Search results for "Alexander polynomial"
showing 6 items of 6 documents
On orderability of fibred knot groups
2003
It is known that knot groups are right-orderable, and that many of them are not bi-orderable. Here we show that certain bred knots in S 3 (or in a homology sphere) do have bi-orderable fundamental group. In particular, this holds for bred knots, such as 41, for which the Alexander polynomial has all roots real and positive. This is an application of the construction of orderings of groups, which are moreover invariant with respect to a certain automorphism.
The Links-Gould invariants as generalizations of the Alexander polynomial
2016
In this thesis we focus on the connections that exist between two link invariants: first the Alexander-Conway invariant ∆ that was the first polynomial link invariant to be discovered, and one of the most thoroughly studied since alongside with the Jones polynomial, and on the other hand the family of Links-Gould invariants LGn,m that are quantum link invariants derived from super Hopf algebras Uqgl(n|m). We prove a case of the De Wit-Ishii-Links conjecture: in some cases we can recover powers of the Alexander polynomial as evaluations of the Links-Gould invariants. So the LG polynomials are generalizations of the Alexander invariant. Moreover we give evidence that these invariants should s…
Detection and visualization of physical knots in macromolecules
2010
Abstract This manuscript provides a pedagogical introduction on how to determine and visualize simple physical knots occurring in polymers, proteins and DNA. We explain how the Alexander polynomial is computed and implemented in a simulation code, and how the structure can be simplified beforehand to save computer time. The concept of knottedness can also be extended in a statistical framework to chains which are not closed. The latter is exemplified by comparing statistics of knots in open random walks and closed random loops.
Protein knot server: detection of knots in protein structures
2007
KNOTS (http://knots.mit.edu) is a web server that detects knots in protein structures. Several protein structures have been reported to contain intricate knots. The physiological role of knots and their effect on folding and evolution is an area of active research. The user submits a PDB id or uploads a 3D protein structure in PDB or mmCIF format. The current implementation of the server uses the Alexander polynomial to detect knots. The results of the analysis that are presented to the user are the location of the knot in the structure, the type of the knot and an interactive visualization of the knot. The results can also be downloaded and viewed offline. The server also maintains a regul…
Finite type invariants of knots in homology 3-spheres with respect to null LP-surgeries
2017
We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky theory for knots in integral homology 3-spheres. We give a partial combinatorial description of the graded space associated with our theory and determine some cases when this description is complete. For null-homologous knots in rational homology 3-spheres with a trivial Alexander polynomial, we show that the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant built from integrals in configuration spaces are universal finite type i…
Varieties of representations of virtual knot groups in SL2(C)
2002
Abstract We study the local structure of the variety of representations of a virtual knot group in SL 2 ( C ) near an abelian representation ρ 0 . To such a representation is attached a complex number ω and there are three cases. If ω and ω −1 are not roots of the Alexander polynomial, there are only abelian representations around ρ 0 . If ω is a root and ω −1 is not, there are only reducible representations. If both ω and ω −1 are roots and certain homological conditions hold, there are irreducible representations.