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Abstract We unify two existing approaches to thetauinvariants in instanton and monopole Floer theories, by identifying , defined by the second author via theminusflavors and of the knot homologies, with , defined by Baldwin and Sivek via cobordism maps of the 3‐manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute and for twist knots. 

In this paper generalizes the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handle-bodies. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for monopole and instanton knot Floer homologies, which were introduced by Kronheimer and Mrowka. The same result was originally proved by Ozsváth and Szabó for link Floer homology. Last, we generalize the construction of minus versions of monopole and instanton knot Floer homology, which was initially done for knots by the second author, to the case of links. Along with the construction of polytopes, we also proved that, for a balanced sutured manifold with vanishing second homology, the rank of the sutured monopole or instanton Floer homology bounds the depth of the balanced sutured manifold. As a corollary, we obtain an independent proof that monopole and instanton knot Floer homologies, as mentioned above, both detect fibred knots in S3. This result was originally achieved by Kronheimer and Mrowka. 

Highland native Andeans have resided at altitude for millennia. They display high aerobic capacity (VO 2 max) at altitude, which may be a reflection of genetic adaptation to hypoxia. Previous genomewide (GW) scans for natural selection have nominated Egl-9 homolog 1 gene ( EGLN1 ) as a candidate gene. The encoded protein, EGLN1/PHD2, is an O 2 sensor that controls levels of the Hypoxia Inducible Factor-α (HIF-α), which regulates the cellular response to hypoxia. From GW association and analysis of covariance performed on a total sample of 429 Peruvian Quechua and 94 US lowland referents, we identified 5 EGLN1 SNPs associated with higher VO 2 max (L⋅min −1 and mL⋅min −1 ⋅kg −1 ) in hypoxia (rs1769793, rs2064766, rs2437150, rs2491403, rs479200). For 4 of these SNPs, Quechua had the highest frequency of the advantageous (high VO 2 max) allele compared with 25 diverse lowland comparison populations from the 1000 Genomes Project. Genotype effects were substantial, with high versus low VO 2 max genotype categories differing by ∼11% (e.g., for rs1769793 SNP genotype TT = 34.2 mL⋅min −1 ⋅kg −1 vs. CC = 30.5 mL⋅min −1 ⋅kg −1 ). To guard against spurious association, we controlled for population stratification. Findings were replicated for EGLN1 SNP rs1769793 in an independent Andean sample collected in 2002. These findings contextualize previous reports of natural selection at EGLN1 in Andeans, and support the hypothesis that natural selection has increased the frequency of an EGLN1 causal variant that enhances O 2 delivery or use during exercise at altitude in Peruvian Quechua.