We introduce a method to detect exotic surfaces without explicitly using a smooth 4-manifold invariant or an invariant of a 4-manifold-surface pair in the construction. Our main tools are two versions of families (Seiberg-Witten) generalizations of Donaldson's diagonalization theorem, including a real and families version of the diagonalization. This leads to an example of a pair of exotically knotted $mathbb{R}P^2$'s embedded in a closed 4-manifold whose complements are diffeomorphic, making it the first example of a non-orientable surface with this property. In particular, any invariant of a 4-manifold-surface pair (including invariants from real Seiberg-Witten theory such as Miyazawa's invariant) fails to detect such an exotic $mathbb{R} P^2$. One consequence of our construction reveals that non-effective embeddings of corks can still be useful in pursuit of exotica. Precisely, starting with an embedding of a cork $C$ in certain a 4-manifold $X$ where the cork-twist does not change the diffeomorphism type of $X$, we give a construction that provides examples of exotically knotted spheres and $mathbb{R}P^2$'s with diffeomorphic complements in $ C # S^2 times S^2 subset X # S^2 times S^2$ or $C # mathbb{C}P^2 subset X # mathbb{C}P^2 $. In another direction, we provide infinitely many exotically knotted embeddings of orientable surfaces, closed surface links, and 3-spheres with diffeomorphic complements in once stabilized corks, and show some of these surfaces survive arbitrarily many internal stabilizations. By combining similar methods with Gabai's 4D light-bulb theorem, we also exhibit arbitrarily large difference between algebraic and geometric intersections of certain family of 2-spheres, embedded in a 4-manifold.
我们介绍了一种检测奇异曲面的方法，而无需在构造中明确使用光滑四芒星不变量或四芒星曲面对的不变量。我们的主要工具是唐纳森对角线化定理的两个族（塞伯格-维滕）广义版本，包括对角线化的实数和族版本。这引出了一个例子：一对外结$mathbb{R}P^2$嵌入到一个封闭的4-manifold中，其复数是差分同构的，这使它成为具有这一性质的非可取向曲面的第一个例子。特别是，4-manifold-曲面对的任何不变式（包括宫泽不变式等来自实塞伯格-维滕理论的不变式）都无法检测到这样一个奇异的$/mathbb{R} P^2$.我们的构造的一个结果揭示出，在追求奇异性时，叉形的非有效嵌入仍然是有用的。确切地说，从软木塞$C$在某个4-manifold $X$中的嵌入开始，软木塞扭转并不改变$X$的衍射类型、我们给出了一种构造，它提供了在 $ C # S^2 times S^2 subset X # S^2 times S^2$ 或 $ C #mathbb{C}P^2 subset X # mathbb{C}P^2$ 中具有差分同构复数的外结球体和 $mathbb{R}P^2$ 的例子。在另一个方向上，我们提供了无限多的可定向曲面、闭合曲面链接、3-球体的外结嵌入，这些嵌入在一次稳定叉中具有差分补集，并证明了其中一些曲面在任意多的内部稳定化中存活下来。通过将类似的方法与加拜的四维光球定理相结合，我们还展示了嵌入四曲面的某些二球体族的代数交集与几何交集之间的任意大差异。
{"title":"Exotically knotted closed surfaces from Donaldson's diagonalization for families","authors":"Hokuto Konno, Abhishek Mallick, Masaki Taniguchi","doi":"arxiv-2409.07287","DOIUrl":"https://doi.org/arxiv-2409.07287","url":null,"abstract":"We introduce a method to detect exotic surfaces without explicitly using a\u0000smooth 4-manifold invariant or an invariant of a 4-manifold-surface pair in the\u0000construction. Our main tools are two versions of families (Seiberg-Witten)\u0000generalizations of Donaldson's diagonalization theorem, including a real and\u0000families version of the diagonalization. This leads to an example of a pair of\u0000exotically knotted $mathbb{R}P^2$'s embedded in a closed 4-manifold whose\u0000complements are diffeomorphic, making it the first example of a non-orientable\u0000surface with this property. In particular, any invariant of a\u00004-manifold-surface pair (including invariants from real Seiberg-Witten theory\u0000such as Miyazawa's invariant) fails to detect such an exotic $mathbb{R} P^2$.\u0000One consequence of our construction reveals that non-effective embeddings of\u0000corks can still be useful in pursuit of exotica. Precisely, starting with an\u0000embedding of a cork $C$ in certain a 4-manifold $X$ where the cork-twist does\u0000not change the diffeomorphism type of $X$, we give a construction that provides\u0000examples of exotically knotted spheres and $mathbb{R}P^2$'s with diffeomorphic\u0000complements in $ C # S^2 times S^2 subset X # S^2 times S^2$ or $C #\u0000mathbb{C}P^2 subset X # mathbb{C}P^2 $. In another direction, we provide\u0000infinitely many exotically knotted embeddings of orientable surfaces, closed\u0000surface links, and 3-spheres with diffeomorphic complements in once stabilized\u0000corks, and show some of these surfaces survive arbitrarily many internal\u0000stabilizations. By combining similar methods with Gabai's 4D light-bulb\u0000theorem, we also exhibit arbitrarily large difference between algebraic and\u0000geometric intersections of certain family of 2-spheres, embedded in a\u00004-manifold.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192032","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}
The mapping class group ${Gamma}_g^ 1$ of a closed orientable surface of genus $g geq 1$ with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation preserving homeomorphims of the circle. This inclusion pulls back the powers of the discrete universal Euler class producing classes $text{E}^n in H^{2n}({Gamma}_g^1;mathbb{Z})$ for all $ngeq 1$. In this paper we study the power $n=g,$ and prove: $text{E}^g$ is a torsion class which generates a cyclic subgroup of $H^{2g}({Gamma}_g^1;mathbb{Z})$ whose order is a positive integer multiple of $4g(2g+1)(2g-1)$.
{"title":"Torsion at the Threshold for Mapping Class Groups","authors":"Solomon Jekel, Rita Jiménez Rolland","doi":"arxiv-2409.07311","DOIUrl":"https://doi.org/arxiv-2409.07311","url":null,"abstract":"The mapping class group ${Gamma}_g^ 1$ of a closed orientable surface of\u0000genus $g geq 1$ with one marked point can be identified, by the Nielsen\u0000action, with a subgroup of the group of orientation preserving homeomorphims of\u0000the circle. This inclusion pulls back the powers of the discrete universal\u0000Euler class producing classes $text{E}^n in H^{2n}({Gamma}_g^1;mathbb{Z})$\u0000for all $ngeq 1$. In this paper we study the power $n=g,$ and prove:\u0000$text{E}^g$ is a torsion class which generates a cyclic subgroup of\u0000$H^{2g}({Gamma}_g^1;mathbb{Z})$ whose order is a positive integer multiple of\u0000$4g(2g+1)(2g-1)$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192030","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}
This paper discusses a generalization of virtual knot theory that we call multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way it arises first in a graph-theoretic setting in relation to generalizing the Penrose evaluation for colorings of planar trivalent graphs to all trivalent graphs, and later by its uses in a virtual knot theory. As a consequence, the paper begins with the graph theory as a basis for our constructions, and then proceeds to the topology of multi-virtual knots and links. The second section of the paper is a review of our previous work (See arXiv:1511.06844). The reader interested in seeing our generalizations of the original Penrose evaluation, can begin this paper at the beginning and see the graph theory. A reader primarily interested in multi-virtual knots and links can begin reading in section 4 with references to the earlier part of the paper.
{"title":"Multi-Virtual Knot Theory","authors":"Louis H Kauffman","doi":"arxiv-2409.07499","DOIUrl":"https://doi.org/arxiv-2409.07499","url":null,"abstract":"This paper discusses a generalization of virtual knot theory that we call\u0000multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of\u0000types of virtual crossings. As we will explain, this multiplicity is motivated\u0000by the way it arises first in a graph-theoretic setting in relation to\u0000generalizing the Penrose evaluation for colorings of planar trivalent graphs to\u0000all trivalent graphs, and later by its uses in a virtual knot theory. As a\u0000consequence, the paper begins with the graph theory as a basis for our\u0000constructions, and then proceeds to the topology of multi-virtual knots and\u0000links. The second section of the paper is a review of our previous work (See\u0000arXiv:1511.06844). The reader interested in seeing our generalizations of the\u0000original Penrose evaluation, can begin this paper at the beginning and see the\u0000graph theory. A reader primarily interested in multi-virtual knots and links\u0000can begin reading in section 4 with references to the earlier part of the\u0000paper.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224771","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 this note we prove that $H^*(text{BSO}(4);mathbb{Q})$ injects into the group cohomology of $text{Diff}^+(S^{3})$ with rational coefficients. The proof is based on an idea of Nariman who proved that the monomials in the Euler and Pontrjagin classes are nontrivial in $H^*(text{BDiff}_+^{delta}(S^{2n-1});mathbb{Q})$.
{"title":"A note on invariants of foliated 3-sphere bundles","authors":"Nils Prigge","doi":"arxiv-2409.06408","DOIUrl":"https://doi.org/arxiv-2409.06408","url":null,"abstract":"In this note we prove that $H^*(text{BSO}(4);mathbb{Q})$ injects into the\u0000group cohomology of $text{Diff}^+(S^{3})$ with rational coefficients. The\u0000proof is based on an idea of Nariman who proved that the monomials in the Euler\u0000and Pontrjagin classes are nontrivial in\u0000$H^*(text{BDiff}_+^{delta}(S^{2n-1});mathbb{Q})$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192018","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}
These are expository notes in which we explain how one can see some exceptional surgeries connecting the suspension of the cat-bat map and the unit tangent bundles to some hyperbolic orbispheres.
{"title":"The cat-bat map, the figure-eight knot, and the five orbifolds","authors":"Pierre Dehornoy","doi":"arxiv-2409.06543","DOIUrl":"https://doi.org/arxiv-2409.06543","url":null,"abstract":"These are expository notes in which we explain how one can see some\u0000exceptional surgeries connecting the suspension of the cat-bat map and the unit\u0000tangent bundles to some hyperbolic orbispheres.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224774","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, in the unit tangent bundle of a hyperbolic orbisphere with cone points of order 3, 3, 4, the lift of the shortest periodic geodesic is homeomorphic to the complement of the figure-eight knot in the 3-sphere. The proof eventually relies on the computation of some linking numbers.
{"title":"La courbe en huit sur les sphères à pointes et le noeud de huit","authors":"Pierre Dehornoy","doi":"arxiv-2409.06532","DOIUrl":"https://doi.org/arxiv-2409.06532","url":null,"abstract":"We show that, in the unit tangent bundle of a hyperbolic orbisphere with cone\u0000points of order 3, 3, 4, the lift of the shortest periodic geodesic is\u0000homeomorphic to the complement of the figure-eight knot in the 3-sphere. The\u0000proof eventually relies on the computation of some linking numbers.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192017","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}
David Aulicino, Aaron Calderon, Carlos Matheus, Nick Salter, Martin Schmoll
We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulas depend only on topological invariants of the cover and number-theoretic properties of the degree: in particular, the ratio of the related Siegel-Veech constants for the locus of covers and for the base stratum component is independent of the number of branch values. One surprising corollary is that this ratio for $area^3$ Siegel-Veech constants is always equal to the reciprocal of the the degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.
{"title":"Siegel-Veech Constants for Cyclic Covers of Generic Translation Surfaces","authors":"David Aulicino, Aaron Calderon, Carlos Matheus, Nick Salter, Martin Schmoll","doi":"arxiv-2409.06600","DOIUrl":"https://doi.org/arxiv-2409.06600","url":null,"abstract":"We compute the asymptotic number of cylinders, weighted by their area to any\u0000non-negative power, on any cyclic branched cover of any generic translation\u0000surface in any stratum. Our formulas depend only on topological invariants of\u0000the cover and number-theoretic properties of the degree: in particular, the\u0000ratio of the related Siegel-Veech constants for the locus of covers and for the\u0000base stratum component is independent of the number of branch values. One\u0000surprising corollary is that this ratio for $area^3$ Siegel-Veech constants is\u0000always equal to the reciprocal of the the degree of the cover. A key ingredient\u0000is a classification of the connected components of certain loci of cyclic\u0000branched covers.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192029","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 this short note, we give examples of binding sums of contact 3-manifolds that do not preserve properties such as tightness or symplectic fillability. We also prove vanishing of the Heegaard Floer contact invariant for an infinite family of binding sums where the summands are Stein fillable. This recovers a result of Wendl and Latschev-Wendl. Along the way, we rectify a subtle computational error in a paper of Juhasz-Kang concerning the spectral order of a neighbourhood of a Giroux torsion domain.
{"title":"On binding sums of contact manifolds","authors":"Miguel Orbegozo Rodriguez","doi":"arxiv-2409.05612","DOIUrl":"https://doi.org/arxiv-2409.05612","url":null,"abstract":"In this short note, we give examples of binding sums of contact 3-manifolds\u0000that do not preserve properties such as tightness or symplectic fillability. We\u0000also prove vanishing of the Heegaard Floer contact invariant for an infinite\u0000family of binding sums where the summands are Stein fillable. This recovers a\u0000result of Wendl and Latschev-Wendl. Along the way, we rectify a subtle\u0000computational error in a paper of Juhasz-Kang concerning the spectral order of\u0000a neighbourhood of a Giroux torsion domain.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192031","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}
A pair $(alpha, beta)$ of simple closed curves on a surface $S_{g,n}$ of genus $g$ and with $n$ punctures is called a filling pair if the complement of the union of the curves is a disjoint union of topological disks and once punctured disks. In this article, we study the length of filling pairs on once-punctured hyperbolic surfaces. In particular, we find a lower bound of the length of filling pairs which depends only on the topology of the surface.
{"title":"Length of Filling Pairs on Punctured Surface","authors":"Bhola Nath Saha, Bidyut Sanki","doi":"arxiv-2409.05483","DOIUrl":"https://doi.org/arxiv-2409.05483","url":null,"abstract":"A pair $(alpha, beta)$ of simple closed curves on a surface $S_{g,n}$ of\u0000genus $g$ and with $n$ punctures is called a filling pair if the complement of\u0000the union of the curves is a disjoint union of topological disks and once\u0000punctured disks. In this article, we study the length of filling pairs on\u0000once-punctured hyperbolic surfaces. In particular, we find a lower bound of the\u0000length of filling pairs which depends only on the topology of the surface.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192038","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 first show that not all boundary points of the fine curve graph of a closed surface are seen via finite approximations, by which we mean via curve graphs of the surface punctured at finitely many points. We then use fine curve graph tools to prove that there exist parabolic isometries of graphs of curves associated to surfaces of infinite type.
{"title":"From curve graphs to fine curve graphs and back","authors":"Federica Fanoni, Sebastian Hensel","doi":"arxiv-2409.05647","DOIUrl":"https://doi.org/arxiv-2409.05647","url":null,"abstract":"We first show that not all boundary points of the fine curve graph of a\u0000closed surface are seen via finite approximations, by which we mean via curve\u0000graphs of the surface punctured at finitely many points. We then use fine curve\u0000graph tools to prove that there exist parabolic isometries of graphs of curves\u0000associated to surfaces of infinite type.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192034","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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