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Anh T. Tran

Publications and source records attributed to Anh T. Tran.

At least 37 records · Page 2Linked to original sources
arXiv2020

Left orderable surgeries of double twist knots

A rational number $r$ is called a left orderable slope of a knot $K \subset S^3$ if the 3-manifold obtained from $S^3$ by $r$-surgery along $K$ has left orderable fundamental group. In this paper we consider the double twist knots $C(k,l)$ in the Conway notation. For any positive integers $m$ and $n$, we show that if $K$ is a double twist knot of the form $C(2m,-2n)$, $C(2m+1, 2n)$ or $C(2m+1, -2n)$ then there is an explicit unbounded interval $I$ such that any rational number $r \in I$ is a left orderable slope of $K$.

math.GT↗
arXiv2020

Left orderable surgeries of double twist knots II

A slope $r$ is called a left orderable slope of a knot $K \subset S^3$ if the 3-manifold obtained by $r$-surgery along $K$ has left orderable fundamental group. Consider two-bridge knots $C(2m, \pm 2n)$ and $C(2m+1, -2n)$ in the Conway notation, where $m \ge 1$ and $n \ge 2$ are integers. By using \textit{continuous} families of hyperbolic $\mathrm{SL}_2(\mathbb{R})$-representations of knot groups, it was shown in \cite{HT-genus1, Tr} that any slope in $(-4n, 4m)$ (resp. $[0, \max\{4m, 4n\})$) is a left orderable slope of $C(2m, 2n)$ (resp. $C(2m, - 2n)$) and in \cite{Ga} that any slope in $(-4n,0]$ is a left orderable slope of $C(2m+1,-2n)$. However, the proofs of these results are incomplete since the \textit{continuity} of the families of representations was not proved. In this paper, we complete these proofs and moreover we show that any slope in $(-4n, 4m)$ is a left orderable slope of $C(2m+1,-2n)$ detected by hyperbolic $\mathrm{SL}_2(\mathbb{R})$-representations of the knot group.

math.GT↗
arXiv2019

Transferability and Hardness of Supervised Classification Tasks

We propose a novel approach for estimating the difficulty and transferability of supervised classification tasks. Unlike previous work, our approach is solution agnostic and does not require or assume trained models. Instead, we estimate these values using an information theoretic approach: treating training labels as random variables and exploring their statistics. When transferring from a source to a target task, we consider the conditional entropy between two such variables (i.e., label assignments of the two tasks). We show analytically and empirically that this value is related to the loss of the transferred model. We further show how to use this value to estimate task hardness. We test our claims extensively on three large scale data sets -- CelebA (40 tasks), Animals with Attributes 2 (85 tasks), and Caltech-UCSD Birds 200 (312 tasks) -- together representing 437 classification tasks. We provide results showing that our hardness and transferability estimates are strongly correlated with empirical hardness and transferability. As a case study, we transfer a learned face recognition model to CelebA attribute classification tasks, showing state of the art accuracy for tasks estimated to be highly transferable.

cs.LG↗
arXiv2019

Twisted Alexander polynomials of torus links

In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL(2, \mathbb C)$-character variety. We also discuss similar things for the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.

math.GT↗
arXiv2019

Kashaev invariants of twice-iterated torus knots

We calculate the asymptotic behavior of the Kashaev invariant of a twice-itarated torus knot and obtain topological interpretation of the formula in terms of the Chern--Simons invariant and the twisted Reidemeister torsion.

math.GT↗
arXiv2019

Non left-orderable surgeries on L-space twisted torus knots

We show that if $K$ is an L-space twisted torus knot $T^{l,m}_{p,pk \pm 1}$ with $p \ge 2$, $k \ge 1$, $m \ge 1$ and $1 \le l \le p-1$, then the fundamental group of the $3$-manifold obtained by $\frac{r}{s}$-surgery along $K$ is not left-orderable whenever $\frac{r}{s} \ge 2 g(K) -1$, where $g(K)$ is the genus of $K$.

math.GT↗
arXiv2018

Left-orderability for surgeries on twisted torus knots

We show that the fundamental group of the $3$-manifold obtained by $\frac{p}{q}$-surgery along the $(n-2)$-twisted $(3,3m+2)$-torus knot, with $n,m \ge 1$, is not left-orderable if $\frac{p}{q} \ge 2n + 6m-3$ and is left-orderable if $\frac{p}{q}$ is sufficiently close to $0$.

math.GT↗
arXiv2016

On the volume of double twist link cone-manifolds

We consider the double twist link $J(2m+1, 2n+1)$ which is the two-bridge link corresponding to the continued fraction $(2m+1)-1/(2n+1)$. It is known that $J(2m+1, 2n+1)$ has reducible nonabelian $SL_2(\mathbb{C})$-character variety if and only if $m=n$. In this paper we give a formula for the volume of hyperbolic cone-manifolds of $J(2m+1,2m+1)$. We also give a formula for the A-polynomial 2-tuple corresponding to the canonical component of the character variety of $J(2m+1,2m+1)$.

math.GT↗
arXiv2016

The asymptotics of the higher dimensional Reidemeister torsion for exceptional surgeries along twist knots

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 SL(2;C)-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coefficients in the higher dimensional Reidemeister torsion explicitly.

math.GT↗
arXiv2016

Twisted Alexander polynomials of hyperbolic links

In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic representations of 2-bridge link groups.

math.GT↗
arXiv2016

On minimality of two-bridge knots

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $\mathfrak{b}(p,q)$ is minimal where $q \leq 6$ or $p \leq 100$.

math.GT↗
arXiv2016

Higher dimensional twisted Alexander polynomials for metabelian representations

We study the asymptotic behavior of the twisted Alexander polynomial for the sequence of SL(n ,C)-representations induced from an irreducible metabelian SL(2, C)-representation of a knot group. We give the limits of the leading coefficients in the asymptotics of the twisted Alexander polynomial and related Reidemeister torsion. The concrete computations for all genus one two-bridge knots are also presented.

math.GT↗
arXiv2016

The A-polynomial 2-tuple of twisted Whitehead links

We compute the A-polynomial 2-tuple of twisted Whitehead links. As applications, we determine canonical components of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds.

math.GT↗
arXiv2015

Volumes of double twist knot cone-manifolds

We give explicit formulae for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation $C(2n,3)$. We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups.

math.GT↗
arXiv2015

Knot Cabling and the Degree of the Colored Jones Polynomial

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 \ref{conj}), and we prove it for the following classes of knots:iterated torus knots and iterated cables of adequate knots, iterated cables of several non-alternating knots with up to nine crossings, pretzel knots of type $(-2, 3, p)$ and their cables, and two-fusion knots.

math.GT↗