On region crossing change and incidence matrix

In a recent work of Ayaka Shimizu, she studied an operation named region crossing change on link diagrams, which was proposed by Kishimoto, and showed that a region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that the region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an n-component link diagram if and only if the rank of the associated incidence matrix equals c n + 1, where c denotes the size of the graph.     

Donaldson's theorem,Heegaard Floer homology,and knots with unknotting number one

We establish an obstruction to unknotting an alternating knot by a single crossing change. The obstruction is lattice-theoretic in nature, and combines Donaldson's diagonalization theorem with an obstruction developed by Ozsváth and Szabó using Heegaard Floer homology. As an application, we enumerate the alternating 3-braid knots with unknotting number one, and show that each has an unknotting crossing in its standard alternating diagram.     

On the unknotting number of minimal diagrams

Answering negatively a question of Bleiler, we give examples of knots where the difference between minimal and maximal unknotting number of minimal crossing number diagrams grows beyond any extent.

     

Triple point cancelling numbers of surface links and quandle cocycle invariants

The unknotting or triple point cancelling number of a surface link F is the least number of 1-handles for F such that the 2-knot obtained from F by surgery along them is unknotted or pseudo-ribbon, respectively. These numbers have been often studied by knot groups and Alexander invariants. On the other hand, quandle colorings and quandle cocycle invariants of surface links were introduced and applied to other aspects, including non-invertibility and triple point numbers. In this paper, we give lower bounds of the unknotting or triple point cancelling numbers of surface links by using quandle colorings and quandle cocycle invariants.     

Union and tangle

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 diagram such that a single unknotting operation on the diagram cannot yield a diagram of a trivial knot.

     

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan's theory

Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the Khovanov-Jacobsson number, by considering the surface-knot as a link cobordism between empty links. In this paper, we study an extension of the Khovanov-Jacobsson number derived from Bar-Natan's theory, and prove that any -knot has trivial Khovanov-Jacobsson number.

     

Unknotting number and number of Reidemeister moves needed for unlinking

Using unknotting number, we introduce a link diagram invariant of type given in Hass and Nowik (2008) [4], which changes at most by 2 under a Reidemeister move. We show that a certain infinite sequence of diagrams of the trivial two-component link need quadratic number of Reidemeister moves for being splitted with respect to the number of crossings.     

Generating functions, Fibonacci numbers and rational knots

We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots. Then we show how to enumerate rational knots of given crossing number depending on genus and/or signature. This allows to determine the asymptotical average value of these invariants among rational knots. We give also an application to the enumeration of lens spaces.     

Common stabilizations of Heegaard splittings of link exteriors

We give a condition for a pair of unknotting tunnels of a non-trivial tunnel number one link to give a genus three Heegaard splitting of the link complement and show that every 2-bridge link has such a pair of unknotting tunnels.     

Relations among the lowest degree of the Jones polynomial and geometric invariants for a closed positive braid

By means of a result due to Fiedler, we obtain a relation between the lowest degree of the Jones polynomial and the unknotting number for any link which has a closed positive braid diagram. Furthermore, we obtain relations between the lowest degree and the slice euler characteristic or the four-dimensional clasp number.     

Knots of genus one or on the number of alternating knots of given genus

We prove that any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number.

     

On Kelley's intersection numbers

We introduce a notion of weak intersection number of a collection of sets, modifying the notion of intersection number due to J.L. Kelley, and obtain an analogue of Kelley's characterization of Boolean algebras which support a finitely additive strictly positive measure. We also consider graph-theoretic reformulations of the notions of intersection number and weak intersection number.

     

Maximal Bennequin numbers and Kauffman polynomials of positive links

By using results of Yamada and of Yokota, concerning link diagrams and link polynomials, we give some relationships between maximal Bennequin numbers and Kauffman polynomials of positive links.

     

Sharp bounds for harmonic numbers

In the paper, we collect some inequalities and establish a sharp double inequality for bounding the n-th harmonic number.     

The almost alternating diagrams of the trivial knot

Bankwitz characterized the alternating diagrams of the trivialknot. A non-alternating diagram is called almost alternatingif one crossing change makes the diagram alternating. We characterizethe almost alternating diagrams of the trivial knot. As a corollary,we determine the unknotting number one alternating knots withthe property that the unknotting operation can be done on itsalternating diagram. Received July 3, 2007. Revised September 29, 2008.     

Knots with unknotting number one and Heegaard Floer homology

We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number equal to one. We also classify alternating knots with 10 crossings and unknotting number equal to one.     

Non-simple links with tunnel number one

We determine all non-simple links which admit an unknotting tunnel, i.e. links which contain an essential annulus or torus in its exterior and have tunnel number one.

     

Computation of class numbers of quadratic number fields

We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields.

     

On the number of generators of Cohen-Macaulay ideals

Several bounds on the number of generators of Cohen-Macaulay ideals known in the literature follow from a simple inequality which bounds the number of generators of such ideals in terms of mixed multiplicities. Results of Cohen and Akizuki, Abhyankar, Sally, Rees and Boratynski-Eisenbud-Rees are deduced very easily from this inequality.

     

The nonquadratic imaginary cyclic fields of -power degrees with class numbers equal to their genus class numbers

It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic fields of -power degrees with class numbers equal to their genus class numbers.