Mostrando recursos 1 - 20 de 758

  1. A minus sign that used to annoy me but now I know why it is there

    Tingley, Peter
    We consider two well known constructions of link invariants. One uses skein theory: you resolve each crossing of the link as a linear combination of things that don’t cross, until you eventually get a linear combination of links with no crossings, which you turn into a polynomial. The other uses quantum groups: you construct a functor from a topological category to some category of representations in such a way that (directed framed) links get sent to endomorphisms of the trivial representation, which are just rational functions. Certain instances of these two constructions give rise to essentially the same invariants, but when one carefully matches them there...

  2. A minus sign that used to annoy me but now I know why it is there

    Tingley, Peter
    We consider two well known constructions of link invariants. One uses skein theory: you resolve each crossing of the link as a linear combination of things that don’t cross, until you eventually get a linear combination of links with no crossings, which you turn into a polynomial. The other uses quantum groups: you construct a functor from a topological category to some category of representations in such a way that (directed framed) links get sent to endomorphisms of the trivial representation, which are just rational functions. Certain instances of these two constructions give rise to essentially the same invariants, but when one carefully matches them there...

  3. From Dominoes to Hexagons

    Thurston, Dylan P.
    There is a natural generalization of domino tilings to tilings of a polygon by hexagons, or, dually, configurations of oriented curves that meet in triples. We show exactly when two such tilings can be connected by a series of moves analogous to the domino flip move. The triple diagrams that result have connections to Legendrian knots, cluster algebras, and planar algebras.

  4. From Dominoes to Hexagons

    Thurston, Dylan P.
    There is a natural generalization of domino tilings to tilings of a polygon by hexagons, or, dually, configurations of oriented curves that meet in triples. We show exactly when two such tilings can be connected by a series of moves analogous to the domino flip move. The triple diagrams that result have connections to Legendrian knots, cluster algebras, and planar algebras.

  5. Mednykh's Formula via Lattice Topological Quantum Field Theories

    Snyder, Noah
    Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for $# \mathrm{Hom}(\pi_1(S),G)$ in terms of the Euler characteristic of $S$ and the dimensions of the irreducible representations of $G$. A similar formula in the nonorientable case was proved by Frobenius and Schur [FS06]. Both of these proofs use character theory and an explicit presentation for $\pi_1$. These results have been reproven using quantum field theory ([FQ93], [MY05], and others). Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to...

  6. Mednykh's Formula via Lattice Topological Quantum Field Theories

    Snyder, Noah
    Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for $# \mathrm{Hom}(\pi_1(S),G)$ in terms of the Euler characteristic of $S$ and the dimensions of the irreducible representations of $G$. A similar formula in the nonorientable case was proved by Frobenius and Schur [FS06]. Both of these proofs use character theory and an explicit presentation for $\pi_1$. These results have been reproven using quantum field theory ([FQ93], [MY05], and others). Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to...

  7. The Hubble diagram in a Bianchi I universe

    Schücker, Thomas
    The Bianchi I metric describes a homogeneous, but anisotropic universe and is commonly used to fit cosmological data. A fit to the angular distribution of 740 supernovae of type Ia with measured redshift and apparent luminosity is presented. It contains an intriguing, yet non-significant signal of a preferred direction in the sky. The Large Synoptic Survey Telescope being built in Chile should measure some 500 000 supernovae within the next 20 years and verify or falsify this signal.

  8. The Hubble diagram in a Bianchi I universe

    Schücker, Thomas
    The Bianchi I metric describes a homogeneous, but anisotropic universe and is commonly used to fit cosmological data. A fit to the angular distribution of 740 supernovae of type Ia with measured redshift and apparent luminosity is presented. It contains an intriguing, yet non-significant signal of a preferred direction in the sky. The Large Synoptic Survey Telescope being built in Chile should measure some 500 000 supernovae within the next 20 years and verify or falsify this signal.

  9. Algebra structures coming from tangles

    Saini, Sohan Lal
    In an attempt to find analogues, for higher relative commutants, of the Bose-Mesner algebra structure on the second relative commutant of a spin-model subfactor, we find that there do indeed exist other algebra structures on the higher relative commutants of any subfactor planar algebra which are induced by the action of tangles; in fact, we show they can only arise in the obvious fashion.

  10. Algebra structures coming from tangles

    Saini, Sohan Lal
    In an attempt to find analogues, for higher relative commutants, of the Bose-Mesner algebra structure on the second relative commutant of a spin-model subfactor, we find that there do indeed exist other algebra structures on the higher relative commutants of any subfactor planar algebra which are induced by the action of tangles; in fact, we show they can only arise in the obvious fashion.

  11. A formula for the Jones-Wenzl projections

    Morrison, Scott
    I present a method of calculating the coefficients appearing in the Jones-Wenzl projections in the Temperley-Lieb algebras. It essentially repeats the approach of Frenkel and Khovanov in [4] published in 1997. I wrote this note mid-2002, not knowing about their work, but then set it aside upon discovering their article. ¶Recently I decided to dust it off and place it on the arXiv — hoping the self-contained and detailed proof I give here may be useful. ¶The proof is based upon a simplification of the Wenzl recurrence relation. I give an example calculation, and compare this method to the formula announced by Ocneanu [13] and partially proved...

  12. A formula for the Jones-Wenzl projections

    Morrison, Scott
    I present a method of calculating the coefficients appearing in the Jones-Wenzl projections in the Temperley-Lieb algebras. It essentially repeats the approach of Frenkel and Khovanov in [4] published in 1997. I wrote this note mid-2002, not knowing about their work, but then set it aside upon discovering their article. ¶Recently I decided to dust it off and place it on the arXiv — hoping the self-contained and detailed proof I give here may be useful. ¶The proof is based upon a simplification of the Wenzl recurrence relation. I give an example calculation, and compare this method to the formula announced by Ocneanu [13] and partially proved...

  13. The generator conjecture for $3^G$ subfactor planar algebras

    Liu, Zhengwei; Penneys, David
    We state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the $3^G$ subfactor planar algebras. We prove the conjecture in the case when $|G|$ is odd. To do so, we find an action of $G$ on the reduced subfactor planar algebra at $f^(2)$, which is obtained from shading the planar algebra of the even half. We also show that this reduced subfactor planar algebra is a Yang-Baxter planar algebra.

  14. The generator conjecture for $3^G$ subfactor planar algebras

    Liu, Zhengwei; Penneys, David
    We state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the $3^G$ subfactor planar algebras. We prove the conjecture in the case when $|G|$ is odd. To do so, we find an action of $G$ on the reduced subfactor planar algebra at $f^(2)$, which is obtained from shading the planar algebra of the even half. We also show that this reduced subfactor planar algebra is a Yang-Baxter planar algebra.

  15. A Cuntz algebra approach to the classification of near-group categories

    Izumi, Masaki
    We classify $C^\ast$ near-group categories by using Vaughan Jones theory of subfactors and the Cuntz algebra endomorphisms. Our results show that there is a sharp contrast between two essentially different cases, integral and irrational cases. When the dimension of the unique non-invertible object is an integer, we obtain a complete classification list, and it turns out that such categories are always group theoretical. When it is irrational, we obtain explicit polynomial equations whose solutions completely classify the $C^\ast$ near-group categories in this class.

  16. A Cuntz algebra approach to the classification of near-group categories

    Izumi, Masaki
    We classify $C^\ast$ near-group categories by using Vaughan Jones theory of subfactors and the Cuntz algebra endomorphisms. Our results show that there is a sharp contrast between two essentially different cases, integral and irrational cases. When the dimension of the unique non-invertible object is an integer, we obtain a complete classification list, and it turns out that such categories are always group theoretical. When it is irrational, we obtain explicit polynomial equations whose solutions completely classify the $C^\ast$ near-group categories in this class.

  17. Mixing and Spectral Gap relative to Pinsker Factors for Sofic Groups

    Hayes, Ben
    Motivated by our previous results, we investigate structural properties of probability measurepreserving actions of sofic groups relative to their Pinsker factor. We also consider the same properties relative to the Outer Pinsker factor, which is another generalization of the Pinsker factor in the nonamenable case. The Outer Pinsker factor is motivated by entropy in the presence, which fixes some of the “pathological” behavior of sofic entropy: namely increase of entropy under factor maps. We show that an arbitrary probability measure-preserving action of a sofic group is mixing relative to its Pinsker and Outer Pinsker factors and, if the group is nonamenable, it has spectral gap relative...

  18. Mixing and Spectral Gap relative to Pinsker Factors for Sofic Groups

    Hayes, Ben
    Motivated by our previous results, we investigate structural properties of probability measurepreserving actions of sofic groups relative to their Pinsker factor. We also consider the same properties relative to the Outer Pinsker factor, which is another generalization of the Pinsker factor in the nonamenable case. The Outer Pinsker factor is motivated by entropy in the presence, which fixes some of the “pathological” behavior of sofic entropy: namely increase of entropy under factor maps. We show that an arbitrary probability measure-preserving action of a sofic group is mixing relative to its Pinsker and Outer Pinsker factors and, if the group is nonamenable, it has spectral gap relative...

  19. $N\subset P \subset M$ planar algebras and the Guionnet-Jones-Shlyakhtenko construction

    Hartglass, Michael

  20. $N\subset P \subset M$ planar algebras and the Guionnet-Jones-Shlyakhtenko construction

    Hartglass, Michael

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