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).

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 NONSURJECTIVE SATELLITE OPERATORS AND PIECEWISE-LINEAR CONCORDANCE

2016 ◽  
Vol 4 ◽  
Author(s):  
ADAM SIMON LEVINE
Keyword(s):  
Piecewise Linear ◽  
Solid Torus ◽  
Satellite Knot

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-linear disk in any homology 4-ball.

Minimal Plat Representations of Prime Knots and Links are not Unique

1976 ◽  
Vol 28 (1) ◽  
pp. 161-167 ◽  
Author(s):  
José M. Montesinos
Keyword(s):  
Infinite Family ◽  
Equivalence Classes ◽  
Covering Space ◽  
Heegaard Splittings ◽  
Cyclic Covering ◽  
Knots And Links

Letdenote the 2-fold cyclic covering space branched over a linkLin S3. We wish to describe an infinite family of prime knots and links in which each memberLexhibits two minimal 6-plat representations, where the associated Heegaard splittings ofare minimal and inequivalent. Thus each knot or link of that family admits at least two equivalence classes of 6-plat representations which are minimal.

TWISTED TORUS KNOTS WITH ESSENTIAL TORI IN THEIR COMPLEMENTS

2013 ◽  
Vol 22 (08) ◽  
pp. 1350041 ◽  
Author(s):  
SANGYOP LEE
Keyword(s):  
Torus Knots ◽  
Torus Knot ◽  
Satellite Knot

A twisted torus knot is a torus knot with a number of full-twists on some adjacent strands. In this paper, we show that if a twisted torus knot is a satellite knot then the number of full-twists is generically at most two.

Homology Invariants

1978 ◽  
Vol 30 (03) ◽  
pp. 655-670 ◽  
Author(s):  
Richard Hartley ◽  
Kunio Murasugi
Keyword(s):  
Homology Group ◽  
Covering Space ◽  
Simple Relationship ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Homology Groups ◽  
Branched Covering ◽  
The Relationship

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

GAUSSIAN RANDOM POLYGONS ARE GLOBALLY KNOTTED

1994 ◽  
Vol 03 (04) ◽  
pp. 455-464 ◽  
Author(s):  
DOUGLAS JUNGREIS
Keyword(s):  
Random Walk ◽  
Small Perturbation ◽  
High Probability ◽  
Random Variables ◽  
Solid Torus ◽  
Line Segments ◽  
Random Polygons ◽  
Straight Line ◽  
Gaussian Random Walk ◽  
Satellite Knot

A Gaussian random walk is a random walk in which each step is a vector whose coordinates are Gaussian random variables. In 3-space, if a Gaussian random walk of n steps begins and ends at the origin, then we can join successive points by straight line segments to get a knot. It is known that if n is large, then the knot is non-trivial with high probability. We give a new proof of this fact. Our proof shows in addition that with high probability the knot is contained as an essential loop in a fat, knotted, solid torus. Therefore the knot is a satellite knot and cannot be unknotted by any small perturbation.

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 DEHN SURGERIES ON 2-BRIDGE LINKS WHICH YIELD REDUCIBLE 3-MANIFOLDS

2009 ◽  
Vol 18 (07) ◽  
pp. 917-956 ◽  
Author(s):  
HIROSHI GODA ◽  
CHUICHIRO HAYASHI ◽  
HYUN-JONG SONG
Keyword(s):  
Torus Knot ◽  
Satellite Knot

We completely determine which Dehn surgeries on 2-bridge links yield reducible 3-manifolds. Further, we consider which surgery on one component of a 2-bridge link yields a torus knot, a cable knot and a satellite knot in this paper.

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.

Sign in / Sign up

Export Citation Format

Share Document