We define the Wirtinger number of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a link equals its bridge number. This equality can be viewed as establishing a weak version of Cappell and Shaneson's Meridional Rank Conjecture, and suggests a new approach to this conjecture. Our result also leads to a combinatorial technique for obtaining strong upper bounds on bridge numbers. This technique has so far allowed us to add the bridge numbers of approximately 50,000 prime knots of up to 14 crossings to the knot table. As another application, we use the Wirtinger number to show there exists a universal constant C with the property that the hyperbolic volume of a prime alternating link L is bounded below by C times the bridge number of L.
Amplification-based gene characterization allows for sensitive and specific quantification of functional genes. There is often a large diversity of genes represented for functional gene groups, and multiple primers may be necessary to target associated genes.
ABSTRACTHigh throughput primer design is needed to simultaneously design primers for multiple genes of interest, such as a group of functional genes. We have developed MetaFunPrimer, a bioinformatic pipeline to design primer targets for genes of interests, with a prioritization based on ranking the presence of gene targets in references, such as metagenomes. MetaFunPrimer takes inputs of protein and nucleotide sequences for gene targets of interest accompanied by a set of reference metagenomes or genomes for determining genes of interest. Its output is a set of primers that may be used to amplify genes of interest. To demonstrate the usage and benefits of MetaFunPrimer, a total of 78 HT-qPCR primer pairs were designed to target observed ammonia monooxygenase subunit A (amoA) genes of ammonia-oxidizing bacteria (AOB) in 1,550 soil metagenomes. We demonstrate that these primers can significantly improve targeting of amoA-AOB genes in soil metagenomes compared to previously published primers.IMPORTANCEAmplification-based gene characterization allows for sensitive and specific quantification of functional genes. Often, there is a large diversity of genes represented for a function of interest, and multiple primers may be necessary to target associated genes. Current primer design tools are limited to designing primers for only a few genes of interest. MetaFunPrimer allows for high throughput primer design for functional genes of interest and also allows for ranking gene targets by their presence and abundance in environmental datasets. This tool enables high throughput qPCR approaches for characterizing functional genes.
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