Research

My current research interests are in geometry and topology, in particular three dimensional topology and knot theory.

Below are some of my writings.

• Bất biến Alexander xoắn của nút (Twisted Alexander invariants, in Vietnamese) bài viết ngắn giới thiệu bất biến Alexander xoắn nhằm phục vụ báo cáo tại hội nghị Đại số-Hình học-Tôpô, Đại học Huế, 24–26 /9/2009.

• A computer routine for computation of the A-polynomials of two-bridge knots, runs on the free computer algebra system Maxima.

• Non-abelian Reidemeister torsion for twist knots, with Jérôme Dubois and Yoshikazu Yamaguchi, Journal of Knot Theory and Its Ramifications, vol 18, (2009), no 3, 303 - 341. pdf; arXiv. The published version is shortened from the preprint version.

  This paper gives an explicit formula for the SL_2(C)-non-abelian Reidemeister torsion in the cases of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot.

• On the twisted Alexander polynomial and the A-polynomial of 2-bridge knots, with Thang T. Q. Le, pdf

  We show that the A-polynomial A(L,M) of a 2-bridge knot b(p,q) is irreducible if p is prime, and if (p-1)/2 is also prime and q\neq 1 then the L-degree of A(L,M) is (p-1)/2. This shows that the AJ conjecture relating the A-polynomial and the colored Jones polynomial holds true for these knots, according to work of the second author. We also study relationships between the A-polynomial of a 2-bridge knot and a twisted Alexander polynomial associated with the adjoint representation of the fundamental group of the knot complement. We show that for twist knots the A-polynomial is a factor of the twisted Alexander polynomial.

• Twisted Alexander polynomial of links in the projective space, with Thang T. Q. Le, Journal of Knot Theory and Its Ramifications, vol 17 (2008), no 4, 411- 438 ; pdf; arXiv

  We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.

• Reidemeister torsion, twisted Alexander polynomial, the A-polynomial, and the colored Jones polynomial of some classes of knots, Ph. D. dissertation, April 2005. pdf.

• On the colored Jones polynomial and the Kashaev invariant, with Thang T. Q. Le, Fundamental and Applied Mathematics, vol 11 (2005), no 5, 57--78. pdf. Or Journal of Mathematical Sciences, vol 146, no 1, 5490-5504, 2007. This paper is also available at the ArXiv.

  We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the q-Weyl algebra of q-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discuss the similarity between our determinant formula of the Kashaev invariant and the determinant formula of the hyperbolic volume of knot complements, hoping it would lead to a proof of the volume conjecture.

• Reidemeister torsion and circle-valued Morse theory, Notes for a graduate student seminar, April 2001. pdf.

• Applications of Rearrangement functions in the equation \Delta u(x)=K(x)e^{u(x)}, senior thesis, June 1997. pdf.