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.

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.

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).

scholarly journals Vassiliev invariants of type one for links in the solid torus

2010 ◽  
Vol 157 (16) ◽  
pp. 2495-2504 ◽  
Author(s):  
Khaled Bataineh ◽  
Mohammad Abu Zaytoon
Keyword(s):  
Solid Torus ◽  
Vassiliev Invariants

scholarly journals VASSILIEV INVARIANTS FROM PARITY MAPPINGS

2013 ◽  
Vol 22 (04) ◽  
pp. 1340008 ◽  
Author(s):  
H. A. DYE
Keyword(s):  
Vassiliev Invariants ◽  
Virtual Knots ◽  
Gauss Diagram

Parity mappings (weights) from the chords of a Gauss diagram to the integers are defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and consequently, virtual knots. Each family forms a set of degree n Vassiliev invariants for n ≥ 1.

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 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.

INVARIANT POLYNOMIAL OF FRAMED KNOTS IN THE SOLID TORUS AND ITS APPLICATION TO WAVE FRONTS AND LEGENDRIAN KNOTS

1996 ◽  
Vol 05 (06) ◽  
pp. 743-778 ◽  
Author(s):  
FRANCESCA AICARDI
Keyword(s):  
Solid Torus ◽  
Invariant Polynomial ◽  
Partial Index ◽  
Legendrian Knots ◽  
Wave Fronts ◽  
Vassiliev Invariants ◽  
Legendrian Knot ◽  
Legendrian Curves

An invariant polynomial s(t) is defined for framed knots in the solid torus. The coefficients are Vassiliev invariants of order one. An invariant polynomial A(t) of Legendrian curves is introduced and it is shown how to calculate it from their fronts. The coefficient of A(t) of the order n term is the restriction to the discriminant of the selftangencies with partial index n of the Arnold invariant J+ of wave fronts. The polynomial A(t) of a Legendrian curve is recovered from the polynomial s(t) of the Legendrian knot, provided with its natural contact framing.

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.

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