Narrow Results

In this paper we extend the idea of Dehn surgery to a singular knot with one singularity, and give conditions under which the surgery manifold of an untangled singular knot is a Haken manifold containing an incompressible torus. We then show that untangled singular knots have Property P.

In this paper we provide a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction presented here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism. A brief discussion of a possibly independent derivation made by Maxwell in 1867 completes this reconstruction. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important role in modern mathematical physics, we offer a direct proof of their equivalence. Explicit examples of its interpretation in terms of oriented area are also provided.

We extend the notion of geometric intersection numbers 0 and 1, for circles in the Dehn quandle, to general quandles, assuming existence of elements having certain algebraic properties in the latter. These properties enable the construction of many useful 2 and 3 cycles in Dehn quandle homology. Particularly, we construct a special 2-cycle homology representative and an operation which promotes it to higher dimensional cycles of the same type, in the Dehn quandle, and show how to do so in any quandles admitting such elements. For such quandles, this gives some non-triviality results in the quandle homology. We look at a secondary rack formed by tuples of elements of the Dehn quandle and using the ideas above, obtain some basic non-triviality results in its homology as well. We also give some initial possibilities for its application in representing monodromy of singularities of elliptic fibrations.

We examine some questions regarding which torus links admit colorings by elements of a Dehn quandle.

We prove that a conjugacy class in the fundamental group of a surface with boundary is represented by a power of a simple curve if and only if the Goldman bracket of two different powers of this class, one of them larger than two, is zero.

The main theorem actually counts self-intersection number of a primitive class by counting the number of terms of the Goldman bracket of two distinct powers, one of them larger than two.

In these lectures we present for the first time a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, and we provide also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction offered here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism and it is complemented by the independent analysis and discussion made by Maxwell in 1867. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important rôle in modern mathematical physics, we offer a direct proof of their equivalence, and we provide some examples of linking number computation based on oriented area information. The material presented in these lectures forms an integral part of a paper that will appear in the Journal of Knot Theory and Its Ramifications.

This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial.