Invariants

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Matthew Hedden - One of the best experts on this subject based on the ideXlab platform.

  • an ozsvath szabo floer homology invariant of knots in a contact manifold
    Advances in Mathematics, 2008
    Co-Authors: Matthew Hedden
    Abstract:

    Abstract Using the knot Floer homology filtration, we define Invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath–Szabo contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston–Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than S 3 with negative maximal Thurston–Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsvath–Szabo Invariants, then any fibered knot can realize the classical Eliashberg–Bennequin bound in at most one of these contact structures.

  • an ozsvath szabo floer homology invariant of knots in a contact manifold
    arXiv: Geometric Topology, 2007
    Co-Authors: Matthew Hedden
    Abstract:

    Using the knot Floer homology filtration, we define Invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston-Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than the three-sphere with negative maximal Thurston-Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsvath-Szabo Invariants, then any fibered knot can realize the classical Eliashberg-Bennequin bound in at most one of these contact structures.

Lijun Zhang - One of the best experts on this subject based on the ideXlab platform.

  • counterexample guided polynomial loop invariant generation by lagrange interpolation
    Computer Aided Verification, 2015
    Co-Authors: Yufang Chen, Chihduo Hong, Bowyaw Wang, Lijun Zhang
    Abstract:

    We apply multivariate Lagrange interpolation to synthesizing polynomial quantitative loop Invariants for probabilistic programs. We reduce the computation of a quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative Invariants. We evaluate our technique by several case studies with polynomial quantitative loop Invariants in the experiments.

  • counterexample guided polynomial loop invariant generation by lagrange interpolation
    arXiv: Software Engineering, 2015
    Co-Authors: Yufang Chen, Chihduo Hong, Bowyaw Wang, Lijun Zhang
    Abstract:

    We apply multivariate Lagrange interpolation to synthesize polynomial quantitative loop Invariants for probabilistic programs. We reduce the computation of an quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative Invariants. We evaluate our technique by several case studies with polynomial quantitative loop Invariants in the experiments.

Zoltan Szabo - One of the best experts on this subject based on the ideXlab platform.

Aneesh V Manohar - One of the best experts on this subject based on the ideXlab platform.

  • algebraic structure of lepton and quark flavor Invariants and cp violation
    Journal of High Energy Physics, 2009
    Co-Authors: Elizabeth E Jenkins, Aneesh V Manohar
    Abstract:

    Lepton and quark flavor Invariants are studied, both in the Standard Model with a dimension five Majorana neutrino mass operator, and in the seesaw model. The ring of Invariants in the lepton sector is highly non-trivial, with non-linear relations among the basic Invariants. The Invariants are classified for the Standard Model with two and three generations, and for the seesaw model with two generations, and the Hilbert series is computed. The seesaw model with three generations proved computationally too difficult for a complete solution. We give an invariant definition of the CP-violating angle in the electroweak sector.

  • algebraic structure of lepton and quark flavor Invariants and cp violation
    arXiv: High Energy Physics - Phenomenology, 2009
    Co-Authors: Elizabeth E Jenkins, Aneesh V Manohar
    Abstract:

    Lepton and quark flavor Invariants are studied, both in the Standard Model with a dimension five Majorana neutrino mass operator, and in the seesaw model. The ring of Invariants in the lepton sector is highly non-trivial, with non-linear relations among the basic Invariants. The Invariants are classified for the Standard Model with two and three generations, and for the seesaw model with two generations, and the Hilbert series is computed. The seesaw model with three generations proved computationally too difficult for a complete solution. We give an invariant definition of the CP-violating angle theta in the electroweak sector.

Ted Stanford - One of the best experts on this subject based on the ideXlab platform.

  • FINITE-TYPE KNOT Invariants BASED ON THE BAND-PASS AND DOUBLED-DELTA MOVES
    2009
    Co-Authors: James Conant, Jacob Mostovoy, Ted Stanford
    Abstract:

    Abstract. We study generalizations of finite-type knot Invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move. First, we represent the local moves by normal subgroups of the pure braid group P∞. Subgroups that are stable under the “strand-tripling ” endomorphisms are shown to produce finite-type Invariants with familiar properties; in particular, generalized Goussarov’s n-equivalence classes of knots form groups under the connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) Treating local moves as surgeries on claspers, we study two particular cases in detail: the band-pass and the doubled delta move. While the band-pass move gives only one “new ” invariant (namely, the Arf invariant), the Invariants corresponding to the doubled-delta move contain information which is not available to any finite collection of Vassiliev Invariants. The complete degree 0 doubled-delta invariant is the S-equivalence class of the knot. In this context, we generalize a result of Murakami and Ohtsuki to show that the only primitive Vassiliev Invariants of S-equivalence taking values in an abelian group with no 2-torsion arise from the Alexander-Conway polynomial. To this end, we introduce a discrete logarithm which transforms the coefficients of the Conway polynomia

  • FINITE TYPE KNOT Invariants BASED ON THE BAND-PASS AND DOUBLED DELTA MOVES
    2005
    Co-Authors: James Conant, Jacob Mostovoy, Ted Stanford
    Abstract:

    Abstract. We study generalizations of finite-type knot Invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move. There are several ways of formalizing the notion of a local move. Representing knots as closed braids, one can define local moves as modifications by elements of some subgroup G of the pure braid group P∞ on an infinite number of strands. We prove that if the subgroup G is stable under the “strand-tripling” endomorphisms, the resulting finite-type Invariants have properties similar to those of the usual finite-type Invariants. In particular, generalized Goussarov’s n-equivalence classes of knots form groups under the connected sum operation. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) These results lead us to ask whether the set of generalized n-equivalence classes, with respect to Garoufalidis and Rozansky’s null-move of pairs (M, k) where M is a homology 3-sphere and k a knot in M, is, in fact, a group. We answer this question in the affirmative for the submonoid where M = S 3. One can also define local moves as surgeries on claspers. Using this description of local moves, we study two particular cases: the band-pass and the doubled delta move. We show that the primitive finite-type Invariants corresponding to the band-pass move coincide with the primitive Vassiliev Invariants in orders greater than zero and that the only band-pass invariant of order zero is the Arf invariant. In the case of the doubled delta move there are many more Invariants of each order. We show that the only primitive Invariants of order zero that are also Vassiliev Invariants, come from the Conway polynomial. (Over the rationals this has been established before by Murakami and Ohtsuki.) We find that there is exactly one Vassiliev invariant in each odd degree which is of doubled delta degree one, whereas in each even degree there is at most a Z2-valued invariant, which we show exists in degree 4. For higher doubled delta degrees, we observe that the Euler degree n + 1 part of the rational lift of the Kontsevich integral is a doubled delta degree 2n invariant. 1