Pub Date : 2019-07-27DOI: 10.2140/PJM.2021.310.355
Daniel Kasprowski, P. Teichner
We show that two closed, connected $4$-manifolds with finite fundamental groups are $mathbb{CP}^2$-stably homeomorphic if and only if their quadratic $2$-types are stably isomorphic and their Kirby-Siebenmann invariant agrees.
{"title":"ℂℙ2-stable classification of 4-manifolds with\u0000finite fundamental group","authors":"Daniel Kasprowski, P. Teichner","doi":"10.2140/PJM.2021.310.355","DOIUrl":"https://doi.org/10.2140/PJM.2021.310.355","url":null,"abstract":"We show that two closed, connected $4$-manifolds with finite fundamental groups are $mathbb{CP}^2$-stably homeomorphic if and only if their quadratic $2$-types are stably isomorphic and their Kirby-Siebenmann invariant agrees.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82337953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic and, for such knots, give an answer to a problem posed by Fiedler.
{"title":"Knots with Hopf crossing number at most one","authors":"M. Mroczkowski","doi":"10.18910/75915","DOIUrl":"https://doi.org/10.18910/75915","url":null,"abstract":"We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic and, for such knots, give an answer to a problem posed by Fiedler.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81532770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-07-15DOI: 10.1142/s0218216520400015
A. A. Akimova, V. Manturov
In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial, the Kuperberg bracket, and the normalised arrow polynomial.
{"title":"Labels instead of coefficients: A label bracket ⋅ℒ which dominates the Jones polynomial 𝒳 ⋅, the Kuperberg bracket ⋅A2, and the normalized arrow polynomial 𝒲⋅","authors":"A. A. Akimova, V. Manturov","doi":"10.1142/s0218216520400015","DOIUrl":"https://doi.org/10.1142/s0218216520400015","url":null,"abstract":"In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial, the Kuperberg bracket, and the normalised arrow polynomial.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72823846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bott, Cattaneo and Rossi defined invariants of long knots $mathbb R^n hookrightarrow mathbb R^{n+2}$ as combinations of configuration space integrals. Here, we give a more flexible definition of these invariants. Our definition allows us to interpret these invariants as counts of diagrams. It extends to long knots inside more general $(n+2)$-manifolds, called parallelized asymptotic homology $mathbb R^{n+2}$, and provides invariants of these knots.
{"title":"Generalized Bott–Cattaneo–Rossi invariants of high-dimensional long knots","authors":"David Leturcq","doi":"10.2969/JMSJ/82908290","DOIUrl":"https://doi.org/10.2969/JMSJ/82908290","url":null,"abstract":"Bott, Cattaneo and Rossi defined invariants of long knots $mathbb R^n hookrightarrow mathbb R^{n+2}$ as combinations of configuration space integrals. Here, we give a more flexible definition of these invariants. Our definition allows us to interpret these invariants as counts of diagrams. It extends to long knots inside more general $(n+2)$-manifolds, called parallelized asymptotic homology $mathbb R^{n+2}$, and provides invariants of these knots.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85026123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds. Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichmuller to curve graph translation length are, in a reasonable sense, ubiquitous in the connected components of strata of Abelian differentials. Finally, we present a further application to filling pairs on punctured surfaces by constructing filling pairs whose algebraic and geometric intersection numbers are equal.
{"title":"Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs","authors":"Luke Jeffreys","doi":"10.1090/TRAN/8374","DOIUrl":"https://doi.org/10.1090/TRAN/8374","url":null,"abstract":"In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds. \u0000Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichmuller to curve graph translation length are, in a reasonable sense, ubiquitous in the connected components of strata of Abelian differentials. Finally, we present a further application to filling pairs on punctured surfaces by constructing filling pairs whose algebraic and geometric intersection numbers are equal.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87829333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-31DOI: 10.1016/J.TOPOL.2021.107753
Antonio F. Costa, Cam Van Quach-Hongler
{"title":"Periodic projections of alternating knots","authors":"Antonio F. Costa, Cam Van Quach-Hongler","doi":"10.1016/J.TOPOL.2021.107753","DOIUrl":"https://doi.org/10.1016/J.TOPOL.2021.107753","url":null,"abstract":"","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84158350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-01DOI: 10.1142/s0219498821501164
M. Elhamdadi, M. Saito, E. Zappala
We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive $n$-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing $2$-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given.
{"title":"Higher arity self-distributive operations in Cascades and their cohomology","authors":"M. Elhamdadi, M. Saito, E. Zappala","doi":"10.1142/s0219498821501164","DOIUrl":"https://doi.org/10.1142/s0219498821501164","url":null,"abstract":"We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive $n$-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing $2$-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"145 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86211391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-16DOI: 10.1142/s0218216520710029
K. Varvarezos
We show that Dehn filling on the manifold $v2503$ results in a non-orderable space for all rational slopes in the interval $(-infty , -1)$. This is consistent with the L-space conjecture, which predicts that all fillings will result in a non-orderable space for this manifold.
{"title":"A note on the orderability of Dehn fillings of the manifold v2503","authors":"K. Varvarezos","doi":"10.1142/s0218216520710029","DOIUrl":"https://doi.org/10.1142/s0218216520710029","url":null,"abstract":"We show that Dehn filling on the manifold $v2503$ results in a non-orderable space for all rational slopes in the interval $(-infty , -1)$. This is consistent with the L-space conjecture, which predicts that all fillings will result in a non-orderable space for this manifold.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78213845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for every $ggeq 2$ there exists a number $c=c(g)>0$ such that the smallest positive eigenvalue of a random closed 3-manifold $M$ of Heegaard genus $g$ is at most $c(g)/{rm vol}(M)^2$.
{"title":"Small eigenvalues of random 3-manifolds","authors":"U. Hamenstaedt, Gabriele Viaggi","doi":"10.1090/tran/8564","DOIUrl":"https://doi.org/10.1090/tran/8564","url":null,"abstract":"We show that for every $ggeq 2$ there exists a number $c=c(g)>0$ such that the smallest positive eigenvalue of a random closed 3-manifold $M$ of Heegaard genus $g$ is at most $c(g)/{rm vol}(M)^2$.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"93 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80261510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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