Knot and link projections in 3-manifolds.
Knot cobordism and amphicheirality.
Knot Floer homology and the four-ball genus.
Knot manifolds with isomorphic spines
We study the relation between the concept of spine and the representation of orientable bordered 3-manifolds by Heegaard diagrams. As a consequence, we show that composing invertible non-amphicheiral knots yields examples of topologically different knot manifolds with isomorphic spines. These results are related to some questions listed in [9], [11] and recover the main theorem of [10] as a corollary. Finally, an application concerning knot manifolds of composite knots with h prime factors completes...
Knot polynomials and Vassiliev's invariants.
Knot theory with the Lorentz group
We analyse perturbative expansions of the invariants defined from unitary representations of the Quantum Lorentz Group in two different ways, namely using the Kontsevich Integral and weight systems, and the R-matrix in the Quantum Lorentz Group defined by Buffenoir and Roche. The two formulations are proved to be equivalent; and they both yield ℂ[[h]]h-valued knot invariants related with the Melvin-Morton expansion of the Coloured Jones Polynomial.
Knots and Involutions.
Knots Fixed by Zp-Actions and Periodic Links.
Knots in derived from Sym(2, ℝ)
We realize closed geodesics on the real conformal compactification of the space V = Sym(2, ℝ) of all 2 × 2 real symmetric matrices as knots or 2-component links in and show that these knots or links have certain types of symmetry of period 2.
Knots in the 4-sphere
Knots of (canonical) genus two
We give a description of all knot diagrams of canonical genus 2 and 3, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic volume for canonical (weak) genus 2 knots. We also study the values of the link polynomials at roots of unity, extending denseness results of Jones. Using these values,...
Knots on a positive template have a bounded number of prime factors.
Knots with property .
Knotted Ball Pairs whose Product with an Interval is Unknotted.
Knotted Hamiltonian cycles in spatial embeddings of complete graphs.
Knotted Lattice-Like Space Fillers.
Kobordismentheorie klassifizierender Räume und Transformationsgruppen.
K-theory, flat bundles and the Borel classes
Using Hausmann and Vogel's homology sphere bundle interpretation of algebraic K-theory, we construct K-theory invariants by a theory of characteristic classes for flat bundles. It is shown that the Borel classes are detected this way, as well as the rational K-theory of integer group rings of finite groups.
K-theory over C*-algebras
The contents of the article represents the minicourse which was delivered at the 7th conference "Geometry and Topology of Manifolds. The Mathematical Legacy of Charles Ehresmann", Będlewo (Poland), 8.05.2005 - 15.05.2005. The article includes the description of the so called Hirzebruch formula in different aspects which lead to a basic list of problems related to noncommutative geometry and topology. In conclusion, two new problems are presented: about almost flat bundles and about the Noether decomposition...
Kurven auf orientierbaren Flächen.