Knot Invariants for Intersecting Loops
D. Armand Ugon, R. Gambini, P. Mora
TL;DR
This paper extends knot theory to intersecting loops by generalizing braid algebra, introducing new Reidemeister moves, and connecting these to quantum gravity via link polynomials and Wilson line operators.
Contribution
It introduces a generalized braid algebra for intersecting loops, develops skein relations for link polynomials, and links these to quantum gravity constraints.
Findings
Derived skein relations for intersecting link polynomials
Connected HOMFLY polynomials to Wilson line operators in Chern-Simons theory
Related intersecting link invariants to solutions of quantum gravity constraints
Abstract
We generalize the braid algebra to the case of loops with intersections. We introduce the Reidemeister moves for 4 and 6-valent vertices to have a theory of rigid vertex equivalence. By considering representations of the extended braid algebra, we derive skein relations for link polynomials, which allow us to generalize any link Polynomial to the intersecting case. We perturbatively show that the HOMFLY Polynomials for intersecting links correspond to the vacuum expectation value of the Wilson line operator of the Chern Simon's Theory. We make contact with quantum gravity by showing that these polynomials are simply related with some solutions of the complete set of constraints with cosmological constant
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