• Kyungpook Mathematical Journal
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• v.48 no.2
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• pp.255-280
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• 2008

We derive a formula for the colored Jones polynomial of 2-bridge knots. For a twist knot, a more explicit formula is given and it leads to a relation between the degree of the colored Jones polynomial and the crossing number.

• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.937-956
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• 2018

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm [26] to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.

• East Asian mathematical journal
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• v.32 no.5
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• pp.767-774
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• 2016

Manturov inroduced the parity bracket polynomial by using a parity of virtual knots, which is an extension of Jones-Kauffman polynomial. We extend Manturov's result to virtual links, so that we obtain the parity bracket polynomial for virtual links and give some examples.

• Journal of the Korean Mathematical Society
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• v.51 no.3
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• pp.443-462
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• 2014

Using a simple recurrence relation, we give a new method to compute the Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for the Jones polynomials. The method is used to estimate the degree of the Jones polynomials for some families of braids and to obtain general qualitative results.

• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.919-947
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• 2009

In this paper, we give a recursive formula for the Jones polynomial of a 2-bridge knot or link with Conway normal form C($-2n_1$, $2n_2$, $-2n_3$, ..., $(-1)_r2n_r$) in terms of $n_1$, $n_2$, ..., $n_r$. As applications, we also give a recursive formula for the Jones polynomial of a 3-periodic link $L^{(3)}$ with rational quotient L = C(2, $n_1$, -2, $n_2$, ..., $n_r$, $(-1)^r2$) for any nonzero integers $n_1$, $n_2$, ..., $n_r$ and give a formula for the span of the Jones polynomial of $L^{(3)}$ in terms of $n_1$, $n_2$, ..., $n_r$ with $n_i{\neq}{\pm}1$ for all i=1, 2, ..., r.

• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.95-113
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• 2018

A mathematical knot is an embedded circle in ${\mathbb{R}}^3$. A fundamental problem in knot theory is classifying knots up to its numbers of crossing points. Knots are often distinguished by using a knot invariant, a quantity which is the same for equivalent knots. Knot polynomials are one of well known knot invariants. In 2006, J. Przytycki showed the effects of a n - move (a local change in a knot diagram) on several knot polynomials. In this paper, the authors review about knot polynomials, especially Jones polynomial, and give an alternative proof to a part of the Przytychi's result for the case n = 3 on the Jones polynomial.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2149-2154
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• 2017

In this paper, we obtain a simple lower bound for the number of positive (resp. negative) crossings in oriented link diagrams in terms of the maximal (resp. minimal) degree of the Jones polynomial.

• Journal of the Korean Mathematical Society
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• v.50 no.3
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• pp.641-693
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• 2013

We show that the optimistic limits of the colored Jones polynomials of the hyperbolic knots coincide with the optimistic limits of the Kashaev invariants modulo $4{\pi}^2$.

• East Asian mathematical journal
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• v.25 no.4
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• pp.469-479
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• 2009

Temperley-Lieb algebras had been generalized to web spaces for rank 2 simple Lie algebras which led us to link invariants for these Lie algebras as a generalization of Jones polynomial. Recently, Geer found a new generalization of Jones polynomial for some Lie superalgebras. In this paper, we study the quantum sl(1|1) representation theory using the web space and find a finite presentation of the representation category (for generic q) of the quantum sl(1|1).

• East Asian mathematical journal
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• v.25 no.4
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• pp.487-494
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• 2009

We show that if two virtual link diagrams which are normal are equivalent under generalized Reidemeister moves, then they are equivalent under generalized Reidemeister moves preserving the normality of diagrams.