# Browsing by Author "Tran, Anh T."

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Item Knot Cabling and the Degree of the Colored Jones Polynomial(University at Albany, 2015-09-16) Kalfagianni, E.; Tran, Anh T.; Tran, Anh T.Show more We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot K satisfies the Slope Conjecture then a (p, q)-cable of K satisfies the conjecture, provided that p/q is not a Jones slope of K. As an application we prove the Slope Conjecture for iterated cables of adequate knots and for iterated torus knots. Furthermore we show that, for these knots, the degree of the colored Jones polynomial also determines the topology of a surface that satisfies the Slope Conjecture. We also state a conjecture suggesting a topological interpretation of the linear terms of the degree of the colored Jones polynomial (Conjecture 5.1), and we prove it for the following classes of knots: iterated torus knots and iterated cables of adequate knots, iterated cables of several nonalternating knots with up to nine crossings, pretzel knots of type (-2, 3, p) and their cables, and two-fusion knots.Show more Item Left-Orderability for Surgeries on Twisted Torus Knots(Japan Academy, 2019-01) Tran, Anh T.; Tran, Anh T.Show more We show that the fundamental group of the 3-manifold obtained by p/q-surgery along the (n - 2)-twisted (3, 3m + 2)-torus knot, with n, m ≥ 1, is not left-orderable if p/q ≥ 2n + 6m - 3 and is left-orderable if p/q is sufficiently close to 0.Show more Item On the AJ Conjecture for Knots(2015-07-07) Le, Thang T. Q.; Tran, Anh T.; Huynh, Vu Q.; Tran, Anh T.Show more We confirm the AJ conjecture [Ga2] that relates the A-polynomial and the colored Jones polynomial for hyperbolic knots satisfying certain conditions. In particular, we show that the conjecture holds true for some classes of two-bridge knots and pretzel knots. This extends the result of the first author in [Le2], who established the AJ conjecture for a large class of two-bridge knots, including all twist knots. Along the way, we explicitly calculate the universal SL₂(C)-character ring of the knot group of the (−2, 3, 2n + 1)-pretzel knot, and show it is reduced for all integers n.Show more Item Proof of the Strong AJ Conjecture for the Figure 8 Knot(2019-05) Nguyen, Hoang-An A.; Tran, Anh T.Show more The AJ conjecture, formulated by Garoufalidis [7], relates the A-polynomial of a knot and the colored Jones polynomial of a knot. The strong AJ conjecture first proposed in [6] and then modified by Sikora [14], relates the orthogonal ideal to the classical peripheral ideal. The orthogonal ideal is an ideal of the skein module of the torus and the classical peripheral ideal is an ideal of the coordinate ring of the SL(2, C) character variety. This conjecture could be seen as the topological and algebraic structure that underlies the AJ conjecture. The strong AJ conjecture has been confirmed for all torus knots and cables over torus knots. As such, the conjecture has only been confirmed for cases of non-hyperbolic knots. It should be noted that most knots fall into the class of being hyperbolic. In this thesis we confirm the strong AJ conjecture for the figure 8 knot which is the simplest hyperbolic knot.Show more Item The Asymptotics of the Higher Dimensional Reidemeister Torsion for Exceptional Surgeries Along Twist Knots(Canadian Mathematical Soc) Tran, Anh T.; Yamaguchi, Yoshikazu; Tran, Anh T.Show more We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible SL2Show more Item The Character Varieties of Rational Links C(2n, 2m + 1, 2)(2019-05) Meyer, Bradley D.; Tran, Anh T.Show more In this thesis we study the nonabelian SL2(C) character varieties of an infinite family of rational links. In chapter 1 we provide background information on rational knots and links and their character varieties. We also provide some properties of Chebyshev polynomials, which will be used in calculating the character varieties. In chapter 2 we first find a presentation for the knot group of C(2n, 2m + 1, 2). We then calculate the nonabelian character variety and prove that the character variety of C(2n, 2m + 1, 2) is irreducible unless n = 1, 1 or m = 1.Show more