Search for: All records

Abstract Let $$T$$ be a satellite knot, link, or spatial graph in a 3-manifold $$M$$ that is either $S^3$ or a lens space. Let $$\b_0$$ and $$\b_1$$ denote genus 0 and genus 1 bridge number, respectively. Suppose that $$T$$ has a companion knot $$K$$ (necessarily not the unknot) and wrapping number $$\omega$$ with respect to $$K$$. When $$K$$ is not a torus knot, we show that $$\b_1(T)\geq \omega \b_1(K)$$. There are previously known counter-examples if $$K$$ is a torus knot. Along the way, we generalize and give a new proof of Schubert's result that $$\b_0(T) \geq \omega \b_0(K)$$. We also prove versions of the theorem applicable to when $$T$$ is a "lensed satellite" and when there is a torus separating components of $$T$$. 

We use thin position of Heegaard splittings to give a new proof of Haken's Lemma that a Heegaard surface of a reducible manifold is reducible and of Scharlemann's ``Strong Haken Theorem'': a Heegaard surface for a 3-manifold may be isotoped to intersect a given collection of essential spheres and discs in a single loop each. We also give a reformulation of Casson and Gordon's theorem on weakly reducible Heegaard splittings, showing that they exhibit additional structure with respect to certain incompressible surfaces. This article could also serve as an introduction to the theory of generalized Heegaard surfaces and it includes a careful study of their behaviour under amalgamation. 

ABSTRACT Hybridization occurs when different species mate and produce offspring. Although hybridization can have negative consequences for cognitive performance, the mechanisms underlying those effects are still poorly understood. A fundamental physiological process found in all animals studied to date that could be disrupted in hybrids is sleep. Given that mechanisms that occur within the brain during sleep may help maintain optimal cognitive performance, here we outline the potential impacts of hybridization on sleep and cognition. We suggest that sleep loss caused by hybridization could lead to negative impacts for neural and molecular mechanisms (e.g. neurogenesis, synaptic plasticity, and brain gene expression) associated with cognition, which may help explain some of the cognitive deficiency recently observed in hybrid birds. However, we acknowledge that these mechanisms may instead be directly impacted by hybridization, which in turn could also disrupt sleep with similar negative consequences for cognition. Limitations in sleep processes apparent in hybrids might influence hybrid fitness and therefore act as a post‐zygotic isolating barrier. 

A bridge trisection of a smooth surface in S4 is a decomposition analogous to a bridge splitting of a link in S3. The Kirby–Thompson invariant of a bridge trisection measures its complexity in terms of distances between disk sets in the pants complex of the trisection surface. We give the first significant bounds for the Kirby–Thompson invariant of spun knots. In particular, we show that the Kirby–Thompson invariant of the spun trefoil is 15. 

In hybrid zones, whether barrier loci experience selection mostly independently or as a unit depends on the ratio of selection to recombination as captured by the coupling coefficient. Theory predicts a sharper transition between an uncoupled and coupled system when more loci affect hybrid fitness. However, the extent of coupling in hybrid zones has rarely been quantified. Here, we use simulations to characterize the relationship between the coupling coefficient and variance in clines across genetic loci. We then re-analyze 25 hybrid zone data sets and find that cline variances and estimated coupling coefficients form a smooth continuum from high variance and weak coupling to low variance and strong coupling. Our results are consistent with low rates of hybridization and a strong genome-wide barrier to gene flow when the coupling coefficient is much greater than 1, but also suggest that this boundary might be approached gradually and at a near constant rate over time. 

Thermoregulatory performance can be modified through changes in various subordinate traits, but the rate and magnitude of change in these traits is poorly understood. We investigated flexibility in traits that affect thermal balance between black-capped chickadees (Poecile atricapillus) acclimated for 6 weeks to cold (−5°C) or control (23°C) environments (n=7 per treatment). We made repeated measurements of basal and summit metabolic rates via flow-through respirometry and of body composition using quantitative magnetic resonance of live birds. At the end of the acclimation period, we measured thermal conductance of the combined feathers and skins. Cold-acclimated birds had a higher summit metabolic rate, reflecting a greater capacity for endogenous heat generation, and an increased lean mass. However, birds did not alter their thermal conductance. These results suggest that chickadees respond to cold stress by increasing their capacity for heat production rather than increasing heat retention, an energetically expensive strategy. 

Abstract We studied hybridization between the Black-crested and Tufted titmouse across two geographically distinct transects that differ in the timing of secondary contact by hundreds to thousands of years. We found that hybridization patterns correspond to localized hybrid swarms and that the titmouse hybrid zone is likely slowly expanding over time, a product of short post-natal dispersal distances coupled with weak or absent selection against admixture. We show the southern part of the hybrid zone located in Texas is four times wider than the northern region of hybridization in Oklahoma, which is likely due to geographic differences in hybrid zone age. Despite differences in width, most individuals in both transects are advanced-generation hybrids and backcrosses, suggesting geographically consistent hybridization dynamics. We documented a strong correlation between genotypes and plumage index, suggesting that hybridization has not yet resulted in the decoupling of plumage and genome-wide ancestry as observed in some other avian hybrid zones. Although our results suggest the ongoing expansion of the hybrid zone, the rate of expansion appears to be slow, on the scale of tens of meters a year, and it will likely take hundreds of thousands to millions of years before homogenization of the parental populations. While we did not find support for partial reproductive isolation in the hybrid zone itself, there is the possibility that ecological or sexual selection limits introgression into allopatric regions. Broadly, the results of our study highlight the value of multiple, geographically distant, transects across a hybrid zone for assessing the evolutionary dynamics of hybridizing lineages.