This paper focuses on intersection of closed curves on translation surfaces. Namely, we investigate the question of determining the intersection of two closed curves of a given length on such surfaces. This question has been investigated in several papers and this paper complement the work of Boulanger, Lanneau and Massart done for double regular polygons, and extend the results to a large family of surfaces which includes in particular Bouw-M"oller surfaces. Namely, we give an estimate for KVol on surfaces based on geometric constraints (angles and indentifications of sides). This estimate is sharp in the case of Bouw-M"oller surfaces with a unique singularity, and it allows to compute KVol on the $SL_2(mathbb{R})$-orbit of such surfaces.
{"title":"Algebraic intersections on Bouw-Möller surfaces, and more general convex polygons","authors":"Julien Boulanger, Irene Pasquinelli","doi":"arxiv-2409.01711","DOIUrl":"https://doi.org/arxiv-2409.01711","url":null,"abstract":"This paper focuses on intersection of closed curves on translation surfaces.\u0000Namely, we investigate the question of determining the intersection of two\u0000closed curves of a given length on such surfaces. This question has been\u0000investigated in several papers and this paper complement the work of Boulanger,\u0000Lanneau and Massart done for double regular polygons, and extend the results to\u0000a large family of surfaces which includes in particular Bouw-M\"oller surfaces.\u0000Namely, we give an estimate for KVol on surfaces based on geometric constraints\u0000(angles and indentifications of sides). This estimate is sharp in the case of\u0000Bouw-M\"oller surfaces with a unique singularity, and it allows to compute KVol\u0000on the $SL_2(mathbb{R})$-orbit of such surfaces.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192053","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 prove a formula for the involutive concordance invariants of the cabled knots in terms of that of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer homology.
{"title":"A note on cables and the involutive concordance invariants","authors":"Kristen Hendricks, Abhishek Mallick","doi":"arxiv-2409.02192","DOIUrl":"https://doi.org/arxiv-2409.02192","url":null,"abstract":"We prove a formula for the involutive concordance invariants of the cabled\u0000knots in terms of that of the companion knot and the pattern knot. As a\u0000consequence, we show that any iterated cable of a knot with parameters of the\u0000form (odd,1) is not smoothly slice as long as either of the involutive\u0000concordance invariants of the knot is nonzero. Our formula also gives new\u0000bounds for the unknotting number of a cabled knot, which are sometimes stronger\u0000than other known bounds coming from knot Floer homology.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192051","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 companion article to [HKK24], we apply the theory of equivariant Poincar'e duality developed there in the special case of cyclic groups $C_p$ of prime order to remove, in a special case, a technical condition given by Davis--L"uck [DL24] in their work on the Nielsen realisation problem for aspherical manifolds. Along the way, we will also give a complete characterisation of $C_p$--Poincar'e spaces as well as introduce a genuine equivariant refinement of the classical notion of virtual Poincar'e duality groups which might be of independent interest.
{"title":"Equivariant Poincaré duality for cyclic groups of prime order and the Nielsen realisation problem","authors":"Kaif Hilman, Dominik Kirstein, Christian Kremer","doi":"arxiv-2409.02220","DOIUrl":"https://doi.org/arxiv-2409.02220","url":null,"abstract":"In this companion article to [HKK24], we apply the theory of equivariant\u0000Poincar'e duality developed there in the special case of cyclic groups $C_p$\u0000of prime order to remove, in a special case, a technical condition given by\u0000Davis--L\"uck [DL24] in their work on the Nielsen realisation problem for\u0000aspherical manifolds. Along the way, we will also give a complete\u0000characterisation of $C_p$--Poincar'e spaces as well as introduce a genuine\u0000equivariant refinement of the classical notion of virtual Poincar'e duality\u0000groups which might be of independent interest.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192050","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 their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a cubic curve. We provide geometric arguments to prove that in the case of a simplicial arrangement, the aforementioned cubic curve cannot be irreducible. It follows that Gr"{u}nbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points.
{"title":"Simplicial arrangements with few double points","authors":"Dmitri Panov, Guillaume Tahar","doi":"arxiv-2409.01892","DOIUrl":"https://doi.org/arxiv-2409.01892","url":null,"abstract":"In their solution to the orchard-planting problem, Green and Tao established\u0000a structure theorem which proves that in a line arrangement in the real\u0000projective plane with few double points, most lines are tangent to the dual\u0000curve of a cubic curve. We provide geometric arguments to prove that in the\u0000case of a simplicial arrangement, the aforementioned cubic curve cannot be\u0000irreducible. It follows that Gr\"{u}nbaum's conjectural asymptotic\u0000classification of simplicial arrangements holds under the additional hypothesis\u0000of a linear bound on the number of double points.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192080","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 undertake a study of topological properties of the real Mordell-Weil group $operatorname{MW}_{mathbb R}$ of real rational elliptic surfaces $X$ which we accompany by a related study of real lines on $X$ and on the "subordinate" del Pezzo surfaces $Y$ of degree 1. We give an explicit description of isotopy types of real lines on $Y_{mathbb R}$ and an explicit presentation of $operatorname{MW}_{mathbb R}$ in the mapping class group $operatorname{Mod}(X_{mathbb R})$. Combining these results we establish an explicit formula for the action of $operatorname{MW}_{mathbb R}$ in $H_1(X_{mathbb R})$.
{"title":"The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree $K^2=1$","authors":"Sergey Finashin, Viatcheslav Kharlamov","doi":"arxiv-2409.01202","DOIUrl":"https://doi.org/arxiv-2409.01202","url":null,"abstract":"We undertake a study of topological properties of the real Mordell-Weil group\u0000$operatorname{MW}_{mathbb R}$ of real rational elliptic surfaces $X$ which we\u0000accompany by a related study of real lines on $X$ and on the \"subordinate\" del\u0000Pezzo surfaces $Y$ of degree 1. We give an explicit description of isotopy\u0000types of real lines on $Y_{mathbb R}$ and an explicit presentation of\u0000$operatorname{MW}_{mathbb R}$ in the mapping class group\u0000$operatorname{Mod}(X_{mathbb R})$. Combining these results we establish an\u0000explicit formula for the action of $operatorname{MW}_{mathbb R}$ in\u0000$H_1(X_{mathbb R})$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192250","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 degree of a map between orientable manifolds is a crucial concept in topology, providing deep insights into the structure and properties of the manifolds and the corresponding maps. This concept has been thoroughly investigated, particularly in the realm of simplicial maps between orientable triangulable spaces. In this paper, we concentrate on constructing simplicial degree $d$ self-maps on $n$-spheres. We describe the construction of several such maps, demonstrating that for every $d in mathbb{Z} setminus {0}$, there exists a degree $d$ simplicial map from a triangulated $n$-sphere with $3|d| + n - 1$ vertices to $mathbb{S}^n_{n+2}$. Further, we prove that, for every $d in mathbb{Z} setminus {0}$, there exists a simplicial map of degree $3 d$ from a triangulated $n$-sphere with $6|d| + n$ vertices, as well as a simplicial map of degree $3d+frac{d}{|d|}$ from a triangulated $n$-sphere with $6|d|+n+3$ vertices, to $mathbb{S}^{n}_{n+2}$. Furthermore, we show that for any $|k| geq 2$ and $n geq |k|$, a degree $k$ simplicial map exists from a triangulated $n$-sphere $K$ with $|k| + n + 3$ vertices to $mathbb{S}^n_{n+2}$. We also prove that for $d = 2$ and 3, these constructions produce vertex-minimal degree $d$ self-maps of $n$-spheres. Additionally, for every $n geq 2$, we construct a degree $n+1$ simplicial map from a triangulated $n$-sphere with $2n + 4$ vertices to $mathbb{S}^{n}_{n+2}$. We also prove that this construction provides facet minimal degree $n+1$ self-maps of $n$-spheres.
{"title":"Simplicial degree $d$ self-maps on $n$-spheres","authors":"Biplab Basak, Raju Kumar Gupta, Ayushi Trivedi","doi":"arxiv-2409.00907","DOIUrl":"https://doi.org/arxiv-2409.00907","url":null,"abstract":"The degree of a map between orientable manifolds is a crucial concept in\u0000topology, providing deep insights into the structure and properties of the\u0000manifolds and the corresponding maps. This concept has been thoroughly\u0000investigated, particularly in the realm of simplicial maps between orientable\u0000triangulable spaces. In this paper, we concentrate on constructing simplicial\u0000degree $d$ self-maps on $n$-spheres. We describe the construction of several\u0000such maps, demonstrating that for every $d in mathbb{Z} setminus {0}$, there\u0000exists a degree $d$ simplicial map from a triangulated $n$-sphere with $3|d| +\u0000n - 1$ vertices to $mathbb{S}^n_{n+2}$. Further, we prove that, for every $d\u0000in mathbb{Z} setminus {0}$, there exists a simplicial map of degree $3 d$\u0000from a triangulated $n$-sphere with $6|d| + n$ vertices, as well as a\u0000simplicial map of degree $3d+frac{d}{|d|}$ from a triangulated $n$-sphere with\u0000$6|d|+n+3$ vertices, to $mathbb{S}^{n}_{n+2}$. Furthermore, we show that for\u0000any $|k| geq 2$ and $n geq |k|$, a degree $k$ simplicial map exists from a\u0000triangulated $n$-sphere $K$ with $|k| + n + 3$ vertices to\u0000$mathbb{S}^n_{n+2}$. We also prove that for $d = 2$ and 3, these constructions\u0000produce vertex-minimal degree $d$ self-maps of $n$-spheres. Additionally, for\u0000every $n geq 2$, we construct a degree $n+1$ simplicial map from a\u0000triangulated $n$-sphere with $2n + 4$ vertices to $mathbb{S}^{n}_{n+2}$. We\u0000also prove that this construction provides facet minimal degree $n+1$ self-maps\u0000of $n$-spheres.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192054","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 $k$-twist spun knot is an $n+1$-dimensional knot in the $n+3$-dimensional sphere which is obtained from an $n$-dimensional knot in the $n+2$-dimensional sphere by applying an operation called a $k$-twist-spinning. This construction was introduced by Zeeman in 1965. In this paper, we show that the $m_2$-twist-spinning of the $m_1$-twist-spinning of a classical knot is a trivial $3$-knot in $S^5$ if $gcd(m_1,m_2)=1$. We also give a sufficient condition for the $m_2$-twist-spinning of the $m_1$-twist-spinning of a classical knot to be non-trivial.
{"title":"Twist spun knots of twist spun knots of classical knots","authors":"Mizuki Fukuda, Masaharu Ishikawa","doi":"arxiv-2409.00650","DOIUrl":"https://doi.org/arxiv-2409.00650","url":null,"abstract":"A $k$-twist spun knot is an $n+1$-dimensional knot in the $n+3$-dimensional\u0000sphere which is obtained from an $n$-dimensional knot in the $n+2$-dimensional\u0000sphere by applying an operation called a $k$-twist-spinning. This construction\u0000was introduced by Zeeman in 1965. In this paper, we show that the\u0000$m_2$-twist-spinning of the $m_1$-twist-spinning of a classical knot is a\u0000trivial $3$-knot in $S^5$ if $gcd(m_1,m_2)=1$. We also give a sufficient\u0000condition for the $m_2$-twist-spinning of the $m_1$-twist-spinning of a\u0000classical knot to be non-trivial.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192056","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 family of right-angled tiling links consists of links built from regular 4-valent tilings of constant-curvature surfaces that contain one or two types of tiles. The complements of these links admit complete hyperbolic structures and contain two totally geodesic checkerboard surfaces that meet at right angles. In this paper, we give a complete characterization of which right-angled tiling links are arithmetic, and which are pairwise commensurable. The arithmeticity classification exploits symmetry arguments and the combinatorial geometry of Coxeter polyhedra. The commensurability classification relies on identifying the canonical decompositions of the link complements, in addition to number-theoretic data from invariant trace fields.
{"title":"Arithmeticity and commensurability of links in thickened surfaces","authors":"David Futer, Rose Kaplan-Kelly","doi":"arxiv-2409.00490","DOIUrl":"https://doi.org/arxiv-2409.00490","url":null,"abstract":"The family of right-angled tiling links consists of links built from regular\u00004-valent tilings of constant-curvature surfaces that contain one or two types\u0000of tiles. The complements of these links admit complete hyperbolic structures\u0000and contain two totally geodesic checkerboard surfaces that meet at right\u0000angles. In this paper, we give a complete characterization of which\u0000right-angled tiling links are arithmetic, and which are pairwise commensurable.\u0000The arithmeticity classification exploits symmetry arguments and the\u0000combinatorial geometry of Coxeter polyhedra. The commensurability\u0000classification relies on identifying the canonical decompositions of the link\u0000complements, in addition to number-theoretic data from invariant trace fields.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192059","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}
Gabai proved that any plumbing, or Murasugi sum, of $pi_1$-essential Seifert surfaces is also $pi_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show that the analogous statement about geometrically essential surfaces is untrue. We then introduce new numerical invariants, the algebraic and geometric essence of a spanning surface $Fsubset S^3$, which measure how far $F$ is from being compressible, and we extend Ozawa's theorem by showing that plumbing respects the algebraic version of this new invariant. We also introduce a ``twisted'' generalization of plumbing and use it to compute essence for many examples, including checkerboard surfaces from reduced alternating diagrams. Finally, we extend all of these results to plumbings and twisted plumbings of spanning surfaces in arbitrary 3-manifolds.
{"title":"How essential is a spanning surface?","authors":"Thomas Kindred","doi":"arxiv-2408.16948","DOIUrl":"https://doi.org/arxiv-2408.16948","url":null,"abstract":"Gabai proved that any plumbing, or Murasugi sum, of $pi_1$-essential Seifert\u0000surfaces is also $pi_1$-essential, and Ozawa extended this result to\u0000unoriented spanning surfaces. We show that the analogous statement about\u0000geometrically essential surfaces is untrue. We then introduce new numerical\u0000invariants, the algebraic and geometric essence of a spanning surface $Fsubset\u0000S^3$, which measure how far $F$ is from being compressible, and we extend\u0000Ozawa's theorem by showing that plumbing respects the algebraic version of this\u0000new invariant. We also introduce a ``twisted'' generalization of plumbing and\u0000use it to compute essence for many examples, including checkerboard surfaces\u0000from reduced alternating diagrams. Finally, we extend all of these results to\u0000plumbings and twisted plumbings of spanning surfaces in arbitrary 3-manifolds.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192061","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 3-manifold is called emph{SU(2)}-abelian if every SU(2)-representation of its fundamental group has abelian image. We classify, in terms of the Seifert coefficients, SU(2)-abelian 3-manifolds among the family of graph manifolds obtained by gluing two Seifert spaces both fibred over a disk and with two singular fibers. Finally, we prove that these SU(2)-abelian manifolds are Heegaard Floer homology L-spaces.
{"title":"A classification of SU(2)-abelian graph manifolds","authors":"Giacomo Bascapè","doi":"arxiv-2408.16635","DOIUrl":"https://doi.org/arxiv-2408.16635","url":null,"abstract":"A 3-manifold is called emph{SU(2)}-abelian if every SU(2)-representation of\u0000its fundamental group has abelian image. We classify, in terms of the Seifert\u0000coefficients, SU(2)-abelian 3-manifolds among the family of graph manifolds\u0000obtained by gluing two Seifert spaces both fibred over a disk and with two\u0000singular fibers. Finally, we prove that these SU(2)-abelian manifolds are\u0000Heegaard Floer homology L-spaces.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192247","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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