58 research outputs found
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simplified" by that application of "elementary moves".Comment: This paper has been withdrawn and replaced by arXiv:math.GT0404602,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 -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
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
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
The curve complex has dead ends
It is proved that the curve graph of a surface
has a local pathology that had not been identified as such: there are vertices
in such that is a dead end of every
geodesic joining to . 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
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
Erratum: On iterated torus knots and transversal knots
In math.GT/0002110 the author's Theorems 1.1 and 1.2, combined, implied that
iterated torus knots are transversally simple.
This result is in error and this erratum pin points the error. In "An
addendum on iterated torus knots" a more subtle result is proven resulting in
giving 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 and math.GT/0610566.)Comment: Erratum to math.GT/000211
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