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 ss-invariant for links

We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen, Beliakova and Wehrli's $s$-invariant. We also study almost positive links, in particular, determine the $s$-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 00-surgery and the slice-ribbon conjecture

Akbulut and Kirby conjectured that two knots with the same $0$-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 $n$, there exist infinitely many mutually distinct knots such that $2$-handle additions along them with framing $n$ yield the same $4$-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 $n$-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 $6_3$ has infinitely many non-characterizing slopes, which affirmatively answers a question by Baker and Motegi. Furthermore, we prove that the knots $6_2$, $6_3$, $7_6$, $7_7$, $8_1$, $8_3$, $8_4$, $8_6$, $8_7$, $8_9$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{17}$,$8_{20}$ and $8_{21}$ 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