arXiv subjects

Dan Guyer

Publications and source records attributed to Dan Guyer.

6 recordsLinked to original sources

arXiv2026

An Efficient Triangulation of $\mathbb{R}P^5$

We present a $6$-dimensional centrally symmetric simplicial polytope for which the antipodal quotient of its boundary forms a $24$-vertex triangulation of the $5$-dimensional real projective space. This $6$-polytope is highly symmetric with an automorphism group of order $192$, and is of independent interest. We conjecture that our construction uses the fewest number of vertices among all triangulations of $\mathbb{R}P^5$. Our method also produces two triangulations of $\mathbb{R}P^6$ on $45$ and $49$ vertices; both improve the previously best known construction in dimension $6$ that used $53$ vertices.

math.CO↗

arXiv2026

Poset Partitions and the Combinatorics of the $\textbf{cd}$-Index

We introduce a new class of Eulerian posets, called S-partitionable posets, which have a non-negative cd-index. These posets are a generalization of S-shellable complexes introduced by Stanley in 1994. We prove that S-partitionable posets have a non-negative cd-index via a recursive formula. Then, we introduce a semi-Eulerian version of S-partitionable posets, which we call SE-partitionable posets. We show that SE-partitionable posets also have a non-negative semi-Eulerian cd-index as defined by Juhnke-Kubitzke, Samper and Venturello in 2024.

math.CO↗

arXiv2025

Monotone Paths on Acyclic 3-Regular Graphs

Motivated by trying to understand the behavior of the simplex method, Athanasiadis, De Loera and Zhang provided upper and lower bounds on the number of the monotone paths on 3-polytopes. For simple 3-polytopes with $2n$ vertices, they showed that the number of monotone paths is bounded above by $(1+\varphi)^n$, with $\varphi$ being the golden ratio. We improve the result and show that for a larger family of graphs the number is bounded above by $c \cdot 1.6779^n$ for some universal constant $c$. Meanwhile, the best known construction and conjectured extremizer is approximately $\varphi^n$.

math.CO↗

arXiv2024

Concordances of sums of alternating torus knots and their mirrors to $L$-space knots

Continuing the work of Zemke, Livingston and Allen, we consider when linear combinations of torus knots are concordant to $L$-space knots. We begin by proving Allen's conjecture for alternating torus knots. That is, we prove that a linear combination of alternating torus knots is concordant to an $L$-space knot if and only if the connected sum is a single torus knot. Then we establish a necessary condition for when a linear combination of torus knots is concordant to an $L$-space knot.

math.GT↗

arXiv2022

Tantalizing properties of subsequences of the Fibonacci sequence modulo 10

The Fibonacci sequence modulo $m$, which we denote $\left(\mathcal{F}_{m,n}\right)_{n=0}^\infty$ where $\mathcal{F}_{m,n}$ is the Fibonacci number $F_n$ modulo $m$, has been a well-studied object in mathematics since the seminal paper by D.~D.~Wall in 1960 exploring a myriad of properties related to the periods of these sequences. Since the time of Lagrange it has been known that $\left(\mathcal{F}_{m,n}\right)_{n=0}^\infty$ is periodic for each $m$. We examine this sequence when $m=10$, yielding a sequence of period length 60. In particular, we explore its subsequences composed of every $r^{\mathrm{th}}$ term of $\left(\mathcal{F}_{10,n}\right)_{n=0}^\infty$ starting from the term $\mathcal{F}_{10,k}$ for some $0 \leq k \leq 59$. More precisely we consider the subsequences $\left(\mathcal{F}_{10,k+rj}\right)_{j=0}^\infty$, which we show are themselves periodic and whose lengths divide 60. Many intriguing properties reveal themselves as we alter the $k$ and $r$ values. For example, for certain $r$ values the corresponding subsequences surprisingly obey the Fibonacci recurrence relation; that is, any two consecutive subsequence terms sum to the next term modulo 10. Moreover, for all $r$ values relatively prime to 60, the subsequence $\left(\mathcal{F}_{10,k+rj}\right)_{j=0}^\infty$ coincides exactly with the original parent sequence $\left(\mathcal{F}_{10,n}\right)_{n=0}^\infty$ (or a cyclic shift of it) running either forward or reverse. We demystify this phenomena and explore many other tantalizing properties of these subsequences.

math.NT↗

arXiv2021

GCD of sums of $k$ consecutive Fibonacci, Lucas, and generalized Fibonacci numbers

We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sums of $k$ consecutive generalized Fibonacci numbers by the symbol $\mathcal{G}_{G_0, G_1}\!(k)$, we give two tantalizing characterizations for these values, one involving a simple formula in $k$ and another involving generalized Pisano periods: $$\mathcal{G}_{G_0, G_1}\!(k) = \gcd(G_{k+1}-G_1,\, G_{k+2}-G_2)\; \mbox{and}$$ $$\mathcal{G}_{G_0, G_1}\!(k) = \mathrm{lcm}\{m \mid π_{G_0,G_1}\!(m) \text{ divides } k\},$$ where $π_{G_0,G_1}\!(m)$ denotes the generalized Pisano period of the generalized Fibonacci sequence modulo $m$. The fact that these vastly different-looking formulas coincide leads to some surprising and delightful new understandings of the Fibonacci and Lucas numbers.

math.NT↗