Abstract
The recently suggested tangle calculus for knot polynomials is intimately related to topological string considerations and can help to build the HOMFLY-PT invariants from the topological vertices. We discuss this interplay in the simplest example of the Hopf link and link $L_{8n8}$. It turns out that the resolved conifold with four different representations on the four external legs, on the topological string side, is described by a special projection of the four-component link $L_{8n8}$, which reduces to the Hopf link colored with two composite representations. Thus, this provides the first explicit example of non-torus link description through the topological vertex. It is not a real breakthrough, because $L_{8n8}$ is just a cable of the Hopf link, still, it can help to intensify the development of the formalism towards more interesting examples.
Highlights
Correlation functions of Wilson loops are the most interesting observables in gauge theories: these are the gauge invariant quantities needed to understand confinement and various phase transitions
In topological vertex theory of [12,13,14,15,17], the Hopf polynomial HHopf is associated with the brane pattern, λ;μ symbolically depicted as According to our logic in the present paper, λ can be a representation from the product of two
We considered an elementary example of the tangle calculus of [8]: the quadratic recursion formula (2) for Hopf polynomials, which immediately follows from pictorial gluing of free ends of the Hopf tangle
Summary
Correlation functions of Wilson loops are the most interesting observables in gauge theories: these are the gauge invariant quantities needed to understand confinement and various phase transitions. The main object in this theory is the topological vertex [12,13] (see [14,15,16,17] for its refined version), and [18] for related network models, which are arbitrary convolutions of vertices It is a long-standing problem to express the knot and link invariants in these terms, and, more generally, in terms of arbitrary tangle blocks. The Appendix contains illustrative examples of the main statement of this paper for first representations
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