Homotopy Groups of the Space of Curves on a Surface

We explicitly calculate the fundamental group of the space $\mathcal F$ of all immersed closed curves on a surface $F$. It is shown that $\pi_n(\mathcal F)=0$, n>1 for $F\neq S^2, RP^2$. It is also proved that $\pi_2(\mathcal F)=\Z$, and $\pi_n(\mathcal F)=\pi_n(S^2)\oplus\pi_{n+1}(S^2)$, n>2, for $F$ equal to $S^2$ or $RP^2$.Comment: 8 pages, 1 figure This paper will appear in Math. Scand. probably in Vol. 86, no. 1, 200

Arnold-type Invariants of Curves on Surfaces

Recently V. Arnold introduced Strangeness and $J^{\pm}$ invariants of generic immersions of an oriented circle to $\R^2$. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface $F$. We explicitly describe all the invariants satisfying axioms, which naturally generalize the axioms used by V. Arnold.Comment: 25 pages, 11 figure

Arnold-type invariants of wave fronts on surfaces

AbstractRecently, Arnold's St and J± invariants of generic planar curves have been generalized to the case of generic planar wave fronts. We generalize these invariants to the case of wave fronts on an arbitrary surface F. All invariants satisfying the axioms which naturally generalize the axioms used by Arnold are explicitly described. We also give an explicit formula for the finest order one J+-type invariant of fronts on an orientable surface F≠S2. We obtain necessary and sufficient conditions for an invariant of nongeneric fronts with one nongeneric singular point to be the Vassiliev-type derivative of an invariant of generic fronts. As a byproduct, we calculate all homotopy groups of the space of Legendrian immersions of S1 into the spherical cotangent bundle of a surface

The universal order one invariant of framed knots in most S^1-bundles over orientable surfaces

It is well-known that self-linking is the only Z valued Vassiliev invariant of framed knots in S^3. However for most 3-manifolds, in particular for the total spaces of S^1-bundles over an orientable surface F not S^2, the space of Z-valued order one invariants is infinite dimensional. We give an explicit formula for the order one invariant I of framed knots in orientable total spaces of S^1-bundles over an orientable not necessarily compact surface F not S^2. We show that if F is not S^2 or S^1 X S^1, then I is the universal order one invariant, i.e. it distinguishes every two framed knots that can be distinguished by order one invariants with values in an Abelian group.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-3.abs.htm

Mapped Null Hypersurfaces and Legendrian Maps

For an $(m+1)$-dimensional space-time $(X^{m+1}, g),$ define a mapped null hypersurface to be a smooth map $\nu:N^{m}\to X^{m+1}$ (that is not necessarily an immersion) such that there exists a smooth field of null lines along $\nu$ that are both tangent and $g$-orthogonal to $\nu.$ We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle $ST^*M$ of an immersed spacelike hypersurface $\mu:M^m\to X^{m+1}.$ We show that a Legendrian map \wt \lambda: L^{m-1}\to (ST^*M)^{2m-1} defines a mapped null hypersurface in $X.$ On the other hand, the intersection of a mapped null hypersurface $\nu:N^m\to X^{m+1}$ with an immersed spacelike hypersurface $\mu':M'^m\to X^{m+1}$ defines a Legendrian map to the spherical cotangent bundle $ST^*M'.$ This map is a Legendrian immersion if $\nu$ came from a Legendrian immersion to $ST^*M$ for some immersed spacelike hypersurface $\mu:M^m\to X^{m+1}.$Comment: 13 pages, 1 figur

New heuristics for the minimum fundamental cut basis problem

Given an undirected connected network and a weight function finding a basis of the cut space with minimum sum of the cut weights is termed Minimum Cut Basis Problem. This problem can be solved, e.g., by the algorithm of Gomory and Hu [GH61]. If, however, fundamentality is required, i.e., the basis is induced by a spanning tree T in G, the problem becomes NP-hard. Theoretical and numerical results on that topic can be found in Bunke et al. [BHMM07] and in Bunke [Bun06]. In the following we present heuristics with complexity O(m log n) and O(mn), where n and m are the numbers of vertices and edges respectively, which obtain upper bounds on the aforementioned problem and in several cases outperform the heuristics of Schwahn [Sch05]

Extracellular Production and Degradation of Superoxide in the Coral Stylophora pistillata and Cultured Symbiodinium

Reactive oxygen species (ROS) are thought to play a major role in cell death pathways and bleaching in scleractinian corals. Direct measurements of ROS in corals are conspicuously in short supply, partly due to inherent problems with ROS quantification in cellular systems.In this study we characterized the dynamics of the reactive oxygen species superoxide anion radical (O(2)(-)) in the external milieu of the coral Stylophora pistillata. Using a sensitive, rapid and selective chemiluminescence-based technique, we measured extracellular superoxide production and detoxification activity of symbiont (non-bleached) and aposymbiont (bleached) corals, and of cultured Symbiodinium (from clades A and C). Bleached and non-bleached Stylophora fragments were found to produce superoxide at comparable rates of 10(-11)-10(-9) mol O(2)(-) mg protein(-1) min(-1) in the dark. In the light, a two-fold enhancement in O(2)(-) production rates was observed in non-bleached corals, but not in bleached corals. Cultured Symbiodinium produced superoxide in the dark at a rate of . Light was found to markedly enhance O(2)(-) production. The NADPH Oxidase inhibitor Diphenyleneiodonium chloride (DPI) strongly inhibited O(2)(-) production by corals (and more moderately by algae), possibly suggesting an involvement of NADPH Oxidase in the process. An extracellular O(2)(-) detoxifying activity was found for bleached and non-bleached Stylophora but not for Symbiodinium. The O(2)(-) detoxifying activity was partially characterized and found to resemble that of the enzyme superoxide dismutase (SOD).The findings of substantial extracellular O(2)(-) production as well as extracellular O(2)(-) detoxifying activity may shed light on the chemical interactions between the symbiont and its host and between the coral and its environment. Superoxide production by Symbiodinium possibly implies that algal bearing corals are more susceptible to an internal build-up of O(2)(-), which may in turn be linked to oxidative stress mediated bleaching