2,538 research outputs found
Torus knot polynomials and susy Wilson loops
We give, using an explicit expression obtained in [V. Jones, Ann. of Math.
126, 335 (1987)], a basic hypergeometric representation of the HOMFLY
polynomial of torus knots, and present a number of equivalent
expressions, all related by Heine's transformations. Using this result the
symmetry and the leading polynomial at large
are explicit. We show the latter to be the Wilson loop of 2d Yang-Mills theory
on the plane. In addition, after taking one winding to infinity, it becomes the
Wilson loop in the zero instanton sector of the 2d Yang-Mills theory, which is
known to give averages of Wilson loops in =4 SYM theory. We also
give, using matrix models, an interpretation of the HOMFLY polynomial and the
corresponding Jones-Rosso representation in terms of -harmonic oscillators.Comment: 17 pages, v2: More concise (published) version; typos correcte
Wilson surfaces and higher dimensional knot invariants
An observable for nonabelian, higher-dimensional forms is introduced, its
properties are discussed and its expectation value in BF theory is described.
This is shown to produce potential and genuine invariants of higher-dimensional
knots.Comment: 31 pages, 9 figure
Chern-Simons Theory and Topological Strings
We review the relation between Chern-Simons gauge theory and topological
string theory on noncompact Calabi-Yau spaces. This relation has made possible
to give an exact solution of topological string theory on these spaces to all
orders in the string coupling constant. We focus on the construction of this
solution, which is encoded in the topological vertex, and we emphasize the
implications of the physics of string/gauge theory duality for knot theory and
for the geometry of Calabi-Yau manifolds.Comment: 46 pages, RMP style, 25 figures, minor corrections, references adde
The Intersection Graph Conjecture for Loop Diagrams
Vassiliev invariants can be studied by studying the spaces of chord diagrams
associated with singular knots. To these chord diagrams are associated the
intersection graphs of the chords. We extend results of Chmutov, Duzhin and
Lando to show that these graphs determine the chord diagram if the graph has at
most one loop. We also compute the size of the subalgebra generated by these
"loop diagrams."Comment: 23 pages, many figures. arXiv admin note: Figures 1, 2, 5 and 11
included in sources but in format not supported by arXi
The Three Loop Isotopy and Framed Isotopy Invariants of Virtual Knots
This paper introduces two virtual knot theory ``analogues'' of a well-known
family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev
finite-type invariants of order two. The first, called the three loop isotopy
invariant, is an invariant of virtual knots while the second, called the three
loop framed isotopy invariant, is a regular isotopy invariant of framed virtual
knots. The properties of these invariants are investigated at length. In
addition, we make precise the informal notion of ``analogue''. Using this
formal definition, it is proved that a generalized three loop invariant is a
virtual knot theory analogue of a generalization of the Grishanov-Vassiliev
invariants of order two
Chern-Simons Invariants of Torus Links
We compute the vacuum expectation values of torus knot operators in
Chern-Simons theory, and we obtain explicit formulae for all classical gauge
groups and for arbitrary representations. We reproduce a known formula for the
HOMFLY invariants of torus links and we obtain an analogous formula for
Kauffman invariants. We also derive a formula for cable knots. We use our
results to test a recently proposed conjecture that relates HOMFLY and Kauffman
invariants.Comment: 20 pages, 5 figures; v2: minor changes, version submitted to AHP. The
final publication is available at
http://www.springerlink.com/content/a2614232873l76h6
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