Twisted Alexander polynomials and surjectivity of a group homomorphism.
Kitano, Teruaki, Suzuki, Masaaki, Wada, Masaaki (2005)
Algebraic & Geometric Topology
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Kitano, Teruaki, Suzuki, Masaaki, Wada, Masaaki (2005)
Algebraic & Geometric Topology
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Kitano, Teruaki, Suzuki, Masaaki (2005)
Experimental Mathematics
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Louis H. Kauffman, Vassily O. Manturov (2005)
Fundamenta Mathematicae
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We describe new approaches for constructing virtual knot invariants. The main background of this paper comes from formulating and bringing together the ideas of biquandle [KR], [FJK], the virtual quandle [Ma2], the ideas of quaternion biquandles by Roger Fenn and Andrew Bartholomew [BF], the concepts and properties of long virtual knots [Ma10], and other ideas in the interface between classical and virtual knot theory. In the present paper we present a new algebraic construction of virtual...
Roger Fenn, Louis H. Kauffman, Vassily O. Manturov (2005)
Fundamenta Mathematicae
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The present paper gives a quick survey of virtual and classical knot theory and presents a list of unsolved problems about virtual knots and links. These are all problems in low-dimensional topology with a special emphasis on virtual knots. In particular, we touch new approaches to knot invariants such as biquandles and Khovanov homology theory. Connections to other geometrical and combinatorial aspects are also discussed.
Khovanov, Mikhail (2003)
Experimental Mathematics
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S. Jablan, R. Sazdanovic (2003)
Visual Mathematics
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Nafaa Chbili (2003)
Annales de la Faculté des sciences de Toulouse : Mathématiques
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Moshe Cohen, Oliver T. Dasbach, Heather M. Russell (2014)
Fundamenta Mathematicae
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We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot depending on a representation of the knot group.
Teruaki Kitano, Takayuki Morifuji (2005)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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We prove that Wada’s twisted Alexander polynomial of a knot group associated to any nonabelian -representation is a polynomial. As a corollary, we show that it is always a monic polynomial of degree for a fibered knot of genus .
Aaron Kaestner, Louis H. Kauffman (2014)
Banach Center Publications
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We use crossing parity to construct a generalization of biquandles for virtual knots which we call parity biquandles. These structures include all biquandles as a standard example referred to as the even parity biquandle. We find all parity biquandles arising from the Alexander biquandle and quaternionic biquandles. For a particular construction named the z-parity Alexander biquandle we show that the associated polynomial yields a lower bound on the number of odd crossings as well as...
Roger Fenn, Denis P. Ilyutko, Louis H. Kauffman, Vassily O. Manturov (2014)
Banach Center Publications
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This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.
Dugopolski, Mark J. (1985)
International Journal of Mathematics and Mathematical Sciences
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Livingston, Charles (2002)
Algebraic & Geometric Topology
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