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Abstract MotivationSampling k-mers is a ubiquitous task in sequence analysis algorithms. Sampling schemes such as the often-used random minimizer scheme are particularly appealing as they guarantee at least one k-mer is selected out of every w consecutive k-mers. Sampling fewer k-mers often leads to an increase in efficiency of downstream methods. Thus, developing schemes that have low density, i.e. have a small proportion of sampled k-mers, is an active area of research. After over a decade of consistent efforts in both decreasing the density of practical schemes and increasing the lower bound on the best possible density, there is still a large gap between the two. ResultsWe prove a near-tight lower bound on the density of forward sampling schemes, a class of schemes that generalizes minimizer schemes. For small w and k, we observe that our bound is tight when k≡1(mod w). For large w and k, the bound can be approximated by 1w+k⌈w+kw⌉. Importantly, our lower bound implies that existing schemes are much closer to achieving optimal density than previously known. For example, with the current default minimap2 HiFi settings w = 19 and k = 19, we show that the best known scheme for these parameters, the double decycling-set-based minimizer of Pellow et al. is at most 3% denser than optimal, compared to the previous gap of at most 50%. Furthermore, when k≡1(mod w) and the alphabet size σ goes to ∞, we show that mod-minimizers introduced by Groot Koerkamp and Pibiri achieve optimal density matching our lower bound. Availability and implementationMinimizer implementations: github.com/RagnarGrootKoerkamp/minimizers ILP and analysis: github.com/treangenlab/sampling-scheme-analysis.
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Gompf’s Cork and Heegaard Floer Homology
Abstract Gompf showed that for $$K$$ in a certain family of double-twist knots, the swallow-follow operation makes $1/n$-surgery on $$K \# -K$$ into a cork boundary. We derive a general Floer-theoretic condition on $$K$$ under which this is the case. Our formalism allows us to produce many further examples of corks, partially answering a question of Gompf. Unlike Gompf’s method, our proof does not rely on any closed 4-manifold invariants or effective embeddings, and also generalizes to other diffeomorphisms.
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- PAR ID:
- 10596138
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 18
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 12663 to 12682
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnae180
