# UNKNOTTING NUMBERS OF DIAGRAMS OF A GIVEN NONTRIVIAL KNOT ARE UNBOUNDED(Knots and soft-matter physics: Topology of polymers and related topics in physics, mathematics and biology)

@inproceedings{Taniyama2008UNKNOTTINGNO, title={UNKNOTTING NUMBERS OF DIAGRAMS OF A GIVEN NONTRIVIAL KNOT ARE UNBOUNDED(Knots and soft-matter physics: Topology of polymers and related topics in physics, mathematics and biology)}, author={Kouki Taniyama}, year={2008} }

We show that for any nontrivial knot K and any natural number n, there is a diagram D of K such that the unknotting number of D is greater than or equal to n. It is well-known that twice the unknotting number of K is less than or equal to the crossing number of K minus one. We show that the equality holds only when K is a (2, p)-torus knot.

#### 8 Citations

Pseudo diagrams of knots, links and spatial graphs

- Physics, Mathematics
- 2009

A pseudo diagram of a spatial graph is a spatial graph project i n on the 2-sphere with over/under information at some of the double points. We introduce the trivializing (resp. knotting) number of a… Expand

ON AN INEQUALITY BETWEEN UNKNOTTING NUMBER AND CROSSING NUMBER OF LINKS

- Mathematics
- 2010

It is well-known that for any link L, twice the unknotting number of L is less than or equal to the crossing number of L. Taniyama characterized the links which satisfy the equality. We characterize… Expand

The Unknotting number and band-unknotting number of a knot

- Mathematics
- 2012

Proof. First, we show the ‘if’ part. If P is one of the projections of the connected sum of a (2, r)-torus knot diagram and a (2, s)-torus knot diagram, it follows from Theorem 2.6 and Proposition… Expand

Unknotting numbers and crossing numbers of spatial embeddings of a planar graph

- Mathematics
- 2020

It is known that the unknotting number $u(L)$ of a link $L$ is less than or equal to half the crossing number $c(L)$ of $L$. We show that there are a planar graph $G$ and its spatial embedding $f$… Expand

Cobordisms with chronologies and a generalisation of the Khovanov complex

- Mathematics
- 2010

There are two categorifications of the Jones polynomial: "even" discovered by M.Khovanov in 1999 and "odd" dicovered by P.Ozsvath, J.Rasmussen and Z.Szabo in 2007. The first one can be fully… Expand

Notes on regular projections of knots

- Mathematics
- 2010

We review the results on knot and link projections and introduce applications of a pseudo diagram.

The warping degree of a link diagram

- Mathematics
- 2009

For an oriented link diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain a monotone diagram from D. We show that d(D)+d(-D)+sr(D) is less than or… Expand

Modelling structure–property relationships in advanced textile materials

- Engineering
- 2014

Abstract: This chapter provides an overview of various methods used for modelling the geometry, structure, and properties of a wide range of textile materials. Methods of experimental design and data… Expand

#### References

SHOWING 1-10 OF 23 REFERENCES

UNKNOTTING NUMBERS AND MINIMAL KNOT DIAGRAMS

- Mathematics
- 1994

We show there is an infinite number of knots whose unknotting numbers can only be realized through a sequence of crossing changes on a nonminimal projection, if ambient isotopies between crossing… Expand

A Partial Ordering of Knots Through Diagrammatic Unknotting

- Mathematics
- 2006

In this paper we define a partial order on the set of all knots and links using a special property derived from their minimal diagrams. A knot or link K 0 is called a predecessor of a knot or link K… Expand

Union and tangle

- Mathematics
- 1996

Shibuya proved that any union of two nontrivial knots without local knots is a prime knot. In this note, we prove it in a general setting. As an application, for any nontrivial knot, we give a knot… Expand

An enumeration of knots and links, and some of their algebraic properties

- Mathematics
- 1970

Publisher Summary This chapter describes knots and links, and some of their algebraic properties. An edge-connected 4-valent planar map is called a polyhedron, and a polyhedron is basic if no region… Expand

Ascending number of knots and links

- Mathematics
- 2007

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. Some fundamental results and an… Expand

The almost alternating diagrams of the trivial knot

- Mathematics
- 2009

Bankwitz characterized the alternating diagrams of the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize the… Expand

The knot book: An elementary introduction to the mathematical theory of knots

- Computer Science
- Complex.
- 1997

This knot book an elementary introduction to the mathematical theory of knots, it will really give you the good idea to be successful. Expand

UNLINKING NUMBER AND UNLINKING GAP

- Mathematics
- 2005

Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard–Jablan conjecture stating that the classical unknotting/unlinking… Expand

Two-bridge links with unlinking number one

- Mathematics
- 1991

Three conditions equivalent to a two-bridge link having unlinking number one are given. As a corollary it is shown that an unknotting crossing appears in the minimal diagram of a two-bridge knot or… Expand

Math. Z

- Math. Z
- 1956