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Moshe Cohen

Publications and source records attributed to Moshe Cohen.

At least 19 recordsLinked to original sources

A central limit theorem for the signatures of 2-bridge knots

Cohen, Lowrance, Madras, and Raanes computed the average (absolute value of) signature over all 2-bridge knots with crossing number $c$ by introducing the number $s(c,\sigma)$ of 2-bridge knots of crossing number $c$ and signature $\sigma$. Here we provide a closed formula for this number. We use these calculations to show that the distribution of the signatures of 2-bridge knots with crossing number $c$ approaches a normal distribution as $c$ tends to infinity.

math.GT

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 $\overline{\Gamma}(c)$ and average crosscap number $\overline{\gamma}(c)$ of all 2-bridge knots with crossing number $c$, and in particular we show that $\displaystyle{\lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overline{\Gamma}(c)\right) = \lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overline{\gamma}(c)\right) = 0}$.

math.GT

Average signature and 4-genus of 2-bridge knots

We show that the average or expected absolute value of the signatures of all 2-bridge knots with crossing number $c$ approaches $\sqrt{{2c}/{\pi}}$. Baader, Kjuchukova, Lewark, Misev, and Ray consider a model for 2-bridge knot diagrams indexed by diagrammatic crossing number $n$ and show that the average 4-genus is sublinear in $n$. We build upon this result in two ways to obtain an upper bound for the average 4-genus of a 2-bridge knot: our model is indexed by crossing number $c$ and gives a specific sublinear upper bound of $9.75c/\log c$.

math.GT

Modeling the effects of adherence to vaccination and health protocols in epidemic dynamics by means of an SIR model

Susceptible-Infected-Recovered (SIR) models have been used for decades to understand epidemic outbreak dynamics. We develop an SIR model specifically designed to study the effects of population behavior with respect to health and vaccination protocols in a generic epidemic. Through a collection of parameters, our model includes the traditional SIR components: population birth, death, infection, recovery and vaccination rates, as well as limited immunity. We first use this simple setup to compare the effects of two vaccination schemes, one in which people are vaccinated at a rate proportional with the population, and one in which vaccines are administered to a fraction of the susceptible people (both of which are know strategies in real life epidemics). We then expand on the model and the analysis by investigating how these two vaccination schemes hold under two scenarios of population behavior: one in which people abide by health protocols and work towards diminishing transmission when infection is high; one in which people relax health protocols when infection is high. We illustrate these two aspects (vaccination and adherence to health protocols) act together to control the epidemic outbreak. While it is ideal that the tow components act jointly, we also show that tight observance of health protocols may diminish the need for vaccination in the effort to clear or mitigate the outbreak. Conversely, an efficient vaccination strategy can compensate for some degree of laxity in people's behavior.

q-bio.PE

The distribution of genera of 2-bridge knots

The average genus of a 2-bridge knot with crossing number $c$ approaches $\frac{c}{4} + \frac{1}{12}$ as $c$ approaches infinity, as proven by Suzuki and Tran and independently Cohen and Lowrance. In this paper, for the genera of $2$-bridge knots of a fixed crossing number $c$, we show that the median and mode are both $\lfloor \frac{c+2}{4} \rfloor$ and that the variance approaches $\frac{c}{16}-\frac{17}{144}$ as $c$ approaches infinity. We prove that the distribution of genera of 2-bridge knots is asymptotically normal.

math.GT

Moduli Spaces of One-Line Extensions of $(10_3)$ Configurations

Two line arrangements in $\mathbb{CP}^2$ can have different topological properties even if they are combinatorially isomorphic. Results by Dan Cohen and Suciu and by Randell show that a reducible moduli space under complex conjugation is a necessary condition. We present a method to produce many examples of combinatorial line arrangements with a reducible moduli space obtained from a set of examples with irreducible moduli spaces. In this paper, we determine the reducibility of the moduli spaces of a family of arrangements of 11 lines constructed by adding a line to one of the ten $(10_3)$ configurations. Out of the four hundred ninety-five combinatorial line arrangements in this family, ninety-five have a reducible moduli space, seventy-six of which are still reducible after the quotient by complex conjugation.

math.AG

The average genus of a 2-bridge knot is asymptotically linear

Experimental work suggests that the Seifert genus of a knot grows linearly with respect to the crossing number of the knot. In this article, we use a billiard table model for $2$-bridge or rational knots to show that the average genus of a $2$-bridge knot with crossing number $c$ asymptotically approaches $c/4+1/12$.

math.GT

A lower bound on the average genus of a 2-bridge knot

Experimental data from Dunfield et al using random grid diagrams suggests that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model of all 2-bridge knots of a given crossing number, almost all counted twice. We present a convenient way to count Seifert circles in this model and use this to compute a lower bound for the average Seifert genus of a 2-bridge knot of a given crossing number.

math.GT

Leveraging human knowledge in tabular reinforcement learning: A study of human subjects

Reinforcement Learning (RL) can be extremely effective in solving complex, real-world problems. However, injecting human knowledge into an RL agent may require extensive effort and expertise on the human designer's part. To date, human factors are generally not considered in the development and evaluation of possible RL approaches. In this article, we set out to investigate how different methods for injecting human knowledge are applied, in practice, by human designers of varying levels of knowledge and skill. We perform the first empirical evaluation of several methods, including a newly proposed method named SASS which is based on the notion of similarities in the agent's state-action space. Through this human study, consisting of 51 human participants, we shed new light on the human factors that play a key role in RL. We find that the classical reward shaping technique seems to be the most natural method for most designers, both expert and non-expert, to speed up RL. However, we further find that our proposed method SASS can be effectively and efficiently combined with reward shaping, and provides a beneficial alternative to using only a single speedup method with minimal human designer effort overhead.

cs.AI

Crossing Numbers of Random Two-Bridge Knots

In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also show that the probability of any given knot appearing in this model decays to zero at an exponential rate as the length of the billiard table goes to infinity. This confirms a conjecture from the previous work.

math.GT

Random knots using Chebyshev billiard table diagrams

We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2. Restricted to this class, we define internal and external reduction moves that decrease the number of crossings of the diagram. We make calculations based on our formula, showing the numerics in graphs and providing evidence for our conjecture that the probability of any knot appearing in this model decays to zero as the number of crossings goes to infinity.

math.GT

The Jones polynomials of 3-bridge knots via Chebyshev knots and billiard table diagrams

This work presents formulas for the Kauffman bracket and Jones polynomials of 3-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation expansion. The subject is introduced by considering the easier case of 2-bridge knots, where some geometric interpretation is provided, as well, via combinatorial tiling problems.

math.GT

A counterexample to Question 1 of "A survey on the Turaev genus of knots"

In "A survey on the Turaev genus of knots," Champanerkar and Kofman propose several open questions. The first asks whether the polynomial whose coefficients count the number of quasi-trees of the all-A ribbon graph obtained from a diagram with minimal Turaev genus is an invariant of the knot. We answer negatively by showing a counterexample obtained from the two diagrams of $8_{21}$ on the KnotAtlas and KnotScape.

math.GT

Combinatorial symmetry of line arrangements and applications

We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples found in previous work by the authors from being Zariski pairs. Robustness is shown by its application to complex cases, as well.

math.AG

Moduli spaces of ten-line arrangements with double and triple points

Two arrangements with the same combinatorial intersection lattice but whose complements have different fundamental groups are called a Zariski pair. This work finds that there are at most nine such pairs amongst all ten line arrangements whose intersection points are doubles or triples. This result is obtained by considering the moduli space of a given configuration table which describes the intersection lattice. A complete combinatorial classification is given of all arrangements of this type under a suitable assumption, producing a list of seventy-one described in a table, most of which do not explicitly appear in the literature. This list also includes other important counterexamples: nine combinatorial arrangements that are not geometrically realizable.

math.AG

Links arising from braid monodromy factorizations

We investigate the local contribution of the braid monodromy factorization in the context of the links obtained by the closure of these braids. We consider plane curves which are arrangements of lines and conics as well as some algebraic surfaces, where some of the former occur as local configurations in degenerated and regenerated surfaces in the latter. In particular we focus on degenerations which involve intersection points of multiplicity two and three. We demonstrate when the same links arise even when the local arrangements are different.

math.AG

A correspondence between complexes and knots

In recent work the author investigates perfect matchings of a bipartite graph obtained from a knot diagram and demonstrates that these correspond to discrete Morse functions on a 2-complex for the 2-sphere. This relationship is expounded below for the opposite audience: those who may be unfamiliar with knots.

math.GT

Kauffman's clock lattice as a graph of perfect matchings: a formula for its height

We give an algorithmic computation for the height of Kauffman's clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by overlaying the two dual Tait graphs of the knot diagram. This setting also makes evident applications to Chebyshev or harmonic knots, whose related bipartite graph is the popular grid graph, and to discrete Morse functions. Furthermore we prove structural properties of the bipartite graph in general.

math.GT