Tran, Anh T.

Permanent URI for this collectionhttps://hdl.handle.net/10735.1/4834

Anh Tran is an Associate Professor of Mathematics. His research interests are focused on quantum topology and knot theory especially as they relate to the Jones polynomial, skein modules, charactyer varieties, the A-polynomial, the Alexander polynomial, and left-orderability of fundamental groups.

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    On the AJ Conjecture for Knots
    (2015-07-07) Le, Thang T. Q.; Tran, Anh T.; Huynh, Vu Q.; Tran, Anh T.
    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.
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    Knot Cabling and the Degree of the Colored Jones Polynomial
    (University at Albany, 2015-09-16) Kalfagianni, E.; Tran, Anh T.; Tran, Anh T.
    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.
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    Left-Orderability for Surgeries on Twisted Torus Knots
    (Japan Academy, 2019-01) Tran, Anh T.; Tran, Anh T.
    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.
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    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.
    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 SL2

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