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.     

Some applications of Tristram-Levine signatures and relation to Vassiliev invariants

Tristram and Levine introduced a continuous family of signature invariants for knots. We show that any possible value of such an invariant is realized by a knot with given Vassiliev invariants of bounded degree. We also show that one can make a knot prime preserving Alexander polynomial and Vassiliev invariants of bounded degree. Finally, the Tristram-Levine signatures are applied to obtain a condition on (signed) unknotting 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.

     

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.     

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.     

Some Examples Related to 4-Genera, Unknotting Numbers and Knot Polynomials

The paper gives examples of knots with equal knot polynomials,but distinct signatures, 4-genera, double branched cover homologygroups and unknotting numbers. This generalizes some examplesgiven by Lickorish and Millett. It is also shown that thereare knots with minimal (crossing number) diagrams of differentunknotting number (thus answering a question of Bleiler), andan alternative proof is given of Rudolph's result that thereare knots of 15n crossings with unit Alexander polynomial and4-genus or unknotting number n.     

Knot Floer Homology of (1, 1)-Knots

We present a combinatorial method for a calculation of the knot Floer homology of (1, l)-knots, and then demonstrate it for nonalternating (1, 1)-knots with 10 crossings and the pretzel knots of type (−2,m, n). Our calculations determine the unknotting numbers and 4-genera of the pretzel knots of this type.Mathematics Subject Classiffications (2000). 57M27, 57M25     

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.     

$H(2)$-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.     

Construction of non-alternating knots

We investigate the behaviour of Rasmussen's invariant  under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into non-alternating knots.

     

Random state transitions of knots: a first step towards modeling unknotting by type II topoisomerases

Type II topoisomerases are enzymes that change the topology of DNA by performing strand-passage. In particular, they unknot knotted DNA very efficiently. Motivated by this experimental observation, we investigate transition probabilities between knots. We use the BFACF algorithm to generate ensembles of polygons in Z³ of fixed knot type. We introduce a novel strand-passage algorithm which generates a Markov chain in knot space. The entries of the corresponding transition probability matrix determine state-transitions in knot space and can track the evolution of different knots after repeated strand-passage events. We outline future applications of this work to DNA unknotting.     

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.

     

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.     

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.

     

Comonotone Adaptive Interpolating Splines

For any given data we propose the construction of an interpolating spline of class C ¹, which is either a quadratic polynomial or a linear/linear rational function between the knots, and preserves the monotonicity of the data on the sections of rational intervals. We prove the uniqueness and existence of this spline. Numerical tests show good approximation properties and flexibility due to the non-coincidence of the given data arguments and the spline knots which can be chosen freely.     

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.     

Counting the 10-point graphs by partition

In this paper we discuss old and new theoretical methods for computing the number of graphs with a given partition. We also show how a judicious combination of these methods gives rise to a procedure that is sufficiently powerful to make possible the enumeration of all graphs on 10 points according to their partitions.