Quantum Groups at a Root of 1 and Tangle Invariants

1993 ◽  
Vol 07 (20n21) ◽  
pp. 3715-3726 ◽  
Author(s):  
MARC ROSSO
Keyword(s):  
Quantum Groups ◽  
Dimensional Representation ◽  
Knot Invariants ◽  
Conway Polynomial

One uses certain representations (in the De Concini-Kac picture) of the quantum groups [Formula: see text] for q a root of 1 to produce for R-matrices depending on rank [Formula: see text] continuous parameters. Using the formalism of Reshetikhin and Turaev, this allows to produce tangle and knot invariants depending on rank [Formula: see text] parameters. The simplest example ([Formula: see text] in the 2-dimensional representation) gives the Alexander-Conway polynomial.

MODULAR CATEGORIES AND 3-MANIFOLD INVARIANTS

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1807-1824 ◽  
Author(s):  
VLADIMIR G. TURAEV
Keyword(s):  
Quantum Groups ◽  
Tensor Category ◽  
Jones Polynomial ◽  
Knot Invariants ◽  
Modular Tensor Category ◽  
Modular Category

The aim of this paper is to give a concise introduction to the theory of knot invariants and 3-manifold invariants which generalize the Jones polynomial and which may be considered as a mathematical version of the Witten invariants. Such a theory was introduced by N. Reshetikhin and the author on the ground of the theory of quantum groups. Here we use more general algebraic objects, specifically, ribbon and modular categories. Such categories in particular arise as the categories of representations of quantum groups. The notion of modular category, interesting in itself, is closely related to the notion of modular tensor category in the sense of G. Moore and N. Seiberg. For simplicity we restrict ourselves in this paper to the case of closed 3-manifolds.

scholarly journals QUANTUM INVARIANTS GIVEN BY EVALUATION OF KNOT POLYNOMIALS

1993 ◽  
Vol 02 (02) ◽  
pp. 195-209 ◽  
Author(s):  
H.R. MORTON
Keyword(s):  
Irreducible Representation ◽  
Quantum Group ◽  
Quantum Groups ◽  
Dimensional Representation ◽  
Fundamental Representation ◽  
Quantum Invariants ◽  
3 Dimensional ◽  
Knot Invariant ◽  
Special Cases ◽  
Homfly Polynomials

It is shown that the knot invariant arising from an irreducible representation of a quantum group is, under certain conditions, an evaluation of the Homfly or Dubrovnik polynomial of the knot. Besides the known cases of the fundamental representation for each of the quantum groups in the series An, Bn, Cn and Dn, the results cover the special cases of the 3-dimensional representation of SU(2) and the 6-dimensional representation of SU(4), which can be viewed as the fundamental representations of SO(3) and SO(6) respectively. The second of these cases leads to a new relation between an evaluation of the Dubrovnik polynomial of a knot and an evaluation of the Homfly polynomials of two 2-cables about the knot.

scholarly journals LOGARITHMIC KNOT INVARIANTS ARISING FROM RESTRICTED QUANTUM GROUPS

2008 ◽  
Vol 19 (10) ◽  
pp. 1203-1213 ◽  
Author(s):  
JUN MURAKAMI ◽  
KIYOKAZU NAGATOMO
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Field Theories ◽  
Projective Modules ◽  
Conformal Field Theories ◽  
Knot Invariants ◽  
Conformal Field ◽  
Alexander Invariants

We construct knot invariants from the radical part of projective modules of the restricted quantum group [Formula: see text] at [Formula: see text], and we also show a relation between these invariants and the colored Alexander invariants. These projective modules are related to logarithmic conformal field theories.

scholarly journals FINITE-TYPE KNOT INVARIANTS BASED ON THE BAND-PASS AND DOUBLED-DELTA MOVES

2010 ◽  
Vol 19 (03) ◽  
pp. 355-384 ◽  
Author(s):  
JAMES CONANT ◽  
JACOB MOSTOVOY ◽  
TED STANFORD
Keyword(s):  
Abelian Group ◽  
Large Class ◽  
Finite Type ◽  
Equivalence Classes ◽  
Connected Sum ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Conway Polynomial ◽  
Band Pass

We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's n-equivalence classes of knots form groups under connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) It turns out that primitive band-pass finite-type invariants essentially coincide with standard primitive finite-type invariants, but things are more interesting for the doubled-delta move. 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. We start analyzing degree one doubled-delta invariants by considering which Vassiliev invariants are of doubled-delta degree one, finding that there is exactly one such invariant in each odd Vassiliev degree, and at most one (which is ℤ2-valued) in each even Vassiliev degree. Analyzing higher doubled-delta degrees, we observe that the Euler degree n + 1 part of Garoufalidis and Kricker's rational lift of the Kontsevich integral is a doubled-delta degree 2n invariant.

Quantum groups and representations of monoidal categories

1990 ◽  
Vol 108 (2) ◽  
pp. 261-290 ◽  
Author(s):  
David N. Yettera
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Quantum Field Theories ◽  
Monoidal Categories ◽  
Knot Invariants ◽  
Intimate Relation ◽  
Relationship Of ◽  
Tannaka Duality ◽  
The Relationship

This paper is intended to make explicit some aspects of the interactions which have recently come to light between the theory of classical knots and links, the theory of monoidal categories, Hopf-algebra theory, quantum integrable systems, the theory of exactly solvable models in statistical mechanics, and quantum field theories. The main results herein show an intimate relation between representations of certain monoidal categories arising from the study of new knot invariants or from physical considerations and quantum groups (that is, Hopf algebras). In particular categories of modules and comodules over Hopf algebras would seem to be much more fundamental examples of monoidal categories than might at first be apparent. This fundamental role of Hopf algebras in monoidal categories theory is also manifest in the Tannaka duality theory of Deligne and Mime [8a], although the relationship of that result and the present work is less clear than might be hoped.

Reconstruction of Objects from their Projections Using Orthogonal Functions

1973 ◽  
Vol 31 ◽  
pp. 268-269
Author(s):  
Elrnar Zeitler
Keyword(s):  
Unit Area ◽  
Three Dimensional ◽  
One Dimension ◽  
Spatial Density ◽  
Dimensional Representation ◽  
Orthogonal Functions ◽  
Object A ◽  
Plane Normal ◽  
Dimensional Object ◽  
Spatial Density Distribution

Considering any finite three-dimensional object, a “projection” is here defined as a two-dimensional representation of the object's mass per unit area on a plane normal to a given projection axis, here taken as they-axis. Since the object can be seen as being built from parallel, thin slices, the relation between object structure and its projection can be reduced by one dimension. It is assumed that an electron microscope equipped with a tilting stage records the projectionWhere the object has a spatial density distribution p(r,ϕ) within a limiting radius taken to be unity, and the stage is tilted by an angle 9 with respect to the x-axis of the recording plane.

Revealing gross three-dimensional structure in the SEM

1987 ◽  
Vol 45 ◽  
pp. 656-657
Author(s):  
Sterling P. Newberry
Keyword(s):  
Freeze Drying ◽  
Three Dimensional ◽  
Dimensional Structure ◽  
Small Object ◽  
Final Solution ◽  
Dimensional Representation ◽  
X Ray ◽  
Three Dimensional Representation ◽  
Freeze Dried ◽  
Critical Point Drying

The beautiful three dimensional representation of small object surfaces by the SEM leads one to search for ways to open up the sample and look inside. Could this be the answer to a better microscopy for gross biological 3-D structure? We know from X-Ray microscope images that Freeze Drying and Critical Point Drying give promise of adequately preserving gross structure. Can we slice such preparations open for SEM inspection? In general these preparations crush more readily than they slice. Russell and Dagihlian got around the problem by “deembedding” a section before imaging. This some what defeats the advantages of direct dry preparation, thus we are reluctant to accept it as the final solution to our problem. Alternatively, consider fig 1 wherein a freeze dried onion root has a window cut in its surface by a micromanipulator during observation in the SEM.

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