517 research outputs found
Optimal Cobordisms between Torus Knots
We construct cobordisms of small genus between torus knots and use them to
determine the cobordism distance between torus knots of small braid index. In
fact, the cobordisms we construct arise as the intersection of a smooth
algebraic curve in with the unit 4-ball from which a 4-ball of
smaller radius is removed. Connections to the realization problem of
-singularities on algebraic plane curves and the adjacency problem for
plane curve singularities are discussed. To obstruct the existence of
cobordisms, we use Ozsv\'ath, Stipsicz, and Szab\'o's -invariant,
which we provide explicitly for torus knots of braid index 3 and 4.Comment: 24 pages, 7 figures. Version 3: Minor corrections, implementation of
referee's recommendations. Comments welcom
A sharp signature bound for positive four-braids
We provide the optimal linear bound for the signature of positive four-braids
in terms of the three-genus of their closures. As a consequence, we improve
previously known linear bounds for the signature in terms of the first Betti
number for all positive braid links. We obtain our results by combining bounds
for positive three-braids with Gordon and Litherland's approach to signature
via unoriented surfaces and their Goeritz forms. Examples of families of
positive four-braids for which the bounds are sharp are provided.Comment: 12 pages, 6 figures, comments welcome! Accepted for publication in Q.
J. Mat
Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space
We provide three 3-dimensional characterizations of the Z-slice genus of a
knot, the minimum genus of a locally-flat surface in 4-space cobounding the
knot whose complement has cyclic fundamental group: in terms of balanced
algebraic unknotting, in terms of Seifert surfaces, and in terms of
presentation matrices of the Blanchfield pairing. This result generalizes to a
knot in an integer homology 3-sphere and surfaces in certain simply connected
signature zero 4-manifolds cobounding this homology sphere. Using the
Blanchfield characterization, we obtain effective lower bounds for the Z-slice
genus from the linking pairing of the double branched cover of the knot. In
contrast, we show that for odd primes p, the linking pairing on the first
homology of the p-fold branched cover is determined up to isometry by the
action of the deck transformation group on said first homology.Comment: 39 pages, 5 figures, comments are welcome! v2: Added generalization
of the main theorem to knots and surfaces in more general 3- and 4-manifolds;
added new corollary showing equality of the Z-slice genus and the superslice
genus; expanded introduction, and added example in last sectio
Braids with as many full twists as strands realize the braid index
We characterize the fractional Dehn twist coefficient of a braid in terms of
a slope of the homogenization of the Upsilon function, where Upsilon is the
function-valued concordance homomorphism defined by Ozsv\'ath, Stipsicz, and
Szab\'o. We use this characterization to prove that -braids with fractional
Dehn twist coefficient larger than realize the braid index of their
closure. As a consequence, we are able to prove a conjecture of Malyutin and
Netsvetaev stating that -times twisted braids realize the braid index of
their closure. We provide examples that address the optimality of our results.
The paper ends with an appendix about the homogenization of knot concordance
homomorphisms.Comment: 26 pages, 5 figures, comments welcome! V2: Implementation of referee
suggestions. Accepted for publication by the Journal of Topolog
Holomorphically Equivalent Algebraic Embeddings
We prove that two algebraic embeddings of a smooth variety in
are the same up to a holomorphic coordinate change, provided
that is smaller than or equal to . This improves an algebraic
result of Nori and Srinivas. For the proof we extend a technique of Kaliman
using generic linear projections of .Comment: 17 pages. Version 2 acknowledges the fact that the main result of
this paper was previously established by Kaliman, see
http://arxiv.org/abs/1309.379
On cobordisms between knots, braid index, and the Upsilon-invariant
We use Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon-invariant to provide bounds
on cobordisms between knots that `contain full-twists'. In particular, we
recover and generalize a classical consequence of the Morton-Franks-Williams
inequality for knots: positive braids that contain a positive full-twist
realize the braid index of their closure. We also establish that quasi-positive
braids that are sufficiently twisted realize the minimal braid index among all
knots that are concordant to their closure. Finally, we provide inductive
formulas for the Upsilon invariant of torus knots and compare it to the
Levine-Tristram signature profile.Comment: 25 pages, 3 figures, comments welcome! Second version: Typos fixed,
implementation of recommendations. Accepted for publication by Mathematische
Annale
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