On composite twisted unknots

Following Mathieu, Motegi and others, we consider the class of possible composite twisted unknots as well as pairs of composite knots related by twisting. At most one composite knot can arise from a particular -twisting of an unknot; moreover a twisting of the unknot cannot be composite if we have applied more than a single full twist. A pair of composite knots can be related through at most one full twist for a particular -twisting, or one summand was unaffected by the twist, or the knots were the right and left handed granny knots. Finally a conjectured characterization of all composite twisted unknots that do arise is given.

     

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.     

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 Automorphisms and Twisted Right Gyrogroups

In this paper we make an attempt to study right loops (S, o) in which, for each y ∈ S, the map σ _y from the inner mapping group G _S of (S, o) to itself given by σ _y (h)(x)o h(y) = h(xoy), x ∈ S, h ∈ G _S is a homomorphism. The concept of twisted automorphisms of a right loop and also the concept of twisted right gyrogroup appears naturally and it turns out that the study is almost equivalent to the study of twisted automorphisms and a twisted right gyrogroup. We also study relationship between twisted gyrotransversals and twisted subgroups.     

Twisted calculus

A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis, the categories of twisted modules and integrable twisted differential modules are equivalent. As particular cases, one recovers classical results from the theory of finite difference equations or q-difference equations.     

Twisted Planes

Let k be a commutative ring. We find and characterize a new family of twisted planes (i.e., associative unitary k-algebra structures on the k-module k[X, Y], having k[X] and k[Y] as subalgebras). Similar results are obtained for the k-module of two variables power series k[[X, Y]].     

Twisted BCH-codes

We develop the theory of a generalization of the notion ofBCH-code to additive codes, which are not necessarily linear. The usefulness of this notion is demonstrated by constructing a large number of record-breaking linear codes via concatenation. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 377–389, 1997     

Twisted analytic torsion

We review the Reidemeister, Ray-Singer’s analytic torsion and the Cheeger-Mller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differential forms on the total space of a principal circle bundle twisted by an invariant flux form. We show that when the dimension is even, such a torsion is invariant under certain deformation of the metric and the flux form. Under T-duality which exchanges the topology of the bundle and the flux form and the radius of the circular fiber with its inverse, the twisted torsion of invariant forms are inverse to each other for any dimension.     

Twisted Steinberg algebras

We introduce twisted Steinberg algebras over a commutative unital ring R. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid G and each locally constant 2-cocycle σ on G taking values in the units $R^{\times }$, we study the algebra $A_R(G,\sigma )$ consisting of locally constant compactly supported R-valued functions on G, with convolution and involution “twisted” by σ. We also introduce a “discretised” analogue of a twist Σ over a Hausdorff étale groupoid G, and we show that there is a one-to-one correspondence between locally constant 2-cocycles on G and discrete twists over G admitting a continuous global section. Given a discrete twist Σ arising from a locally constant 2-cocycle σ on an ample Hausdorff groupoid G, we construct an associated twisted Steinberg algebra $A_R(G;\mathrm{\Sigma })$, and we show that it coincides with $A_R(G,{\sigma }^{?1})$. Given any discrete field ${\mathbb{F}}_d$, we prove a graded uniqueness theorem for $A_{{\mathbb{F}}_d}(G,\sigma )$, and under the additional hypothesis that G is effective, we prove a Cuntz–Krieger uniqueness theorem and show that simplicity of $A_{{\mathbb{F}}_d}(G,\sigma )$ is equivalent to minimality of G.     

Twisted support varieties

We define and study twisted support varieties for modules over an Artin algebra, where the twist is induced by an automorphism of the algebra. Under a certain finite generation hypothesis we show that the twisted variety of a module satisfies Dade’s Lemma and is one dimensional precisely when the module is periodic with respect to the twisting automorphism. As a special case we obtain results on DTr-periodic modules over Frobenius algebras.     

Twisted Equivariant Matter

We show how general principles of symmetry in quantum mechanics lead to twisted notions of a group representation. This framework generalizes both the classical threefold way of real/complex/ quaternionic representations as well as a corresponding tenfold way which has appeared in condensed matter and nuclear physics. We establish a foundation for discussing continuous families of quantum systems. Having done so, topological phases of quantum systems can be defined as deformation classes of continuous families of gapped Hamiltonians. For free particles, there is an additional algebraic structure on the deformation classes leading naturally to notions of twisted equivariant K-theory. In systems with a lattice of translational symmetries, we show that there is a canonical twisting of the equivariant K-theory of the Brillouin torus. We give precise mathematical definitions of two invariants of the topological phases which have played an important role in the study of topological insulators. Twisted equivariant K-theory provides a finer classification of topological insulators than has been previously available.     

Twisted Semigroup Crossed Products and Twisted Toeplitz Algebras of Ordered Groups

Let г^＋ be the positive cone of a totally ordered abelian group г, and σa cocycle in г. We study the twisted crossed products by actions of г＋ as endomorphisms of C^＊-algebras, and use this to generalize the theorem of Ji.     

Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” A^T, T sumHom(H,End (A)). If ^T is convolution invertible, the categories of relative right Hopf modules over A and A^Tare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products     

Twisted cubics, bis

We consider the smooth compactification constructed in [12] for a space of varieties like twisted cubics. We show this compactification embeds naturally in a product of flag varieties.Partially supported by CNPq, Pronex (ALGA)