An official website of the United States government
Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant singular instanton theory, and which is closely related to the Chern-Simons functional. This also answers a conjecture of Livingston about slicing numbers. Also studied is the singular instanton Frøyshov invariant of a knot. If defined with integer coefficients, this gives a lower bound for the unoriented slice genus, and is computed for quasi-alternating and torus knots. In contrast, for certain other coefficient rings, the invariant is identified with a multiple of the knot signature. This result is used to address a conjecture by Poudel and Saveliev about traceless SU(2) representations of torus knots. Further, for a concordance between knots with non-zero signature, it is shown that there is a traceless representation of the concordance complement which restricts to non-trivial representations of the knot groups. Finally, some evidence towards an extension of the slice-ribbon conjecture to torus knots is provided.
more »
« less
This content will become publicly available on June 3, 2025
Dynamics of Double-Knotted DNA Molecules under Nanochannel Confinement
Langevin dynamics simulations of double-knotted DNA molecules in a nanochannel reveal that the interactions between the two knots differ with the degree of channel confinement. In relatively wide channels, the two knots can intertwine with each other, forming a persistently intertwined knot. Moreover, the two knots can pass through each other in large channels. In contrast, for small channel sizes, the knots tend to remain separated, and their crossing is inhibited. The change in knot–knot interactions as the channel size decreases is rationalized through an analysis of the magnitude of the transverse fluctuations, which must be large enough to allow one knot to swell to accommodate the intertwined state.
more »
« less
- Award ID(s):
- 2016879
- PAR ID:
- 10512810
- Publisher / Repository:
- ACS
- Date Published:
- Journal Name:
- Macromolecules
- Volume:
- 57
- Issue:
- 11
- ISSN:
- 0024-9297
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into lR2, such that no more than two points project to the same point in lR2. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in lR3, so their projections should be smooth curves in lR2 with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defned by circular-arc edges and perfect angular resolution). We show that several knots do not allow crossing-minimal plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have plane Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a crossing-minimal plane 2-Lombardi drawing (where edges are composed of two circular arcs). Further, every knot is near-Lombardi, that is, it can be drawn as a plane Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset ε, while maintaining a 180◦ angle between opposite edges.more » « less
-
Paiva, Vitor Hugo (Ed.)Understanding factors that influence a species’ distribution and abundance across the annual cycle is required for range-wide conservation. Thousands of imperiled red knots ( Calidris cantus rufa ) stop on Virginia’s barrier islands each year to replenish fat during spring migration. We investigated the variation in red knot presence and flock size, the effects of prey on this variation, and factors influencing prey abundance on Virginia’s barrier islands. We counted red knots and collected potential prey samples at randomly selected sites from 2007–2018 during a two-week period during early and peak migration. Core samples contained crustaceans (Orders Amphipoda and Calanoida), blue mussels ( Mytilus edulis) , coquina clams ( Donax variabilis ), and miscellaneous prey (horseshoe crab eggs ( Limulus polyphemus ), angel wing clams ( Cyrtopleura costata ), and other organisms (e.g., insect larvae, snails, worms)). Estimated red knot peak counts in Virginia during 21–27 May were highest in 2012 (11,959) and lowest in 2014 (2,857; 12-year peak migration x ¯ = 7,175, SD = 2,869). Red knot and prey numbers varied across sampling periods and substrates (i.e., peat and sand). Red knots generally used sites with more prey. Miscellaneous prey ( x ¯ = 2401.00/m 2 , SE = 169.16) influenced red knot presence at a site early in migration, when we only sampled on peat banks. Coquina clams ( x ¯ = 1383.54/m 2 , SE = 125.32) and blue mussels ( x ¯ = 777.91/m 2 , SE = 259.31) affected red knot presence at a site during peak migration, when we sampled both substrates. Few relationships between prey and red knot flock size existed, suggesting that other unmeasured factors determined red knot numbers at occupied sites. Tide and mean daily water temperature affected prey abundance. Maximizing the diversity, availability, and abundance of prey for red knots on barrier islands requires management that encourages the presence of both sand and peat bank intertidal habitats.more » « less
-
Yong, Xin (Ed.)Knots in proteins and DNA are known to have significant effect on their equilibrium and dynamic properties as well as on their function. While knot dynamics and thermodynamics in electrically neutral and uniformly charged polymer chains are relatively well understood, proteins are generally polyampholytes, with varied charge distributions along their backbones. Here we use simulations of knotted polymer chains to show that variation in the charge distribution on a polyampholyte chain with zero net charge leads to significant variation in the resulting knot dynamics, with some charge distributions resulting in long-lived metastable knots that escape the (open-ended) chain on a timescale that is much longer than that for knots in electrically neutral chains. The knot dynamics in such systems can be described, quantitatively, using a simple one-dimensional model where the knot undergoes biased Brownian motion along a “reaction coordinate”, equal to the knot size, in the presence of a potential of mean force. In this picture, long-lived knots result from charge sequences that create large electrostatic barriers to knot escape. This model allows us to predict knot lifetimes even when those times are not directly accessible by simulations.more » « less
-
In the past decade, topological data analysis has emerged as a powerful algebraic topology approach in data science. Although knot theory and related subjects are a focus of study in mathematics, their success in practical applications is quite limited due to the lack of localization and quantization. We address these challenges by introducing knot data analysis (KDA), a paradigm that incorporates curve segmentation and multiscale analysis into the Gauss link integral. The resulting multiscale Gauss link integral (mGLI) recovers the global topological properties of knots and links at an appropriate scale and offers a multiscale geometric topology approach to capture the local structures and connectivities in data. By integration with machine learning or deep learning, the proposed mGLI significantly outperforms other state-of-the-art methods across various benchmark problems in 13 intricately complex biological datasets, including protein flexibility analysis, protein–ligand interactions, human Ether-à-go-go-Related Gene potassium channel blockade screening, and quantitative toxicity assessment. Our KDA opens a research area—knot deep learning—in data science.more » « less
