• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.303-318
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• 2022

n-writhes denoted by J_n(K) are virtual knot invariants for n ≠ 0 and are closely associated with coefficients of some polynomial invariants of virtual knots. In this work, we investigate the variations of J_n(K) under arc shift move and conclude that n-writhes J_n(K) vary randomly in the sense that it may change by any random integer value under one arc shift move. Also, for each n ≠ 0 we provide an infinite family of virtual knots which can be distinguished by n-writhes J_n(K), whereas odd writhe J(K) fails to do so.

• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.469-474
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• 2011

The Kauffman bracket skein module $K_t$(M) of a 3-manifold M becomes an algebra for t = -1. We prove that this algebra has no non-trivial nilpotent elements for M being the exterior of the twist knot in 3-sphere and, therefore, it is isomorphic to the $SL_2(\mathbb{C})$-character ring of the fundamental group of M. Our proof is based on some properties of Chebyshev polynomials.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.519-529
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• 2006

The Property P Conjecture States that the 3-manifold $Y_r$ obtained by Dehn surgery on a non-trivial knot in $S^3$ with surgery coefficient ${\gamma}{\in}Q$ has the non-trivial fundamental group (so not simply connected). Recently Kronheimer and Mrowka provided a proof of the Property P conjecture for the case ${\gamma}={\pm}2$ that was the only remaining case to be established for the conjecture. In particular, their results show that the two phenomena of having a cyclic fundamental group and having a homomorphism with non-cyclic image in SU(2) are quite different for 3-manifolds obtained by Dehn filings. In this paper we extend their results to some other Dehn surgeries via the A-polynomial, and provide more evidence of the ubiquity of the above mentioned phenomena.

• Journal of the Korean Institute of Intelligent Systems
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• v.21 no.2
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• pp.268-275
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• 2011

This paper proposes two optimal path planning methods of a mobile robot using uDEAS (univariate Dynamic Encoding Algorithm for Searches). Before start of autonomous traveling, a self-controlled mobile robot must generate an optimal global path as soon as possible. To this end, numerical optimization method is applied to real time path generation of a mobile robot with an obstacle avoidance scheme and the basic path generation method based on the concept of knot and node points between start and goal points. The first improvement in the present work is to generate diagonal paths using three node points in the basic path. The second innovation is to make a smooth path plotted with the blending polynomial using uDEAS. Effectiveness of the proposed schemes are validated for several environments through simulation.