Ascending number of knots and links

We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number.Comment: 11 pages, 30 figure

Closed incompressible surfaces in the complements of positive knots

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splitability and the primeness of positive knots and links can be seen from their positive diagrams.Comment: 6 pages, 6 figure

Non-triviality of generalized alternating knots

In this article, we consider alternating knots on a closed surface in the 3-sphere, and show that these are not parallel to any closed surface disjoint from the prescribed one.Comment: 8 pages, 4 figure

Impossibility of obtaining split links from split links via twistings

We show that if a split link is obtained from a split link $L$ in $S^3$ by $1/n$-Dehn surgery along a trivial knot $C$, then the link $L\cup C$ is splittable. That is to say, it is impossible to obtain a split link from a split link via a non-trivial twisting. As its corollary, we completely determine when a trivial link is obtained from a trivial link via a twisting.Comment: 3 page

Morse position of knots and closed incompressible surfaces

In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both of knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have following results. *Smallness of Montesinos tangles with length two and Kinoshita's theta curve *Classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions.Comment: 20 pages, 17 figures. This version (v6) is a final version to appear in J. Knot Theory and its Ramification

Essential state surfaces for knots and links

We study a canonical spanning surface obtained from a knot or link diagram depending on a given Kauffman state, and give a sufficient condition for the surface to be essential. By using the essential surface, we can see the triviality and splittability of a knot or link from its diagrams. This has been done on the extended knot or link class which includes all of semiadequate, homogeneous, and most of algebraic knots and links. In the process of the proof of main theorem, Gabai's Murasugi sum theorem is extended to the case of nonorientable spanning surfaces.Comment: In version 2, main theorem was expanded and the proof was refined. In version 3, Theorem 1.2 was added. In version 4, Theorem 1.2 (3) in v3 was removed, and many examples and problems are added. In version 5, many places are rewritten due to referee reports, and some new references are added. The version 6 is a final version which was accepted by J. Austral. Math. So

Edge number of knots and links

We introduce a new numerical invariant of knots and links made from the partitioned diagrams. It measures the complexity of knots and links.Comment: 7 pages, 6 figure

A property of diagrams of the trivial knot

In this paper, we give a necessary condition for a diagram to represent the trivial knot.Comment: 11 pages, 15 figure

Knots and surfaces

This article is an English translation of Japanese article "Musubime to Kyokumen", Math. Soc. Japan, Sugaku Vol. 67, No. 4 (2015) 403--423. It surveys a specific area in Knot Theory concerning surfaces in knot exteriors. In version 2, we added comments on the solutions or counterexamples for Conjecture 3.5, Conjecture 3.7 and Conjecture 5.30.Comment: Any comment would be highly appreciate

Multibranched surfaces in 3-manifolds

This is a latest survey article on embeddings of multibranched surfaces into 3-manifolds.Comment: Comments are welcom