, 53%, and 54% after 1, 5 and 10, minutes, respectively. Increasing aldosterone concentration to 10 nmol/l produced a further 56% decrease in channel activity after five minutes. Aldosterone 1–10 nmol/l had no effect on channel activity in the presence of 20 µmol/l ethylisopropylamiloride, an inhibitor of Na+:H+ exchange. RT-PCR identified KCNN4 mRNA, which is likely to encode the 27 pS K+ channel inhibited by aldosterone. CONCLUSION: Intermediate conductance K+ channels (KCNN4) present in the basolateral membranes of human colonic crypt cells are a target for the non-genomic inhibitory effect of aldosterone, which involves stimulation of Na+:H+ exchange, thereby reducing the capacity of the colon for Cl− secretion

Non-axisymmetric oscillations of stratified coronal magnetic loops with elliptical cross-sections

We study non-axisymmetric oscillations of a straight magnetic tube with an elliptic cross-section and density varying along the tube. The governing equations for kink and fluting modes in the thin tube approximation are derived. We found that there are two kink modes, polarised along the large and small axes of the elliptic cross-section. We have shown that the ratio of frequencies of the first overtone and fundamental harmonic is the same for both kink modes and independent of the ratio of the ellipse axes. On the basis of this result we concluded that the estimates of the atmospheric scale height obtained using simultaneous observations of the fundamental harmonic and first overtone of the coronal loop kink oscillations are independent of the ellipticity of the loop cross-section