Jim Hoste
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View article: On the nonorientable 4-genus of double twist knots
On the nonorientable 4-genus of double twist knots Open
We investigate the nonorientable 4-genus $γ_4$ of a special family of 2-bridge knots, the twist knots and double twist knots $C(m,n)$. Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that $γ_4(C(m,n))…
View article: Crosscap number and epimorphisms of two-bridge knot groups
Crosscap number and epimorphisms of two-bridge knot groups Open
We consider the relationship between the crosscap number [Formula: see text] of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots [Formula: see text] and [Formula: see text], we say [Formul…
View article: Crosscap number and the partial order on two-bridge knots
Crosscap number and the partial order on two-bridge knots Open
We consider the relationship between the crosscap number $\gamma$ of knots and the partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K \geq J$ if there exists an epimorphism $f:\pi…
View article: Triple-crossing number and moves on triple-crossing link diagrams
Triple-crossing number and moves on triple-crossing link diagrams Open
Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We…
View article: A note on Alexander polynomials of 2-bridge links
A note on Alexander polynomials of 2-bridge links Open
A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-component, 2-bridg…
View article: Finite n-quandles of torus and two-bridge links
Finite n-quandles of torus and two-bridge links Open
We compute Cayley graphs and automorphism groups for all finite [Formula: see text]-quandles of two-bridge and torus knots and links, as well as torus links with an axis.
View article: Remarks on Suzuki's Knot Epimorphism Number
Remarks on Suzuki's Knot Epimorphism Number Open
A partial order on prime knots can be defined by declaring $J\ge K$ if there exists an epimorphism from the knot group of $J$ onto the knot group of $K$. Suppose that $J$ is a 2-bridge knot that is strictly greater than $m$ distinct, nontr…
View article: An enumeration process for racks
An enumeration process for racks Open
Given a presentation for a rack $\mathcal R$, we define a process which systematically enumerates the elements of $\mathcal R$. The process is modeled on the systematic enumeration of cosets first given by Todd and Coxeter. This generalize…
View article: Involutory quandles of (2,2,r)-Montesinos links
Involutory quandles of (2,2,r)-Montesinos links Open
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 d…