2007
DOI: 10.2140/agt.2007.7.47
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Configuration space integral for longn–knots and the Alexander polynomial

Abstract: There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is constructed by Bott for degree 2 and by Cattaneo-Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon n-knots and characterize the Bott-Cattaneo-Rossi invariant as a … Show more

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Cited by 16 publications
(47 citation statements)
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“…Equation (27) is simply (26) rewritten using γ = v γ v v. To prove (28), take its right hand side and use (27) and (24) to get u back again, and hence our formula for RC γ u indeed inverts the formula already established for C −γ u . Equation (29) amounts to writing the group law of a 2D Lie group in terms of its 2D Lie algebra, L 0 := span(α, β), and this is again an exercise in 2 × 2 matrix algebra, though a slightly harder one.…”
Section: Proof (Sketch) Equationmentioning
confidence: 76%
See 2 more Smart Citations
“…Equation (27) is simply (26) rewritten using γ = v γ v v. To prove (28), take its right hand side and use (27) and (24) to get u back again, and hence our formula for RC γ u indeed inverts the formula already established for C −γ u . Equation (29) amounts to writing the group law of a 2D Lie group in terms of its 2D Lie algebra, L 0 := span(α, β), and this is again an exercise in 2 × 2 matrix algebra, though a slightly harder one.…”
Section: Proof (Sketch) Equationmentioning
confidence: 76%
“…Most likely the work of Watanabe [28] is a proof of Conjecture 8.1 for the case of a single balloon and no hoops, and very likely, it contains all key ideas necessary for a complete proof of Conjecture 8.1.…”
Section: Conjecture 81 If γ Is An Rkbh Then O γ Bf = E τ (ζ (γ ))mentioning
confidence: 99%
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“…If , is such a graph, then configurations are attached to its vertices, while momentum are attached to edges in the two dual representations (Feynman rules in position and momentum spaces). This duality is represented by a pairing between a "configuration functor" (typically C  , (configuration space of subgraphs and strings (Watanabe, 2007), and a "Lagrangian" (e.g. , determined by its value on an edge, e.g.…”
Section: Classic Feynman Integrals and Their Propertiesmentioning
confidence: 99%
“…But we require their electromagnetic mean into the context of X, for we obtain their corresponding diagnosis using an electromagnetic device that establish an univocal correspondence between detected anomalies and Feynman diagrams used to the spectral encoding through of the integrals of Feynman-Bulnes. If we consider the space C 0  () (Watanabe, 2007), as space of configuration associate with sub-graph (, ), where , is the corresponding smooth embedding to n-knot that which is identified as a , in an integral as the given in (6), we can define rules to sub-graphs that coincides with the rules of signs in the calculate of integrals like (36). Thus we can identify the three fundamental forms given for () = sgn(z), (figure 5).…”
Section: Combination Of Quantum Factors and Programming Diagrams Of Pmentioning
confidence: 99%