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Patrick D. Shanahan

Publications and source records attributed to Patrick D. Shanahan.

14 recordsLinked to original sources
arXiv2025

Average crosscap number of a 2-bridge knot

We determine a simple condition on a particular state graph of an alternating knot or link diagram that characterizes when the unoriented genus and crosscap number coincide, extending work of Adams and Kindred. Building on this same work and using continued fraction expansions, we provide a new formula for the unoriented genus of a 2-bridge knot or link. We use recursion to obtain exact formulas for the average unoriented genus $\overlineΓ(c)$ and average crosscap number $\overlineγ(c)$ of all 2-bridge knots with crossing number $c$, and in particular we show that $\displaystyle{\lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overlineΓ(c)\right) = \lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overlineγ(c)\right) = 0}$.

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arXiv2023

On the nonorientable 4-genus of double twist knots

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 $γ_4(C(m,n)) \le 3$. By using explicit constructions to obtain upper bounds on $γ_4$ and known obstructions derived from Donaldson's diagonalization theorem to obtain lower bounds on $γ_4$, we produce infinite subfamilies of $C(m,n)$ where $γ_4=0,1,2,$ and $3$, respectively. However, there remain infinitely many double twist knots where our work only shows that $γ_4$ lies in one of the sets $\{1,2\}, \{2,3\}$, or $\{1,2,3\}$. We tabulate our results for all $C(m,n)$ with $|m|$ and $|n|$ up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara.

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arXiv2021

Crosscap number and epimorphisms of two-bridge knot groups

We consider the relationship between the crosscap number $γ$ of knots and a 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 $f:π_1(S^3-K) \longrightarrow π_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K> J$, then $γ(K) \geq 3γ(J) -4$. We also classify all pairs $(K,J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K >J$, then $g(K) \geq 3 g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.

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arXiv2018

Remarks on Suzuki's Knot Epimorphism Number

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, nontrivial knots. In this paper we determine a lower bound on the crossing number of $J$ in terms of $m$. Using this bound we answer a question of Suzuki regarding the 2-bridge epimorphism number $\mbox{EK}(n)$ which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number $n$. We establish our results using techniques associated to parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot.

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arXiv2018

An enumeration process for racks

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 generalizes and improves the diagramming method for $n$-quandles introduced by Winker. We provide pseudocode that is similar to that given by Holt for the Todd-Coxeter process. We prove that the process terminates if and only if $\mathcal R$ is finite, in which case, the procedure outputs an operation table for the finite rack. We conclude with an application to knot theory.

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arXiv2017

Links with finite $n$-quandles

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.

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arXiv2016

Involutory quandles of (2,2,r)-Montesinos links

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.

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arXiv2012

Twisted Alexander polynomials of 2-bridge knots

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.

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arXiv2011

Upper bounds in the Ohtsuki-Riley-Sakuma partial order on 2-bridge knots

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.

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arXiv2010

Epimorphisms and Boundary Slopes of 2-Bridge Knots

In this article we study a partial ordering on knots in the 3-sphere where K_1 is greater than or equal to K_2 if there is an epimorphism from the knot group of K_1 onto the knot group of K_2 which preserves peripheral structure. If K_1 is a 2-bridge knot and K_1 > K_2, then it is known that K_2 must also be 2-bridge. Furthermore, Ohtsuki, Riley, and Sakuma give a construction which, for a given 2-bridge knot K_{p/q}, produces infinitely 2-bridge knots K_{p'/q'} with K_{p'/q'}>K_{p/q}. After characterizing all 2-bridge knots with 4 or less distinct boundary slopes, we use this to prove that in any such pair, K_{p'/q'} is either a torus knot or has 5 or more distinct boundary slopes. We also prove that 2-bridge knots with exactly 3 distinct boundary slopes are minimal with respect to the partial ordering. This result provides some evidence for the conjecture that all pairs of 2-bridge knots with K_{p'/q'}>K_{p/q} arise from the Ohtsuki-Riley-Sakuma construction.

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arXiv2006

Boundary slopes of 2-bridge links determine the crossing number

A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the diameter of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of diagonal surfaces, is equal to the crossing number.

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arXiv2005

Computing boundary slopes of 2-bridge links

We describe an algorithm for computing boundary slopes of 2-bridge links. As an example, we work out the slopes of the links obtained by 1/k surgery on one component of the Borromean rings. A table of all boundary slopes of all 2-bridge links with 10 or less crossings is also included.

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arXiv2004

Commensurability classes of twist knots

In this paper we prove that if $M_K$ is the complement of a non-fibered twist knot $K$ in $\mathbb S^3$, then $M_K$ is not commensurable to a fibered knot complement in a $\mathbb Z/ 2 \mathbb Z$-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.

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