arXiv subjects

Cornelia A. Van Cott

Publications and source records attributed to Cornelia A. Van Cott.

13 recordsLinked to original sources

arXiv2026

There and back again with Kaprekar 2-cycles

A curious property of integers, first observed by D.~R. Kaprekar in 1949, is as follows: write the digits of a 4-digit number in both descending and ascending order, and then find the positive difference between these integers. Iterating this process on any 4-digit number - except for multiples of 1111 - eventually produces the number 6174. The process ends here, since 6174 is a fixed point of the procedure. In general, if we start with any integer and iterate this process, it either ends at a fixed point (as with 4 digits) or enters a cycle of numbers, called a Kaprekar cycle. In 2011, Dolan classified all fixed points of this process. We study Kaprekar cycles of length 2. We find general properties of numbers in a Kaprekar 2-cycle, and we classify all 2-cycles such that the two integers have the same smallest digit.

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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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arXiv2025

The change-making problem for six coin values and beyond

The change-making problem asks: given a positive integer $v$ and a collection $C$ of integer coin values $c_1=1<c_2< c_3< \cdots< c_n$, what is the minimum number of coins needed to represent $v$ with coin values from $C$? For some coin systems $C$, the greedy algorithm finds a representation with a minimum number of coins for all $v$. We call such coin systems orderly. However, there are coin systems where the greedy algorithm fails to always produce a minimal representation. Over the past fifty years, progress has been made on the change-making problem, including finding a characterization of all orderly coin systems with 3, 4, and 5 coin values. We characterize orderly coin systems with 6 coin values, and we make generalizations to orderly coin systems with $n$ coin values.

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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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arXiv2023

Unshuffling a deck of cards

We investigate the mathematics behind unshuffles, a type of card shuffle closely related to classical perfect shuffles. To perform an unshuffle, deal all the cards alternately into two piles and then stack the one pile on top of the other. There are two ways this stacking can be done (left stack on top or right stack on top), giving rise to the terms left shuffle ($L$) and right shuffle ($R$), respectively. We give a solution to a generalization of Elmsley's Problem (a classic mathematical card trick) using unshuffles for decks with $2^k$ cards. We also find the structure of the permutation groups $\langle L, R \rangle$ for a deck of $2n$ cards for all values of $n$. We prove that the group coincides with the perfect shuffle group unless $n\equiv 3 \pmod 4$, in which case the group $\langle L, R \rangle$ is equal to $B_n$, the group of centrally symmetric permutations of $2n$ elements, while the perfect shuffle group is an index 2 subgroup of $B_n$.

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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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arXiv2020

A look at generalized perfect shuffles

Standard perfect shuffles involve splitting a deck of $2n$ cards into two stacks and interlacing the cards from the stacks. There are two ways that this interlacing can be done, commonly referred to as an in shuffle and an out shuffle, respectively. In 1983, Diaconis, Graham, and Kantor determined the permutation group generated by in and out shuffles on a deck of $2n$ cards for all $n$. Diaconis et al. concluded their work by asking whether similar results can be found for so-called generalized perfect shuffles. For these new shuffles, we split a deck of $mn$ cards into $m$ stacks and similarly interlace the cards with an in $m$-shuffle or out $m$-shuffle (denoted $I_m$ and $O_m$, respectively). In this paper, we find the structure of the group generated by these two shuffles for a deck of $m^k$ cards, together with $m^y$-shuffles, for all possible values of $m$, $k$, and $y$. The group structure is completely determined by $k/\gcd(y,k)$ and the parity of $y/\gcd(y,k)$. In particular, the group structure is independent of the value of $m$.

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arXiv2020

Comparing nonorientable three genus and nonorientable four genus of torus knots

We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let T(p,q) be any torus knot with p even and q odd. The difference between these two invariants on T(p,q) is at least k/2, where p = qk + a and 0 < a < q and $k\geq 0$. Hence, the difference between the two invariants on torus knots T(p,q) grows arbitrarily large for any fixed odd q, as p ranges over values of a fixed congruence class modulo q. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot T(p,q) is (p-1)(q-1)/2, and Kronheimer and Mrowka later proved that the orientable four genus of T(p,q) is also this same value.

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arXiv2019

On a Nonorientable Analogue of the Milnor Conjecture

The nonorientable 4-genus $γ_4(K)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot $K$. We study a conjecture proposed by Batson about the value of $γ_4$ for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjecture for the orientable 4-genus of torus knots. We prove the conjecture for many infinite families of torus knots, by relying on a lower bound for $γ_4$ formulated by Ozsváth, Stipsicz, and Szabó. As a side product we obtain new closed formulas for the signature of torus knots.

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arXiv2015

The four-genus of connected sums of torus knots

We study the four-genus of linear combinations of torus knots: aT(p,q) # -bT(p',q'). Fixing positive p, q, p', and q', our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram-Levine signature function; for the second, the recently defined Upsilon function of Ozsvath-Stipsicz-Szabo determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.

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arXiv2009

An obstruction to slicing iterated Bing doubles

Let K be a knot in S^3. We study the iterated Bing doubles of K, giving a new proof for the following statement: If BD_n(K) is slice for some n, then K is algebraically slice. This result was first proved by Cha and Kim using covering link calculus. We also use this tool, but our proof is substantially simpler and illuminates several generalizations.

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arXiv2008

Ozsvath-Szabo and Rasmussen invariants of cable knots

We study the behavior of the Ozsvath-Szabo and Rasmussen knot concordance invariants tau and s on K(m,n), the (m,n)-cable of a knot K where m and n are relatively prime. We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on K(m,n) differ from their value on the torus knot T(m,n) by a fixed constant for all but finitely many n>0. Combining this result together with Hedden's extensive work on the behavior of tau on (m,mr+1)-cables yields bounds on the value of tau on any (m,n)-cable of K. In addition, several of Hedden's obstructions for cables bounding complex curves are extended.

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arXiv2006

Relationships between braid length and the number of braid strands

For a knot K, let b_n(K) be the minimum length of an n-stranded braid representative of K. Examples of knots exist for which b_n(K) is a non-increasing function. We investigate the behavior of b_n(K). We develop bounds on the function in terms of the genus of K, with stronger results for homogeneous knots and braid positive knots. For knots of nine or fewer crossings, we show that b_n(K) is an increasing function and determine it completely.

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