NONSURJECTIVE SATELLITE OPERATORS AND PIECEWISE-LINEAR CONCORDANCE

Forum of Mathematics Sigma ◽

10.1017/fms.2016.31 ◽

2016 ◽

Vol 4 ◽

Cited By ~ 7

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.

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

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216505004226 ◽

2005 ◽

Vol 14 (08) ◽

pp. 1029-1044 ◽

Cited By ~ 2

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).

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GAUSSIAN RANDOM POLYGONS ARE GLOBALLY KNOTTED

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216594000332 ◽

1994 ◽

Vol 03 (04) ◽

pp. 455-464 ◽

Cited By ~ 14

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.

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KNOTS, SATELLITE OPERATIONS AND INVARIANTS OF FINITE ORDER

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216506004944 ◽

2006 ◽

Vol 15 (08) ◽

pp. 1061-1077 ◽

Cited By ~ 2

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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