774 research outputs found
The Rasmussen invariant of a homogeneous knot
A homogeneous knot is a generalization of alternating knots and positive
knots. We determine the Rasmussen invariant of a homogeneous knot. This is a
new class of knots such that the Rasmussen invariant is explicitly described in
terms of its diagrams. As a corollary, we obtain some characterizations of a
positive knot. In particular, we recover Baader's theorem which states that a
knot is positive if and only if it is homogeneous and strongly quasipositive.Comment: 12pages, 6 figure
Characterization of positive links and the -invariant for links
We characterize positive links in terms of strong quasipositivity,
homogeneity and the value of Rasmussen, Beliakova and Wehrli's -invariant.
We also study almost positive links, in particular, determine the
-invariants of almost positive links. This result suggests that all almost
positive links might be strongly quasipositive. On the other hand, it implies
that almost positive links are never homogeneous links.Comment: 18 pages, 10 figures, v5:the statements of Corollary 1.7 and
Corollary 7.4 were corrected, Question 10.5 and two references were added,
v6:this is the published version, survey parts are delete
Fibered knots with the same -surgery and the slice-ribbon conjecture
Akbulut and Kirby conjectured that two knots with the same -surgery are
concordant. In this paper, we prove that if the slice-ribbon conjecture is
true, then the modified Akbulut-Kirby's conjecture is false. We also give a
fibered potential counterexample to the slice-ribbon conjecture.Comment: 15 pages, 9 figures; Final version. In version 4, Abstract is
mistakenly delete
Ribbon disks with the same exterior
We construct an infinite family of slice disks with the same exterior, which
gives an affirmative answer to an old question asked by Hitt and Sumners in
1981. Furthermore, we prove that these slice disks are ribbon disks.Comment: 9 pages, 14 figures. Statements of Lemmas 4.1 and 4.2 are clarifie
Unoriented band surgery on knots and links
We consider a relation between two kinds of unknotting numbers defined by
using a band surgery on unoriented knots; the band-unknotting number and
H(2)-unknotting number, which we may characterize in terms of the first Betti
number of surfaces in S^3 spanning the knot and the trivial knot. We also give
several examples for these numbers.Comment: 22 page
A construction of slice knots via annulus twists
We give a new construction of slice knots via annulus twists. The simplest
slice knots obtained by our method are those constructed by Omae. In this
paper, we introduce a sufficient condition for given slice knots to be ribbon,
and prove that all Omae's knots are ribbon.Comment: 26 pages and 28 figures. Comments are welcome.Version 2: A new
section added. Version 3: The definition of an annulus twist was clarified.
The anonymous referee pointed out a gap of Theorem 3.1. The statement of
Theorem 3.1 was weakened. Version 4: Abstract, Introduction and Section 6 are
rewritte
Annulus twist and diffeomorphic 4-manifolds II
We solve a strong version of Problem 3.6 (D) in Kirby's list, that is, we
show that for any integer , there exist infinitely many mutually distinct
knots such that -handle additions along them with framing yield the same
-manifold.Comment: 16 pages, 23 figure
Annulus twist and diffeomorphic 4-manifolds
We give a method for obtaining infinitely many framed knots which represent a
diffeomorphic 4-manifold. We also study a relationship between the -shake
genus and the 4-ball genus of a knot. Furthermore we give a construction of
homotopy 4-spheres from a slice knot with unknotting number one.Comment: 19 pages, 17 figure
Knots with infinitely many non-characterizing slopes
Using the techniques on annulus twists, we observe that has infinitely
many non-characterizing slopes, which affirmatively answers a question by Baker
and Motegi. Furthermore, we prove that the knots , , , ,
, , , , , , , , ,
, , , and have infinitely many
non-characterizing slopes. We also introduce the notion of trivial annulus
twists and give some possible applications. Finally, we completely determine
which knots have special annulus presentations up to 8-crossings.Comment: v2: Added Appendix with a complete proof of Theorem 3.1. This paper
has been accepted by Kodai Mathematical Journa
The dealternating number and the alternation number of a closed 3-braid
We give an upper bound for the dealternating number of a closed 3-braid. As
applications, we determine the dealternating numbers, the alternation numbers
and the Turaev genera of some closed positive 3-braids. We also show that there
exist infinitely many positive knots with any dealternating number (or any
alternation number) and any braid index.Comment: We corrected Section
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