Notes on quasipositivity and combinatorial knot invariants

A characteristic property of a knot is a criterion that allows us to recognize this knot. For example, the trivial knot is the only knot which bounds a disk, whence it is characterized by this property. Thsi does not mean it is easy to recognize the trivial knot, since it might still be difficult to tell whether a given knot bounds a disk or not. In general, characteristic properties of knots are hard to handle. Therefore we usually content ourselves looking at weaker properties of knots, in particular at knot invariants, that allow us to distinguish certain knots from others. Kurt Reidemeister's diagrammatical formulation of knot theor4y in his famous book Knotentheorei ([39]) gave rise to a variety of combinatorial knot properties, such as the minimal crossing number, alternation or the Jones polynomial. Some of them do also have a topological interpretation, notably the fundamental group. In the present doctoral thesis I discuss relations between various knot properties, with a special emphasis on quasipositivity.
A link is called quasipositive if it has a special braid diagram, namely a product of conjugates of the positive standard generators of the braid group. If this product contains words of the form
σi,j = (σi ... σj-2)σj-1 ( σi ... σj-2)-1
only then we call the link strongly quasipositive. Here σi is the i-th positive standard generator of the braid group. The concept of quasipositive links is due to Lee Rudolph. He showed that every quasipositive link is a transverse C-link, i.e. a transverse intersection of a complex plane curve with the standard sphere S3 ⊂ C2. Recently, M. Boileau and S. Orevkov proved the converse, namely, that every transverse C-link is a quasipositive link. However, most of the considerations in my thesis are based upon a diagrammatical point of view.
In the first chapter I introduce the class of track knots and, at the same time, a method to construct knots with prescribed unknotting numbers. Track knots are closely related to the class of divide knots, which were introduced by Norert A'Campo and served as a starting point for my studies in knot theory. Furthermore, track knots share a basic property with quasipositive knots. Actually they are quasipositive, as we shall see in the second chapter. For this purpose I introduce a new diagrammatical description of quasipositive knots.
In the third chapter I prove that a knot is positive if and only if it is homogeneous and strongly quasipositive. This result follows quite easily from three famous inequalities due to D. Bennequin, H. Morton and P. Cromwell. The fourth chapter contains the main result of my thesis: any finite number of Vassiliev invariants of a knot can be realized by a quasipositive knot. This is closely related to Lee Rudolph's result which says that any Alexander polynomial of a knot can be realized by a quasipositive knot.
Crossing changes play an important part in my thesis, two. They give rise not only to the notion of unknotting numbers, but also to the Gordian complex of knots. The Gordian complex of knots is a simplicial complex whose certex set consists of all the siotopy classes of smooth oriented knots in S3. An edge of the Gordian complex is a pair of knots of Gordian distance one, i.e. a pair of knots whcih differe by one crossing change. Similarly, an n-simplex is a set of (n+1) knots whose pairwise Gordian distance is one. In the fifth chapter I prove that every n-simplex of the Gordian complex of knots is contained in an (n+1)-simplex. This is a generalization of M. Hirasawa and Y. Uchida's result, who showed that every edge of the Gordian complex is contained in a simplex of infinite dimension. Further we shall see that a knot of unknotting number two can be unknotted via infinitely many different knots of unknotting number one.
The sixth chapter is devoted to the study of a special class of knots, namely arborescent knots arising from plumbing positive and negative Hopf bands along any tree. In particular, I determine the minimal crossing numbers of these knots. This is kind of a counterpart to a result of W. B. B. Lickorish and M. B. Thistlethwaite on the minimal crossing number of Montesinos links, i.e. links associated with starshaped trees.
At last, some problems appear in the seventh and last chapter. Appendix A contains a table of all quasipositive knots up to ten crossings. Some examples of track knots are presented in Appendix B, while Appendix C contains a table of all special fibered arborescent knots up to ten crossings.
Although the succession of the chapters follows a certain logical and chronological order, most of them can be read independently, notably chapters 3 and 6. However, chapter 4 relies upon the diagrammatical description of quasipositive knots presented in chapter 2.