The Self-Linking of Torus Knots (original) (raw)
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Cornell University - arXiv, 2022
We give a complete coarse classification of Legendrian and transverse torus knots in any contact structure on 3. CONTENTS 2 JOHN B. ETNYRE, HYUNKI MIN, AND ANUBHAV MUKHERJEE 7.4. The extra torus knot when < 0 97 7.5. The Giroux torsion of the examples above 97 7.6. Non-loose torus knots with convex Giroux torsion 98 7.7. Proof that the algorithm gives a complete classification 100 8. General results of non-loose torus knots 101 References 105
On iterated torus knots and transversal knots
2001
A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transversal knot in the standard contact structure for S^3 is defined to be transversally simple if it is characterized up to transversal isotopy by its topological knot type and its self-linking number. Theorem 2 of Birman and Wrinkle [op cit] establishes that exchange reducibility implies transversally simplicity. The main result in this note, establishes that iterated torus knots are exchange reducible. It then follows as a Corollary that iterated torus knots are transversally simple.
Twisting of composite torus knots
The Michigan Mathematical Journal, 2017
We prove that the family of connected sums of torus knots T (2, p) # T (2, q) # T (2, r) is nontwisted for any odd positive integers p, q, r ≥ 3, partially answering in the positive a conjecture of Teragaito [22].
Classical invariants of Legendrian knots in the 3-dimensional torus
Topology and its Applications, 2015
All knots in R 3 possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on R 3 can be defined. The definitions extend easily to nullhomologous knots in any 3-manifold M endowed with a contact structure ξ. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not nullhomologous, in a contact structure on the 3-torus T 3. We show how to compute the Thurston-Bennequin and rotation invariants in a tight oriented contact structure on T 3 using projections.
The L^2 signature of torus knots
2010
We find a formula for the L2 signature of a (p,q) torus knot, which is the integral of the omega-signatures over the unit circle. We then apply this to a theorem of Cochran-Orr-Teichner to prove that the n-twisted doubles of the unknot, for n not 0 or 2, are not slice. This is a new proof of the result first proved by Casson and Gordon.
Vassiliev Invariants for Torus Knots
Journal of Knot Theory and Its Ramifications, 1996
Vassiliev. invariants up to order six for arbitrary torus knots {n, m}, with n and m coprime integers, are computed. These invariants are polynomials in n and m whose degree coincide with their order. Furthermore, they turn out to be integer-valued in a normalization previously proposed by the authors.
On Rational Knots and Links in the Solid Torus
Mediterranean Journal of Mathematics
We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.
Non-triviality of welded knots and ribbon torus-knots
arXiv (Cornell University), 2024
In this paper we study welded knots and their invariants. We focus on generating examples of non-trivial knotted ribbon tori as the tube of welded knots that are obtained from classical knot diagrams by welding some of the crossings. Non-triviality is shown by determining the fundamental group of the concerned welded knot. Sample examples under consideration are the standard diagrams of the family of (2, q) torus knots and the twist knots. Standard diagrams of knots from Rolfsen's tables with 6 crossings are also discussed which are not in the family of torus and twist knots.
Knots Obtained by Twisting Unknots
Let K be the unknot in the 3-sphere S3, and D a disk in S3 meeting K transversely in the interior, at least twice (after all isotopies). We denote by KD,n a knot obtained from K by n twistings along the disk D. We describe for which pairs (K,D) and integers n, KD,n is a torus knot, a satellite knot or a hyperbolic knot.