416 research outputs found
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 in by
-Dehn surgery along a trivial knot , then the link 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
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
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
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