Displaying 701 – 720 of 1631

Showing per page

Knot theory with the Lorentz group

João Faria Martins (2005)

Fundamenta Mathematicae

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 in S 2 x S 1 derived from Sym(2, ℝ)

Sang Lee, Yongdo Lim, Chan-Young Park (2000)

Fundamenta Mathematicae

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 S 2 × S 1 and show that these knots or links have certain types of symmetry of period 2.

Knots of (canonical) genus two

A. Stoimenow (2008)

Fundamenta Mathematicae

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,...

ℓ²-homology and planar graphs

Timothy A. Schroeder (2013)

Colloquium Mathematicae

In his 1930 paper, Kuratowski proves that a finite graph Γ is planar if and only if it does not contain a subgraph that is homeomorphic to K₅, the complete graph on five vertices, or K 3 , 3 , the complete bipartite graph on six vertices. This result is also attributed to Pontryagin. In this paper we present an ℓ²-homological method for detecting non-planar graphs. More specifically, we view a graph Γ as the nerve of a related Coxeter system and construct the associated Davis complex, Σ Γ . We then use a...

L²-homology and reciprocity for right-angled Coxeter groups

Boris Okun, Richard Scott (2011)

Fundamenta Mathematicae

Let W be a Coxeter group and let μ be an inner product on the group algebra ℝW. We say that μ is admissible if it satisfies the axioms for a Hilbert algebra structure. Any such inner product gives rise to a von Neumann algebra μ containing ℝW. Using these algebras and the corresponding von Neumann dimensions we define L ² μ -Betti numbers and an L ² μ -Euler charactersitic for W. We show that if the Davis complex for W is a generalized homology manifold, then these Betti numbers satisfy a version of Poincaré...

Currently displaying 701 – 720 of 1631