4,809 research outputs found
Rational cobordisms and integral homology
We consider the question of when a rational homology 3-sphere is rational
homology cobordant to a connected sum of lens spaces. We prove that every
rational homology cobordism class in the subgroup generated by lens spaces is
represented by a unique connected sum of lens spaces whose first homology
embeds in any other element in the same class. As a first consequence, we show
that several natural maps to the rational homology cobordism group have
infinite rank cokernels. Further consequences include a divisibility condition
between the determinants of a connected sum of 2-bridge knots and any other
knot in the same concordance class. Lastly, we use knot Floer homology combined
with our main result to obstruct Dehn surgeries on knots from being rationally
cobordant to lens spaces.Comment: 19 pages, final version to appear in Compositio Mathematic
Quasi-alternating Montesinos links
The aim of this article is to detect new classes of quasi-alternating links.
Quasi-alternating links are a natural generalization of alternating links.
Their knot Floer and Khovanov homology are particularly easy to compute. Since
knot Floer homology detects the genus of a knot as well as whether a knot is
fibered, as provided bounds on unknotting number and slice genus,
characterization of quasi-alternating links becomes an interesting open
problem. We show that there exist classes of non-alternating Montesinos links,
which are quasi-alternating.Comment: 11 pages, 7 figure
On the Coloring of Pseudoknots
Pseudodiagrams are diagrams of knots where some information about which
strand goes over/under at certain crossings may be missing. Pseudoknots are
equivalence classes of pseudodiagrams, with equivalence defined by a class of
Reidemeister-type moves. In this paper, we introduce two natural extensions of
classical knot colorability to this broader class of knot-like objects. We use
these definitions to define the determinant of a pseudoknot (i.e. the
pseudodeterminant) that agrees with the classical determinant for classical
knots. Moreover, we extend Conway notation to pseudoknots to facilitate the
investigation of families of pseudoknots and links. The general formulae for
pseudodeterminants of pseudoknot families may then be used as a criterion for
p-colorability of pseudoknots.Comment: 22 pages, 24 figure
Concordance groups of links
We define a notion of concordance based on Euler characteristic, and show that it gives rise to a concordance group of links in the 3-sphere, which has the concordance group of knots as a direct summand with infinitely generated complement. We consider variants of this using oriented and nonoriented surfaces as well as smooth and locally flat embeddings
Quasi-alternating links with small determinant
Quasi-alternating links of determinant 1, 2, 3, and 5 were previously
classified by Greene and Teragaito, who showed that the only such links are
two-bridge. In this paper, we extend this result by showing that all
quasi-alternating links of determinant at most 7 are connected sums of
two-bridge links, which is optimal since there are quasi-alternating links not
of this form for all larger determinants. We achieve this by studying their
branched double covers and characterizing distance-one surgeries between lens
spaces of small order, leading to a classification of formal L-spaces with
order at most 7
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