Journal of Knot Theory and Its Ramifications

We investigate a family of (reducible) representations of the braid groups corresponding to a specific solution to the Yang–Baxter equation. The images of under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of factoring over Temperley–Lieb algebras and the corresponding link invariants.

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Möbius Energy of the smooth knot as the polygons converge to the smooth knot. For this to work, the polygons must converge in a "nice" way, and the energies must be correctly regularized. We determine an explicit error bound for the convergence.

The classical 3j symbol can be obtained from the representation theory of . It can be generalized to using web spaces. We find its general shape and some trihedron coefficients of the representation theory of .

Kervaire's conjecture is equivalent to the following one: If F is a compact orientable nonseparating surface properly embedded in a knot exterior E then π₁(E/F) ≈ Z.

We investigate the Reshetikhin–Turaev invariants associated to SU(2) for the 3-manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a formula for the leading asymptotics of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian SL(2, ℂ)-representations of the fundamental group of M and prove that the values of these phase functions at the relevant stationary points equals the classical Chern–Simons invariants of the corresponding flat SU(2)-connections. Our findings are in agreement with the asymptotic expansion conjecture. Moreover, we calculate the leading asymptotics of the colored Jones polynomial of the figure 8 knot following Kashaev [14]. This leads to a slightly finer asymptotic description of the invariant than predicted by the volume conjecture [24].