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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 $$T$$. 

ABSTRACT As graduate students transition into advanced academic environments, the physical and social contexts in which they engage play a critical role in shaping their sense of belonging, academic success, and personal development. Using a qualitative approach, this study explores how an immersive and place‐based fieldwork program impacted community building and self‐efficacy in incoming graduate students in an Ecology and Evolutionary Biology (EEB) program. Data were collected through surveys, focus groups, and in‐depth interviews with students over the program's duration. Our findings reveal that the remote location of the program played an important role in community development and fostered autonomy and competence. We also found that choosing a discipline‐focused location for fieldwork can positively impact student experiences. Opportunities for interdisciplinary collaboration and mentorship emerged as key components of fostering a supportive academic community. The study demonstrates a positive role for place‐based strategies in graduate program design, suggesting that creating spaces that nurture collaboration, allow students to enact disciplinary skills, and present students with formative challenges can enhance academic resilience and self‐confidence. The findings offer implications for institutions looking to cultivate stronger, more cohesive graduate communities and for future research on the intersection of place, identity, and academic success in higher education. 

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. 

Incubation is a crucial part of the avian life cycle; eggs must be kept warm enough for embryos to develop. As egg temperatures drop rapidly when not being actively incubated, birds must balance the thermal requirements of their developing offspring with their own energy requirements when deciding to make a feeding trip. Incubation behavior can vary with ambient temperature, and across the breeding cycle. Here, we examine the incubation behavior of black-capped (Poecile atricapillus) and mountain (P. gambeli) chickadees where they live together in the Front Range of the Colorado Rocky Mountains. We deployed iButton temperature monitors in and around chickadee nests to record nest and ambient temperatures during the incubation period. For each deployment, we visually inspected temperature data to determine whether incubation was detected. For each deployment in which incubation was detected we quantified incubation behavior using the incR R package. Across the four nests for which incubation was detected, females (the incubating sex) spent about 85% of the day incubating, and left the nest about 32 times per day for an average of 6.7 minutes. On average, nest temperatures were maintained around 35˚C. 

We show that if a composite θ-curve has (proper rational) unknotting number one, then it is the order 2 sum of a (proper rational) unknotting number one knot and a trivial θ-curve. We also prove similar results for 2-strand tangles and knotoids. 

A central issues in ecology is the underrepresentation of individuals from diverse backgrounds. Using the Phenomenological Variant Ecological Systems Theory, we present findings from the evaluation of a field-based graduate training program. Three cases describe different students belonging outcomes, providing critical constructive perspectives. 

Abstract Determining the genetic architecture of traits involved in adaptation and speciation is one of the key components of understanding the evolutionary mechanisms behind biological diversification. Hybrid zones provide a unique opportunity to use genetic admixture to identify traits and loci contributing to partial reproductive barriers between taxa. Many studies have focused on the temporal dynamics of hybrid zones, but geographical variation in hybrid zones that span distinct ecological contexts has received less attention. We address this knowledge gap by analyzing hybridization and introgression between black-capped and Carolina chickadees in two geographically remote transects across their extensive hybrid zone, one located in eastern and one in central North America. Previous studies demonstrated that this hybrid zone is moving northward as a result of climate change but is staying consistently narrow due to selection against hybrids. In addition, the hybrid zone is moving ~5× slower in central North America compared to more eastern regions, reflecting continent-wide variation in the rate of climate change. We use whole genome sequencing of 259 individuals to assess whether variation in the rate of hybrid zone movement is reflected in patterns of hybridization and introgression, and which genes and genomic regions show consistently restricted introgression in distinct ecological contexts. Our results highlight substantial similarities between geographically remote transects and reveal large Z-linked chromosomal rearrangements that generate measurable differences in the degree of gene flow between transects. We further use simulations and analyses of climatic data to examine potential factors contributing to continental-scale nuances in selection pressures. We discuss our findings in the context of speciation mechanisms and the importance of sex chromosome inversions in chickadees and other species. 

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.