Unknotting tunnels in two-bridge knot and link complements

We give a complete classification of the unknotting tunnels in 2-bridge link complements, proving that only the upper and lower tunnels are unknotting tunnels. Moreover, we show that the only strongly parabolic tunnels in 2-cusped hyperbolic 3-manifolds are exactly the upper and lower tunnels in 2-bridge knot and link complements. From this, it follows that the upper and lower tunnels in 2-bridge knot and link complements must be isotopic to geodesics of length at most ln(4), where length is measured relative to maximal cusps. Moreover, the four dual unknotting tunnels in a 2-bridge knot complement, which together with the upper and lower tunnels form the set of all known unknotting tunnels for these knots, must each be homotopic to a geodesic of length at most 6ln(2). First author supported by NSF Grant DMS-93028943, second author supported by the Royal Society.     

Detecting Incompressibility of Boundary in 3-Manifolds

A construction is presented which can be utilized to prove incompressibility of boundary in a 3-manifold W. One constructs a new 3-manifold DW by doubling W along a subsurface in its boundary. If DW is hyperbolic, and if W has compressible boundary, then DW must have a longitude of 'length' less than 4. This can be applied to show that an arc that is a candidate for an unknotting tunnel in a 3-manifold cannot be an unknotting tunnel. It can also be used to show that a 'tubed surface' is incompressible. For knot and link complements in S ³, and an unknotting tunnel, DW is almost always hyperbolic. Empirically, this construction appears to provide a surprisingly effective procedure for demonstrating that specific arcs are not unknotting tunnels.     

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.     

On unknotting numbers and four-dimensional clasp numbers of links

In this paper, we estimate the unknotting number and the four-dimensional clasp number of a link, considering the greatest euler characteristic for an oriented two-manifold in the four-ball bounded by the link. Combining with a result due to Rudolph, we prove that an inequality stronger than the Bennequin unknotting inequality actually holds for any link diagram. As an application we show the equality conjectured by Boileau and Weber for a closed positive braid diagram.

     

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.     

NOTE ON CROSSING CHANGES

For any pair of knots of Gordian distance two, we constructan infinite family of knots which are ‘between’these two knots, that is, which differ from the given two knotsby one crossing change. In particular, we prove that every knotof unknotting number two can be unknotted via infinitely manydifferent knots of unknotting number one.     

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.     

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.     

A Lower Bound on Unknotting Number

In this paper the authors use a modified Wirtinger presentation to give a lower bound on the unknotting number of a knot in S3.     

The Canonical Decompositions of Some Family of Compact Orientable Hyperbolic 3-Manifolds with Totally Geodesic Boundary

L. Paoluzzi and B. Zimmermann constructed a family of compact orientable hyperbolic 3-manifolds with totally geodesic boundary, and classified them up to homeomorphism. Our main purpose is to determine the canonical decompositions of these manifolds. Using the result, we can obtain an alternative proof of the classification theorem of these manifolds and determine their isometry groups. We also determine their unknotting tunnels. Some of these manifolds are related to certain spatial graphs, so-called Suzukis Brunnian graphs. The properties of these manifolds enable us to obtain those of the graphs. Moreover, we give an affirmative answer to Kinoshitas problem concerning these graphs. In the Appendix, we calculate the volume of these manifolds.     

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.

     

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.     

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.     

On a class of hyperbolic 3-orbifolds of small volume and small heegaard genus associated to 2-bridge links

We consider a class of hyperbolic 3-orbifoldsO(α/β); the underlying topological space of such an orbifold is the 3-sphere and the singular set is obtained by adding the two standard (upper and lower) unknotting tunnels to a 2-bridge linkL(α/β) (and associating branching order two to both unknotting tunnels). These 3-orbifolds are extremal with respect to the notion of Heegaard genus or Heegaard number of 3-orbifolds; it is to be expected that they are also extremal with respect to the volume, that is the smallest volume hyperbolic 3-orbifolds should belong to this or some closely related class. We show that an orbifoldO(α/β) has a uniqueD ₂-covering by an orbifoldℒ _n(α/β) wose space is the 3-sphere and whose singular set is the same 2-bridge linkL(α/β) used for the construction ofO(α/β); moreoverO(α/β) is hyperbolic if and only ifℒ _n(α/β) is hyperbolic. As the volumes of the orbifoldsℒ _n(α/β) are known resp. can be computed, this allows to compute the volumes of the orbifoldsO(α/β). The problem of computation of volumes remains open for some closely related classes of 3-orbifolds which are also extremal with respect to the Heegaard genus (for example associating a branching order bigger than two to one or both unknotting tunnels).     

Destabilizing Heegaard splittings of knot exteriors

We give a necessary and sufficient condition for Heegaard splittings of knot exteriors to admit destabilizations. As an application, we show the following: let K₁ and K₂ be a pair of knots which is introduced by Morimoto as an example giving degeneration of tunnel number under connected sum. The Heegaard splitting of the exterior of K₁#K₂ derived from certain minimal unknotting tunnel systems of K₁ and K₂ is stabilized.     

On a class of hyperbolic 3-orbifolds of small volume and small heegaard genus associated to 2-bridge links

We consider a class of hyperbolic 3-orbifoldsO(α/β); the underlying topological space of such an orbifold is the 3-sphere and the singular set is obtained by adding the two standard (upper and lower) unknotting tunnels to a 2-bridge linkL(α/β) (and associating branching order two to both unknotting tunnels). These 3-orbifolds are extremal with respect to the notion of Heegaard genus or Heegaard number of 3-orbifolds; it is to be expected that they are also extremal with respect to the volume, that is the smallest volume hyperbolic 3-orbifolds should belong to this or some closely related class. We show that an orbifoldO(α/β) has a uniqueD ₂-covering by an orbifold? _n(α/β) wose space is the 3-sphere and whose singular set is the same 2-bridge linkL(α/β) used for the construction ofO(α/β); moreoverO(α/β) is hyperbolic if and only if? _n(α/β) is hyperbolic. As the volumes of the orbifolds? _n(α/β) are known resp. can be computed, this allows to compute the volumes of the orbifoldsO(α/β). The problem of computation of volumes remains open for some closely related classes of 3-orbifolds which are also extremal with respect to the Heegaard genus (for example associating a branching order bigger than two to one or both unknotting tunnels).     

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.

     

$H(2)$-Unknotting Number of a Knot

An $H(2)$-move is a local move of a knot which is performed by adding a half-twisted band. It is known an $H(2)$-move is an unknotting operation. We define the $H(2)$-unknotting number of a knot $K$ to be the minimum number of $H(2)$-moves needed to transform K into a trivial knot. We give several methods to estimate the $H(2)$-unknotting number of a knot. Then we give tables of $H(2)$-unknotting numbers of knots with up to 9 crossings.     

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.