293 research outputs found
The Surgery Unknotting Number of Legendrian Links
The surgery unknotting number of a Legendrian link is defined as the minimal
number of particular oriented surgeries that are required to convert the link
into a Legendrian unknot. Lower bounds for the surgery unknotting number are
given in terms of classical invariants of the Legendrian link. The surgery
unknotting number is calculated for every Legendrian link that is topologically
a twist knot or a torus link and for every positive, Legendrian rational link.
In addition, the surgery unknotting number is calculated for every Legendrian
knot in the Legendrian knot atlas of Chongchitmate and Ng whose underlying
smooth knot has crossing number 7 or less. In all these calculations, as long
as the Legendrian link of components is not topologically a slice knot, its
surgery unknotting number is equal to the sum of and twice the smooth
4-ball genus of the underlying smooth link.Comment: 26 pages, 27 figure
Knots with unknotting number one and Heegaard Floer homology
We use Heegaard Floer homology to give obstructions to unknotting a knot with
a single crossing change. These restrictions are particularly useful in the
case where the knot in question is alternating. As an example, we use them to
classify all knots with crossing number less than or equal to nine and
unknotting number equal to one. We also classify alternating knots with ten
crossings and unknotting number equal to one.Comment: Minor revisions, updated reference
Unknotting information from Heegaard Floer homology
We use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsváth and Szabó's obstruction to unknotting number one. We determine the unknotting numbers of 910, 913, 935, 938, 1053, 10101 and 10120; this completes the table of unknotting numbers for prime knots with crossing number nine or less. Our obstruction uses a refined version of Montesinos' theorem which gives a Dehn surgery description of the branched double cover of a knot
Knots with unknotting number 1 and essential Conway spheres
For a knot K in S^3, let T(K) be the characteristic toric sub-orbifold of the
orbifold (S^3,K) as defined by Bonahon and Siebenmann. If K has unknotting
number one, we show that an unknotting arc for K can always be found which is
disjoint from T(K), unless either K is an EM-knot (of Eudave-Munoz) or (S^3,K)
contains an EM-tangle after cutting along T(K). As a consequence, we describe
exactly which large algebraic knots (ie algebraic in the sense of Conway and
containing an essential Conway sphere) have unknotting number one and give a
practical procedure for deciding this (as well as determining an unknotting
crossing). Among the knots up to 11 crossings in Conway's table which are
obviously large algebraic by virtue of their description in the Conway
notation, we determine which have unknotting number one. Combined with the work
of Ozsvath-Szabo, this determines the knots with 10 or fewer crossings that
have unknotting number one. We show that an alternating, large algebraic knot
with unknotting number one can always be unknotted in an alternating diagram.
As part of the above work, we determine the hyperbolic knots in a solid torus
which admit a non-integral, toroidal Dehn surgery. Finally, we show that having
unknotting number one is invariant under mutation.Comment: This is the version published by Algebraic & Geometric Topology on 19
November 200
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