We use bordered Floer theory to study properties of twisted Mazur pattern satellite knots $Q_{n}(K)$. We prove that $Q_n(K)$ is not Floer homologically thin, with two exceptions. We calculate the 3-genus of $Q_{n}(K)$ in terms of the twisting parameter $n$ and the 3-genus of the companion $K$, and we determine when $Q_n(K)$ is fibered. As an application to our results on Floer thickness and 3-genus, we verify the Cosmetic Surgery Conjecture for many of these satellite knots.
{"title":"Twisted Mazur Pattern Satellite Knots & Bordered Floer Theory","authors":"I. Petkova, Biji Wong","doi":"10.1307/mmj/20205927","DOIUrl":"https://doi.org/10.1307/mmj/20205927","url":null,"abstract":"We use bordered Floer theory to study properties of twisted Mazur pattern satellite knots $Q_{n}(K)$. We prove that $Q_n(K)$ is not Floer homologically thin, with two exceptions. We calculate the 3-genus of $Q_{n}(K)$ in terms of the twisting parameter $n$ and the 3-genus of the companion $K$, and we determine when $Q_n(K)$ is fibered. As an application to our results on Floer thickness and 3-genus, we verify the Cosmetic Surgery Conjecture for many of these satellite knots.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"5 5","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72390086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we survey some mathematical developments that followed from the discovery of simple supercuspidal representations of p-adic groups.
本文综述了p进群的简单超尖表示的一些数学进展。
{"title":"Simple Supercuspidals and the Langlands Correspondence","authors":"B. Gross","doi":"10.1307/mmj/20207202","DOIUrl":"https://doi.org/10.1307/mmj/20207202","url":null,"abstract":"In this paper, we survey some mathematical developments that followed from the discovery of simple supercuspidal representations of p-adic groups.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"33 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79039672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander-Nirenberg theorem under the presence of a $C^2$ strictly pseudoconvex boundary. When a given formally integrable complex structure $X$ is defined on the closure of a bounded strictly pseudoconvex domain with $C^2$ boundary $Dsubset mathbb{C}^n$, we show the existence of global holomorphic coordinate systems defined on $overline{D}$ that transform $X$ into the standard complex structure provided that $X$ is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small $C^2$ perturbation of $partial D$. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a $C^2$ real hypersurface $Msubset mathbb{C}^n$, we prove the existence of local one-sided holomorphic coordinate systems provided that $M$ is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.
{"title":"Global Newlander–Nirenberg Theorem for Domains with C2 Boundary","authors":"Chun Gan, Xianghong Gong","doi":"10.1307/mmj/20216084","DOIUrl":"https://doi.org/10.1307/mmj/20216084","url":null,"abstract":"The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander-Nirenberg theorem under the presence of a $C^2$ strictly pseudoconvex boundary. When a given formally integrable complex structure $X$ is defined on the closure of a bounded strictly pseudoconvex domain with $C^2$ boundary $Dsubset mathbb{C}^n$, we show the existence of global holomorphic coordinate systems defined on $overline{D}$ that transform $X$ into the standard complex structure provided that $X$ is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small $C^2$ perturbation of $partial D$. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a $C^2$ real hypersurface $Msubset mathbb{C}^n$, we prove the existence of local one-sided holomorphic coordinate systems provided that $M$ is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"20 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89764845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Existence of Euclidean Ideal Classes in Real Cubic and Quadratic Fields with Cyclic Class Group","authors":"S. Gun, J. Sivaraman","doi":"10.1307/mmj/1580180457","DOIUrl":"https://doi.org/10.1307/mmj/1580180457","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"14 1","pages":"429-448"},"PeriodicalIF":0.9,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73413743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we collect various structural results to determine when an integral homology $3$--sphere bounds an acyclic smooth $4$--manifold, and when this can be upgraded to a Stein embedding. In a different direction we study whether smooth embedding of connected sums of lens spaces in $mathbb{C}^2$ can be upgraded to a Stein embedding, and determined that this never happens.
{"title":"Homology Spheres Bounding Acyclic Smooth Manifolds and Symplectic Fillings","authors":"John B. Etnyre, B. Tosun","doi":"10.1307/mmj/20206003","DOIUrl":"https://doi.org/10.1307/mmj/20206003","url":null,"abstract":"In this paper, we collect various structural results to determine when an integral homology $3$--sphere bounds an acyclic smooth $4$--manifold, and when this can be upgraded to a Stein embedding. In a different direction we study whether smooth embedding of connected sums of lens spaces in $mathbb{C}^2$ can be upgraded to a Stein embedding, and determined that this never happens.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"21 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87225725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Following Haken and Casson-Gordon, it was shown in [Sc] that given a reducing sphere or boundary-reducing disk S in a Heegaard split manifold M in which every sphere separates, the Heegaard surface T can be isotoped so that it intersects S in a single circle. Here we show that when this is achieved by two different positionings of T, one can be moved to the other by a sequence of 1) isotopies of T rel S 2) pushing a stabilizing pair of T through S and 3) eyegelass twists of T. The last move is inspired by one of Powell's proposed generators for the Goeritz group.
{"title":"Uniqueness in Haken’s Theorem","authors":"M. Freedman, M. Scharlemann","doi":"10.1307/mmj/20216081","DOIUrl":"https://doi.org/10.1307/mmj/20216081","url":null,"abstract":"Following Haken and Casson-Gordon, it was shown in [Sc] that given a reducing sphere or boundary-reducing disk S in a Heegaard split manifold M in which every sphere separates, the Heegaard surface T can be isotoped so that it intersects S in a single circle. Here we show that when this is achieved by two different positionings of T, one can be moved to the other by a sequence of 1) isotopies of T rel S 2) pushing a stabilizing pair of T through S and 3) eyegelass twists of T. The last move is inspired by one of Powell's proposed generators for the Goeritz group.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"137 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79738803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Every $L$-space knot is fibered and strongly quasi-positive, but this does not hold for $L$-space links. In this paper, we use the so called H-function, which is a concordance link invariant, to introduce a subfamily of fibered strongly quasi-positive $L$-space links. Furthermore, we present an infinite family of $L$-space links which are not quasi-positive.
{"title":"Fibered and Strongly Quasi-Positive L-Space Links","authors":"A. Cavallo, Beibei Liu","doi":"10.1307/mmj/20205947","DOIUrl":"https://doi.org/10.1307/mmj/20205947","url":null,"abstract":"Every $L$-space knot is fibered and strongly quasi-positive, but this does not hold for $L$-space links. In this paper, we use the so called H-function, which is a concordance link invariant, to introduce a subfamily of fibered strongly quasi-positive $L$-space links. Furthermore, we present an infinite family of $L$-space links which are not quasi-positive.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"171 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72527856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a classical link, Milnor defined a family of isotopy invariants, called Milnor $overline{mu}$-invariants. Recently, Chrisman extended Milnor $overline{mu}$-invariants to welded links by a topological approach. The aim of this paper is to show that Milnor $overline{mu}$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof for the invariance of the original $overline{mu}$-invariants of classical links.
{"title":"Combinatorial Approach to Milnor Invariants of Welded Links","authors":"H. A. Miyazawa, K. Wada, A. Yasuhara","doi":"10.1307/mmj/20205905","DOIUrl":"https://doi.org/10.1307/mmj/20205905","url":null,"abstract":"For a classical link, Milnor defined a family of isotopy invariants, called Milnor $overline{mu}$-invariants. Recently, Chrisman extended Milnor $overline{mu}$-invariants to welded links by a topological approach. The aim of this paper is to show that Milnor $overline{mu}$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof for the invariance of the original $overline{mu}$-invariants of classical links.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86675464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety which is a toric variety. We give a complete classification of Fano and weak Fano regular semisimple Hessenberg varieties in type A in terms of combinatorics of Hessenberg functions. In particular, we show that if the anti-canonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.
{"title":"Fano and Weak Fano Hessenberg Varieties","authors":"Hiraku Abe, Naoki Fujita, Haozhi Zeng","doi":"10.1307/mmj/20205971","DOIUrl":"https://doi.org/10.1307/mmj/20205971","url":null,"abstract":"Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety which is a toric variety. We give a complete classification of Fano and weak Fano regular semisimple Hessenberg varieties in type A in terms of combinatorics of Hessenberg functions. In particular, we show that if the anti-canonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"31 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72792351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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