Floer Theory of Higher Rank Quiver 3-folds

We study threefolds Y fibred by $$A_m$$ A m -surfaces over a curve S of positive genus. An ideal triangulation of S defines, for each rank m, a quiver $$Q(\Delta _m)$$ Q ( Δ m ) , hence a $$CY_3$$ C Y 3 -category $$\mathcal {C}(W)$$ C ( W ) for any potential W on $$Q(\Delta _m)$$ Q ( Δ m ) . We show that for $$\omega $$ ω in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of $$(Y,\omega )$$ ( Y , ω ) is quasi-isomorphic to $$(\mathcal {C},W_{[\omega ]})$$ ( C , W [ ω ] ) for a certain generic potential $$W_{[\omega ]}$$ W [ ω ] . This partially establishes a conjecture of Goncharov (in: Algebra, geometry, and physics in the 21st century, Birkhäuser/Springer, Cham, 2017) concerning ‘categorifications’ of cluster varieties of framed $${\mathbb {P}}GL_{m+1}$$ P G L m + 1 -local systems on S, and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class $${\mathcal {S}}$$ S ’.

Incompleteness of the pressure metric on the Teichmüller space of a bordered surface

We prove that the pressure metric on the Teichmüller space of a bordered surface is incomplete and that a completion can be given by the moduli space of metrics on a graph (dual to a special ideal triangulation of the same bordered surface) equipped with pressure metric. In contrast to the closed surface case, we obtain as a corollary that the pressure metric is not bi-Lipschitz to the Weil–Petersson metric.

THE COMPLEX VOLUMES OF TWIST KNOTS VIA COLORED JONES POLYNOMIALS

For a hyperbolic knot, an ideal triangulation of the knot complement corresponding to the colored Jones polynomial was introduced by Thurston. Considering this triangulation of a twist knot, we find a function which gives the hyperbolicity equations and the complex volume of the knot complement, using Zickert's theory of the extended Bloch group and the complex volume. We also consider a formal approximation of the colored Jones polynomial. Following Ohnuki's theory of 2-bridge knots, we define another function which comes from the approximation. We show that this function is essentially the same as the previous function, and therefore it also gives the same hyperbolicity equations and the complex volume. Finally we compare this result with our previous one which dealt with Yokota theory, and, as an application to Yokota theory, present a refined formula of the complex volumes for any twist knots.

SEIFERT SURFACES IN KNOT COMPLEMENTS

In the ordinary normal surface for a compact 3-manifold, any incompressible, ∂-incompressible, compact surface can be moved by an isotopy to a normal surface [9]. But in a non-compact 3-manifold with an ideal triangulation, the existence of a normal surface representing an incompressible surface cannot be guaranteed. The figure-8 knot complement is presented in a counterexample in [12]. In this paper, we show the existence of normal Seifert surface under some restriction for a given ideal triangulation of the knot complement.

THE COLORED JONES POLYNOMIALS OF 2-BRIDGE LINK AND HYPERBOLICITY EQUATIONS OF IT'S COMPLEMENTS

In this paper, we discuss the relation between the colored Jones polynomial of a 2-bridge link and the ideal triangulation of it's complement in S3. The aim of this paper is to describe the ideal triangulation of a 2-bridge link complement and to show that the hyperbolicity equations coincide with the equations obtained from the colored Jones polynomial of a 2-bridge link, and to compare this triangulation with the canonical decomposition of the 2-bridge link complement introduced by Sakuma and Weeks in [10].