Only single twists on unknots can produce composite knots

Let be a knot in the -sphere , and a disc in meeting transversely more than once in the interior. For non-triviality we assume that over all isotopy of . Let () be a knot obtained from by cutting and -twisting along the disc (or equivalently, performing -Dehn surgery on ). Then we prove the following: (1) If is a trivial knot and is a composite knot, then ; (2) if is a composite knot without locally knotted arc in and is also a composite knot, then . We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

     

Twisted unknots

Let K be a knot in the 3-sphere S³, and D a disk in S³ meeting K transversely in the interior. For non-triviality we assume that |D∩K|?2 over all isotopies of K in S³??D. Let K_D,n(?S³) be the knot obtained from K by n twisting along the disk D. If the original knot is unknotted in S³, we call K_D,n a twisted unknot. We describe for which pairs (K,D) and integers n, the twisted unknot K_D,n is a torus knot, a satellite knot or a hyperbolic knot. To cite this article: M. A??t Nouh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On the stressed state in a fiber-reinforced composite material with twisted filaments

Using the model of a piecewise-homogeneous body in the framework of the linear theory of elasticity we study the distribution of internal stresses in a fiber-reinforced composite material with twisted filaments under loading at infinity by uniformly distributed normal strains in the direction of the filaments. It is assumed that the concentration of filaments is rather sparse and their interaction is not taken into account.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 74–79.     

Obtaining graph knots by twisting unknots

Let K be a knot in the 3-sphere S³ and D a disk in S³ meeting K transversely more than once in the interior. For nontriviality we assume that |D∩K|⩾2 over all isotopies of K in S³−∂D. Let K_D,n (⊂S³) be a knot obtained from K by n twisting along the disk D. We prove that if K is a trivial knot and K_D,n is a graph knot, then |n|⩽1 or K and D form a special pair which we call an “exceptional pair”. As a corollary, if (K,D) is not an exceptional pair, then by twisting unknot K more than once (in the positive or the negative direction) along the disk D, we always obtain a knot with positive Gromov volume. We will also show that there are infinitely many graph knots each of which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained in such a manner.     

On twisted character sums

In this paper we give a simple proof of a result by Burgess about short sums involving Dirichlet characters and exponentials. Indeed we establish a slightly stronger and more general bound that applies to sums of the form \({\sum_{n=M+1}^{M+N}f(\alpha n)\chi(n)}\), where χ is a non-principal character to the modulus p and f is a smooth 1-periodic function.     

On spherical twisted conjugacy classes

Let G be a simple algebraic group over an algebraically closed field of good odd characteristic, and let ?? be an automorphism of G arising from an involution of its Dynkin diagram. We show that the spherical ??-twisted conjugacy classes are precisely those intersecting only Bruhat cells corresponding to twisted involutions in the Weyl group. We show how the analogue of this statement fails in the triality case. As a byproduct, we obtain a dimension formula for spherical twisted conjugacy classes that was originally obtained by J.-H. Lu in characteristic zero.     

On twisted contact groupoids and on integration of twisted Jacobi manifolds

We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic groupoids with twisted contact ones. Some examples for each structure are presented.     

On twisted representations of vertex algebras

In this paper, we develop a formalism for working with representations of vertex and conformal algebras by generalized fields—formal power series involving non-integer powers of the variable. The main application of our technique is the construction of a large family of representations for the vertex superalgebra corresponding to an integer lattice Λ. For an automorphism coming from a finite-order automorphism we find the conditions for existence of twisted modules of . We show that the category of twisted representations of is semisimple with finitely many isomorphism classes of simple objects.     

On the symmetric square Unstable twisted characters

We provide a purely local computation of the (elliptic) twisted (by “transpose-inverse”) character of the representationπ=I(1) of PGL(3) over ap-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads — see [FK] — to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals. The new case in [FK] is the unstable one. A direct local proof of the fundamental lemma is given in [V].     

On solutions to the twisted Yang-Baxter equation

Solutions to the twisted Yang-Baxter equation given by intertwiners for cyclic representations of are described via two coupled lattice current algebras. Bibliography: 4 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Published inZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 301–308. Translated by V. O. Tarasov.     

On divisible designs and twisted field planes

Starting from desarguesian and twisted field planes we construct and study some classes of divisible designs admitting an automorphism group which is 2-transitive on the set of point classes.     

On Tutte polynomial uniqueness of twisted wheels

A graph G is called T-unique if any other graph having the same Tutte polynomial as G is isomorphic to G. Recently, there has been much interest in determining T-unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105-119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57-65] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all T-unique. In this paper, we prove that the twisted wheels are also T-unique.