The SL(2, C) - character varieties of torus knots.
The SL(2,C) character variety of a class of torus knots.
The superpolynomial for knot homologies.
The systolic constant of orientable Bieberbach 3-manifolds
A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds are aspherical, therefore the supremum of their systolic ratio, over the set of Riemannian metrics, is finite by a fundamental result of M. Gromov. We study the optimal systolic ratio of compact -dimensional orientable Bieberbach manifolds which are not tori, and prove that it cannot be realized by a flat metric. We also highlight a metric that we construct on one type of such manifolds () which...
The universal order one invariant of framed knots in most -bundles over orientable surfaces.
The virtual Haken conjecture: Experiments and examples.
Three-manifolds having complexity at most 9.
Torsion in graph homology
Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsion. When the underlying algebra is ℤ[x]/(x²), we determine precisely those graphs whose cohomology contains torsion. For a large class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. Our computation could potentially be used to predict the Khovanov-Rozansky...
Torsion in Khovanov homology of semi-adequate links
The goal of this paper is to address A. Shumakovitch's conjecture about the existence of ℤ₂-torsion in Khovanov link homology. We analyze torsion in Khovanov homology of semi-adequate links via chromatic cohomology for graphs, which provides a link between link homology and the well-developed theory of Hochschild homology. In particular, we obtain explicit formulae for torsion and prove that Khovanov homology of semi-adequate links contains ℤ₂-torsion if the corresponding Tait-type graph has a cycle...
Torsion in one-term distributive homology
The one-term distributive homology was introduced in [Prz] as an atomic replacement of rack and quandle homology, which was first introduced and developed by Fenn-Rourke-Sanderson [FRS] and Carter-Kamada-Saito [CKS]. This homology was initially suspected to be torsion-free [Prz], but we show in this paper that the one-term homology of a finite spindle may have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their homology precisely....
Torsion of Khovanov homology
Khovanov homology is a recently introduced invariant of oriented links in ℝ³. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of Khovanov homology is a version of the Jones polynomial for links. In this paper we study torsion of Khovanov homology. Based on our calculations, we formulate several conjectures about the torsion and prove weaker versions of the first two of them. In particular, we prove that all non-split alternating links have their integer Khovanov...
Torsion, TQFT, and Seiberg-Witten invariants of 3-manifolds.
Toward a general theory of linking invariants.
Toward optimality in discrete Morse theory.
Triangulations of 3-dimensional pseudomanifolds with an application to state-sum invariants.
Twisted Alexander polynomials of periodic knots.
Twisted Alexander polynomials, symplectic 4-manifolds and surfaces of minimal complexity
Let M be a 4-manifold which admits a free circle action. We use twisted Alexander polynomials to study the existence of symplectic structures and the minimal complexity of surfaces in M. The results on the existence of symplectic structures summarize previous results of the authors in [FV08a,FV08,FV07]. The results on surfaces of minimal complexity are new.
Two applications of elementary knot theory to Lie algebras and Vassiliev invariants.
Une construction intrinsèque du cœur de l'invariant de Casson
Nous donnons une nouvelle construction cohomologique du cœur de l'invariant de Casson et nous donnons un nouveau calcul de ses valeurs sur les générateurs du sous- groupe de Johnson du mapping class group.
Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion
For every rational homology 3-sphere with H₁(M,ℤ) = (ℤ/2ℤ)ⁿ we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring) such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root, and at any even root of unity the SU(2) quantum invariant. Moreover, this unified invariant splits into a sum of the refined unified invariants dominating spin and cohomological refinements of...