{"title":"A quantitative Birman–Menasco finiteness theorem and its application to crossing number","authors":"Tetsuya Ito","doi":"10.1112/topo.12259","DOIUrl":"10.1112/topo.12259","url":null,"abstract":"<p>Birman–Menasco proved that there are finitely many knots having a given genus and braid index. We give a quantitative version of the Birman–Menasco finiteness theorem, an estimate of the crossing number of knots in terms of genus and braid index. As applications, we give a solution of the braid index problem, the problem to determine the braid index of a given link, and provide estimates of the crossing number of connected sums or satellites.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44059372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"R\u0000 $mathbb {R}$\u0000 -motivic stable stems","authors":"Eva Belmont, Daniel C. Isaksen","doi":"10.1112/topo.12256","DOIUrl":"10.1112/topo.12256","url":null,"abstract":"<p>We compute some <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math>-motivic stable homotopy groups. For <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mo>−</mo>\u0000 <mi>w</mi>\u0000 <mo>⩽</mo>\u0000 <mn>11</mn>\u0000 </mrow>\u0000 <annotation>$s - w leqslant 11$</annotation>\u0000 </semantics></math>, we describe the motivic stable homotopy groups <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>π</mi>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mo>,</mo>\u0000 <mi>w</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$pi _{s,w}$</annotation>\u0000 </semantics></math> of a completion of the <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math>-motivic sphere spectrum. We apply the <math>\u0000 <semantics>\u0000 <mi>ρ</mi>\u0000 <annotation>$rho$</annotation>\u0000 </semantics></math>-Bockstein spectral sequence to obtain <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math>-motivic <math>\u0000 <semantics>\u0000 <mo>Ext</mo>\u0000 <annotation>$operatorname{Ext}$</annotation>\u0000 </semantics></math> groups from the <math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$mathbb {C}$</annotation>\u0000 </semantics></math>-motivic <math>\u0000 <semantics>\u0000 <mo>Ext</mo>\u0000 <annotation>$operatorname{Ext}$</annotation>\u0000 </semantics></math> groups, which are well understood in a large range. These <math>\u0000 <semantics>\u0000 <mo>Ext</mo>\u0000 <annotation>$operatorname{Ext}$</annotation>\u0000 </semantics></math> groups are the input to the <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math>-motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by <math>\u0000 <semantics>\u0000 <mi>ρ</mi>\u0000 <annotation>$rho$</annotation>\u0000 </semantics></math>, 2, and <math>\u0000 <semantics>\u0000 <mi>η</mi>\u0000 <annotation>$eta$</annotation>\u0000 </semantics></math>. As a consequence of our computations, we recover Mahowald invariants of many low-dimensional classical stable homotopy elements.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43583287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}