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ON THE CHARACTER RINGS OF TWIST KNOTS
Nagasato, Fumikazu Korean Mathematical Society 2011 대한수학회보 Vol.48 No.3
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.
On the character rings of twist knots
Fumikazu Nagasato 대한수학회 2011 대한수학회보 Vol.48 No.3
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(C)-character ring of the fundamental group of M. Our proof is based on some properties of Chebyshev polynomials. 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(C)-character ring of the fundamental group of M. Our proof is based on some properties of Chebyshev polynomials.
On a Background of the Existence of Multi-variable Link Invariants
Nagasato, Fumikazu,Hamai, Kanau Department of Mathematics 2008 Kyungpook mathematical journal Vol.48 No.2
We present a quantum theorical background of the existence of multi-variable link invariants, for example the Kauffman polynomial, by observing the quantum (sl(2,$\mathbb{C}$), ad)-invariant from the Kontsevich invariant point of view. The background implies that the Kauffman polynomial can be studied by using the sl(N,$\mathbb{C}$)-skein theory similar to the Jones polynomial and the HOMFLY polynomial.