An addendum on iterated torus knots

In Theorem 1.2 of the paper math.GT/0002110 the author claimed to have proved that all transversal knots whose topological knot type is that of an iterated torus knot (we call them cable knots) are transversally simple. That theorem is false, and the Erratum math.GT/0610565 identifies the gap. The purpose of this paper is to explore the situation more deeply, in order to pinpoint exactly which cable knots are {\it not} transversally simple. The class is subtle and interesting. We will recover the strength of the main theorem in math.GT/0002110, in the sense that we will be able to prove a strong theorem about cable knots, but the theorem itself is more subtle than Theorem 1.2 of math.GT/0002110. In particular, we give a geometric realization of the Honda-Etnyre transverse (2,3)-cable of the (2,3)-torus knot example (Appendix joint with H. Matsuda). (See math.SG/0306330)Comment: 33 pages, 16 figures, an addendum to "On iterated torus knots and transversal knots", math.GT/000211

Monotonic Simplification and Recognizing Exchange Reducibility

The Markov Theorem Without Stabilization (MTWS) (see math.GT/0310279) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isotopies that were identified and analyzed--destabilization, exchange moves and elementary braid preserving flypes. One of the critical open problems left in the wake of the MTWS is the "recognition problem"--determining when a given closed n-braid admits a specified move of the calculus. In this note we give an algorithmic solution to the recognition problem for three isotopies of the MTWS calculus--destabilization, exchange moves and braid preserving flypes. The algorithm is directed by a complexity measure that can be "monotonic simplified" by that application of "elementary moves".Comment: This paper has been withdrawn and replaced by arXiv:math.GT0404602, 26 JAN 2012 which is titled " Recognizing destabilization, exchange moves and flypes

Recognizing destabilization, exchange moves and flypes

The Markov Theorem Without Stabilization (MTWS) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isotopies that were identified and analyzed---destabilization, exchange moves and elementary braid preserving flypes. One of the critical open problems left in the wake of the MTWS is the "recognition problem"---determining when a given closed $n$-braid admits a specified move of the calculus. In this note we give an algorithmic solution to the recognition problem for these three key isotopies of the MTWS calculus. The algorithm is "directed" by a complexity measure that can be {\em monotonically simplified} by the application of "elementary moves".Comment: 66 pages, 27 figures (some figures use color). This is a replacement for arXiv:math.GT/0507124 titled "Monotonic Simplification and Recognizing Exchange Reducibility", which has been withdraw

On rectangular diagrams, Legendrian knots and transverse knots

A correspondence is studied by H. Matsuda between front projections of Legendrian links in the standard contact structure for 3-space and rectangular diagrams. In this paper, we introduce braided rectangular diagrams, and study a relationship with Legendrian links in the standard contact structure for 3-space. We show Alexander and Markov Theorems for Legendrian links in 3-space.Comment: 15 pages, 11 figure

Embedded annuli and Jones' conjecture

We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the problem by Kawamuro.Comment: 10 pages, 13 figures; expanded background, added figure

Positive knots and knots with braid index three have property_p

We prove that positive knots and knots with braid index three in the 3-sphere satisfy the Property P conjecture.Comment: 24 pages, 12 figure

The curve complex has dead ends

It is proved that the curve graph $C^1(\Sigma)$ of a surface $\Sigma_{g,n}$ has a local pathology that had not been identified as such: there are vertices $\alpha,\beta$ in $C^1(\Sigma)$ such that $\beta$ is a dead end of every geodesic joining $\alpha$ to $\beta$. It also has double dead-ends. Every dead end has depth 1.Comment: Final version, published on-line in Geometriae Dedicata. 4 pages, 1 figur

On Markov's Theorem

We give a new proof of Markov's classical theorem relating any two closed braid representations of the same knot or link. The proof is based upon ideas in a forthcoming paper by the authors, "Stabilization in the braid groups". The new proof of the classical Markov theorem is used by Nancy Wrinkle in her forthcoming manuscript "The Markov Theorem for transverse knots".Comment: 16 pages, 10 figure

Heegaard splittings and virtually Haken Dehn filling

We use Heegaard splittings to give some examples of virtually Haken 3-manifolds.Comment: 18 pages, 19 figure

Stabilization in the Braid Groups (with applications to transverse knots)

Withdrawn and replaced by two related manuscripts: (1) "Stabilization in the braid groups I:MTWS", published in Geometry and Topology Volume 10 (2006), 413-540, arXiv:math.GT/0310279, and (2) "Stabilization in the braid groups II: Transversal simplicity of knots", Geometry and Topology Volume 10 (2006), to appear, arXiv:math,GT/0310280.Comment: Withdraw