scholarly journals Polynomial hull of a torus fibered over the circle

2018 ◽  
Vol 29 (04) ◽  
pp. 1850031
Author(s):  
Julien Duval ◽  
Mark Lawrence
Keyword(s):  
Solid Torus ◽  
Winding Number ◽  
Polynomial Hull ◽  
Holomorphic Discs

Given a 2-sheeted torus over the circle with winding number 1, we prove that its polynomial hull is a union of 2-sheeted holomorphic discs. Moreover, when the hull is non-degenerate its boundary is a Levi-flat solid torus foliated by such discs.

THE CASSON INVARIANT OF THE CYCLIC COVERING BRANCHED OVER SOME SATELLITE KNOT

2005 ◽  
Vol 14 (08) ◽  
pp. 1029-1044 ◽  
Author(s):  
YASUYOSHI TSUTSUMI
Keyword(s):  
Solid Torus ◽  
Covering Space ◽  
Winding Number ◽  
Cyclic Covering ◽  
Torus Knot ◽  
Satellite Knot

Let V be the standard solid torus in S3. Let Kp, 2 be the (p, 2)-torus knot in V such that Kp, 2 meets a meridian disk D of V in two points with the winding number zero and the 2-string tangle TKp, 2 obtained by cutting along D is a rational tangle. We compute the Casson invariant of the cyclic covering space of S3 branched over a satellite knot whose companion is any 2-bridge knot D(b1,…,b2m) and pattern is (V, Kp, 2).

Type-two invariants for knots in the solid torus

2016 ◽  
Vol 25 (08) ◽  
pp. 1650051 ◽  
Author(s):  
Khaled Bataineh
Keyword(s):  
Linear Combination ◽  
Solid Torus ◽  
Winding Number ◽  
Vassiliev Invariants ◽  
Natural Filtration ◽  
Gauss Diagram ◽  
Singular Knots

We introduce a natural filtration in the space of knots and singular knots in the solid torus, and start the study of the type-two Vassiliev invariants with respect to this filtration. The main result of the work states that any such invariant within the second term of this filtration in the space of knots with zero winding number is a linear combination of seven explicitly described Gauss diagram invariants. This introduces a basis (and a universal invariant) for the type-two Vassiliev invariants for knots with zero winding number. Then we formalize the problem of exploring the set of all type-two invariants for knots with zero winding number.

THE MODULE OF TWO-CHORD DIAGRAMS FOR ORIENTED KNOTS OF ZERO WINDING NUMBER IN THE SOLID TORUS

2012 ◽  
Vol 21 (07) ◽  
pp. 1250064 ◽  
Author(s):  
KHALED BATAINEH
Keyword(s):  
Abelian Group ◽  
Infinite Number ◽  
Solid Torus ◽  
Winding Number ◽  
Chord Diagrams ◽  
Direct Sum ◽  
Number Of Generators ◽  
Free Modules

In this paper we study the ℤ-module A2of two-chord diagrams for knots with zero winding number in the solid torus KST0, which is needed in studying the type-two invariants for knots in KST0. We show that this module (or abelian group), which is given as a presentation with infinite number of generators and an infinite number of relations, is a free infinitely generated module. Moreover, we show that this module is isomorphic to the direct sum of three free modules that are easier to understand.

KNOTS, SATELLITE OPERATIONS AND INVARIANTS OF FINITE ORDER

2006 ◽  
Vol 15 (08) ◽  
pp. 1061-1077 ◽  
Author(s):  
L. PLACHTA
Keyword(s):  
Finite Order ◽  
Nonnegative Integer ◽  
Solid Torus ◽  
Winding Number ◽  
Knot Invariant ◽  
Satellite Knot

Let sQ be the satellite operation on knots defined by a pattern (V, Q), where V is a standard solid torus in S3 and Q ⊂ V is a knot that is geometrically essential in V. It is known (Kuperberg [5]) that if v is any knot invariant of order n ≥ 0, then v ◦ sQ is also a knot invariant of order ≤ n. We show that if the knot Q has the winding number zero in V, then the satellite map [Formula: see text] passes n-equivalent knots into (n + 1)-equivalent ones. Kalfagianni [4] has defined for each nonnegative integer n surgery n-trivial knots and studied their properties. It is known that for each n every surgery n-trivial knot is n-trivial. We show that for each n there are n-trivial knots which do not admit a non-unitary n-trivializer that show they to be surgery n-trivial. Przytycki showed [12] that if a knot Q is trivial in S3 and is embedded in V in such a way that it is k-trivial inside V and if a knot [Formula: see text] is m-trivial, then the satellite knot [Formula: see text] is (k + m + 1)-trivial. We establish a version of aforementioned Przytycki's result for surgery n-triviality, refining thus a construction for surgery n-trivial knots suggested by Kalfagianni.

scholarly journals On numerical invariants for knots in the solid torus

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Khaled Bataineh
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Solid Torus ◽  
Geometric Features ◽  
Winding Number ◽  
Vassiliev Invariants ◽  
Numerical Invariants ◽  
Bridge Number

AbstractWe define some new numerical invariants for knots with zero winding number in the solid torus. These invariants explore some geometric features of knots embedded in the solid torus. We characterize these invariants and interpret them on the level of the knot projection. We also find some relations among some of these invariants. Moreover, we give lower bounds for some of these invariants using Vassiliev invariants of type one. We connect our invariants to the bridge number in the solid torus. We give a lower bound and an upper bound of the wrap of a knot in the solid torus in terms of our new invariants.

scholarly journals Simulations of a bubble wall interacting with an electroweak plasma

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Zong-Gang Mou ◽  
Paul M. Saffin ◽  
Anders Tranberg
Keyword(s):  
Large Scale ◽  
Baryon Asymmetry ◽  
Winding Number ◽  
Bubble Wall ◽  
Electroweak Phase Transition ◽  
First Order ◽  
The Standard Model ◽  
Technical Details ◽  
Real Time Simulations ◽  
Chern Simons

Abstract We perform large-scale real-time simulations of a bubble wall sweeping through an out-of-equilibrium plasma. The scenario we have in mind is the electroweak phase transition, which may be first order in extensions of the Standard Model, and produce such bubbles. The process may be responsible for baryogenesis and can generate a background of primordial cosmological gravitational waves. We study thermodynamic features of the plasma near the advancing wall, the generation of Chern-Simons number/Higgs winding number and consider the potential for CP-violation at the wall generating a baryon asymmetry. A number of technical details necessary for a proper numerical implementation are developed.

scholarly journals Winding Numbers, Unwinding Numbers, and the Lambert W Function

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Complex Number ◽  
Lambert W Function ◽  
Complex Numbers ◽  
Winding Number ◽  
Line Integral ◽  
Complex Functions

AbstractThe unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.

scholarly journals A theory of giant vortices

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Alexander A. Penin ◽  
Quinten Weller
Keyword(s):  
Asymptotic Expansion ◽  
Analytic Form ◽  
Scalar Fields ◽  
Winding Number ◽  
Asymptotic Results ◽  
Convergence Region ◽  
Ginzburg Landau ◽  
Landau Model ◽  
Neutral Scalar ◽  
Vortex Solutions

Abstract We elaborate a theory of giant vortices [1] based on an asymptotic expansion in inverse powers of their winding number n. The theory is applied to the analysis of vortex solutions in the abelian Higgs (Ginzburg-Landau) model. Specific properties of the giant vortices for charged and neutral scalar fields as well as different integrable limits of the scalar self-coupling are discussed. Asymptotic results and the finite-n corrections to the vortex solutions are derived in analytic form and the convergence region of the expansion is determined.

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