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Articles 1 - 6 of 6
Full-Text Articles in Physical Sciences and Mathematics
Involutory Quandles Of (2,2,R)-Montesinos Links, Jim Hoste, Patrick D. Shanahan
Involutory Quandles Of (2,2,R)-Montesinos Links, Jim Hoste, Patrick D. Shanahan
Mathematics Faculty Works
In this paper we show that Montesinos links of the form L(1/2, 1/2, p/q;e), which we call (2,2,r)-Montesinos links, have finite involutory quandles. This generalizes an observation of Winker regarding the (2, 2, q)-pretzel links. We also describe some properties of these quandles.
Links With Finite N-Quandles, Jim Hoste, Patrick D. Shanahan
Links With Finite N-Quandles, Jim Hoste, Patrick D. Shanahan
Mathematics Faculty Works
We prove a conjecture of Przytycki which asserts that the n-quandle of a link L in the 3-sphere is finite if and only if the fundamental group of the n-fold cyclic branched cover of the 3-sphere, branched over L, is finite.
Twisted Alexander Polynomials Of 2-Bridge Knots, Jim Hoste, Patrick D. Shanahan
Twisted Alexander Polynomials Of 2-Bridge Knots, Jim Hoste, Patrick D. Shanahan
Mathematics Faculty Works
We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.
Upper Bounds In The Ohtsuki-Riley-Sakuma Partial Order On 2-Bridge Knots, Scott M. Garrabrant, Jim Hoste, Patrick D. Shanahan
Upper Bounds In The Ohtsuki-Riley-Sakuma Partial Order On 2-Bridge Knots, Scott M. Garrabrant, Jim Hoste, Patrick D. Shanahan
Mathematics Faculty Works
In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K we characterize all other 2-bridge knots J such that {K, J} has an upper bound. As an application we answer a question of Suzuki, showing that there is no upper bound for the set consisting of the trefoil and figure-eight knots.
Virtual Spatial Graphs, Thomas Fleming, Blake Mellor
Virtual Spatial Graphs, Thomas Fleming, Blake Mellor
Mathematics Faculty Works
Two natural generalizations of knot theory are t he study of spatially embedded graphs, and Kauffman's theory of virtual knots. In this paper we combine these approaches to begin the study of virtual spat ial graphs.
Commensurability Classes Of Twist Knots, Jim Hoste, Patrick D. Shanahan
Commensurability Classes Of Twist Knots, Jim Hoste, Patrick D. Shanahan
Mathematics Faculty Works
In this paper we prove that if MK is the complement of a non-fibered twist knot K in S3, then MK is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.