{"title":"On determinant of checkerboard colorable virtual knots","authors":"Tomoaki Hatano, Yuta Nozaki","doi":"arxiv-2408.01891","DOIUrl":"https://doi.org/arxiv-2408.01891","url":null,"abstract":"For classical knots, it is well known that their determinants mod $8$ are\u0000classified by the Arf invariant. Boden and Karimi introduced a determinant of\u0000checkerboard colorable virtual knots. We prove that their determinant mod $8$\u0000is classified by the coefficient of $z^2$ in the ascending polynomial which is\u0000an extension of the Conway polynomial for classical knots.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942496","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}
{"title":"Thurston geodesics: no backtracking and active intervals","authors":"Anna Lenzhen, Babak Modami, Kasra Rafi, Jing Tao","doi":"arxiv-2408.01632","DOIUrl":"https://doi.org/arxiv-2408.01632","url":null,"abstract":"We develop the notion of the active interval for a subsurface along a\u0000geodesic in the Thurston metric on Teichmuller space of a surface S. That is,\u0000for any geodesic in the Thurston metric and any subsurface R of S, we find an\u0000interval of times where the length of the boundary of R is uniformly bounded\u0000and the restriction of the geodesic to the subsurface R resembles a geodesic in\u0000the Teichmuller space of R. In particular, the set of short curves in R during\u0000the active interval represents a reparametrized quasi-geodesic in the curve\u0000graph of R (no backtracking) and the amount of movement in the curve graph of R\u0000outside of the active interval is uniformly bounded which justifies the name\u0000active interval. These intervals provide an analogue of the active intervals\u0000introduced by the third author in the setting of Teichmuller space equipped\u0000with the Teichmuller metric.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942497","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}
{"title":"The Giroux Correspondence in dimension 3","authors":"Joan Licata, Vera Vértesi","doi":"arxiv-2408.01079","DOIUrl":"https://doi.org/arxiv-2408.01079","url":null,"abstract":"In an earlier paper, the authors proved the Giroux Correspondence for tight\u0000contact $3$-manifolds via convex Heegaard surfaces. Simultaneously, Breen,\u0000Honda and Huang gave an all-dimensions proof of the Giroux Correspondence by\u0000generalising convex surface theory to higher dimensions. This paper uses a key\u0000result about relations of bypasses to complete the $3$-dimensional proof for\u0000arbitrary (not necessarily tight) contact 3-manifolds. This presentation\u0000features low-dimensional techniques and further clarifies the relationship\u0000between contact manifolds and their Heegaard splittings.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"103 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942500","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}
{"title":"On Watanabe's theta graph diffeomorphism in the 4-sphere","authors":"David T. Gay","doi":"arxiv-2408.01324","DOIUrl":"https://doi.org/arxiv-2408.01324","url":null,"abstract":"Watanabe's theta graph diffeomorphism, constructed using Watanabe's clasper\u0000surgery construction which turns trivalent graphs in 4-manifolds into\u0000parameterized families of diffeomorphisms of 4-manifolds, is a diffeomorphism\u0000of $S^4$ representing a potentially nontrivial smooth mapping class of $S^4$.\u0000The \"(1,2)-subgroup\" of the smooth mapping class group of $S^4$ is the subgroup\u0000represented by diffeomorphisms which are pseudoisotopic to the identity via a\u0000Cerf family with only index 1 and 2 critical points. This author and Hartman\u0000showed that this subgroup is either trivial or has order 2 and explicitly\u0000identified a diffeomorphism that would represent the nontrivial element if this\u0000subgroup is nontrivial. Here we show that the theta graph diffeomorphism is\u0000isotopic to this one possibly nontrivial element of the (1,2)-subgroup. To\u0000prove this relation we develop a diagrammatic calculus for working in the\u0000smooth mapping class group of $S^4$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942498","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}
{"title":"Systems of curves on non-orientable surfaces","authors":"Xiao Chen","doi":"arxiv-2408.00369","DOIUrl":"https://doi.org/arxiv-2408.00369","url":null,"abstract":"We show that the order of the cardinality of maximal complete $1$-systems of\u0000loops on non-orientable surfaces is $sim |chi|^{2}$. In particular, we\u0000determine the exact cardinality of maximal complete $1$-systems of loops on\u0000punctured projective planes. To prove these results, we show that the\u0000cardinality of maximal systems of arcs pairwise-intersecting at most once on a\u0000non-orientable surface is $2|chi|(|chi|+1)$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885194","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}
{"title":"Pointed Quandle Coloring Quivers of Linkoids","authors":"Jose Ceniceros, Max Klivans","doi":"arxiv-2407.21606","DOIUrl":"https://doi.org/arxiv-2407.21606","url":null,"abstract":"We enhance the pointed quandle counting invariant of linkoids through the use\u0000of quivers analogously to quandle coloring quivers. This allows us to\u0000generalize the in-degree polynomial invariant of links to linkoids.\u0000Additionally, we introduce a new linkoid invariant, which we call the in-degree\u0000quiver polynomial matrix. Lastly, we study the pointed quandle coloring quivers\u0000of linkoids of $(p,2)$-torus type with respect to pointed dihedral quandles.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863592","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}
{"title":"Shadow-complexity and trisection genus","authors":"Hironobu Naoe, Masaki Ogawa","doi":"arxiv-2407.21265","DOIUrl":"https://doi.org/arxiv-2407.21265","url":null,"abstract":"The shadow-complexity is an invariant of closed $4$-manifolds defined by\u0000using $2$-dimensional polyhedra called Turaev's shadows, which, roughly\u0000speaking, measures how complicated a $2$-skeleton of the $4$-manifold is. In\u0000this paper, we define a new version $mathrm{sc}_{r}$ of shadow-complexity\u0000depending on an extra parameter $rgeq0$, and we investigate the relationship\u0000between this complexity and the trisection genus $g$. More explicitly, we prove\u0000an inequality $g(W) leq 2+2mathrm{sc}_{r}(W)$ for any closed $4$-manifold $W$\u0000and any $rgeq1/2$. Moreover, we determine the exact values of\u0000$mathrm{sc}_{1/2}$ for infinitely many $4$-manifolds, and also we classify all\u0000the closed $4$-manifolds with $mathrm{sc}_{1/2}leq1/2$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"87 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863591","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}
{"title":"An analogue of Turaev comultiplication for knots in non-orientable thickening of a non-orientable surface","authors":"Vladimir Tarkaev","doi":"arxiv-2407.20715","DOIUrl":"https://doi.org/arxiv-2407.20715","url":null,"abstract":"This paper concerns pseudo-classical knots in the non-orientable manifold\u0000$hat{Sigma} =Sigma times [0,1]$, where $Sigma$ is a non-orientable surface\u0000and a knot $K subset hat{Sigma}$ is called pseudo-classical if $K$ is\u0000orientation-preserving path in $hat{Sigma}$. For this kind of knot we\u0000introduce an invariant $Delta$ that is an analogue of Turaev comultiplication\u0000for knots in a thickened orientable surface. As its classical prototype,\u0000$Delta$ takes value in a polynomial algebra generated by homotopy classes of\u0000non-contractible loops on $Sigma$, however, as a ground ring we use some\u0000subring of $mathbb{C}$ instead of $mathbb{Z}$. Then we define a few homotopy,\u0000homology and polynomial invariants, which are consequences of $Delta$,\u0000including an analogue of the affine index polynomial.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"197 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863593","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}
{"title":"Four-manifolds defined by vector-colorings of simple polytopes","authors":"Nikolai Erokhovets","doi":"arxiv-2407.20575","DOIUrl":"https://doi.org/arxiv-2407.20575","url":null,"abstract":"We consider (non-necessarily free) actions of subgroups $Hsubset mathbb\u0000Z_2^m$ on the real moment-angle manifold $mathbb Rmathcal{Z}_P$ over a simple\u0000$n$-polytope $P$. The orbit space $N(P,H)=mathbb Rmathcal{Z}_P/H$ has an\u0000action of $mathbb Z_2^m/H$. For general $n$ we introduce the notion of a\u0000Hamiltonian $mathcal{C}(n,k)$-subcomplex generalizing the three-dimensional\u0000notions of a Hamiltonian cycle, theta- and $K_4$-subgraphs. Each\u0000$mathcal{C}(n,k)$-subcomplex $Csubset partial P$ corresponds to a subgroup\u0000$H_C$ such that $N(P,H_C)simeq S^n$. We prove that in dimensions $nleqslant\u00004$ this correspondence is a bijection. Any subgroup $Hsubset mathbb Z_2^m$\u0000defines a complex $mathcal{C}(P,H)subset partial P$. We prove that each\u0000Hamiltonian $mathcal{C}(n,k)$-subcomplex $Csubset mathcal{C}(P,H)$ inducing\u0000$H$ corresponds to a hyperelliptic involution $tau_Cinmathbb Z_2^m/H$ on the\u0000manifold $N(P,H)$ (that is, an involution with the orbit space homeomorphic to\u0000$S^n$) and in dimensions $nleqslant 4$ this correspondence is a bijection. We\u0000prove that for the geometries $mathbb X= mathbb S^4$, $mathbb\u0000S^3timesmathbb R$, $mathbb S^2times mathbb S^2$, $mathbb S^2times\u0000mathbb R^2$, $mathbb S^2times mathbb L^2$, and $mathbb L^2times mathbb\u0000L^2$ there exists a compact right-angled $4$-polytope $P$ with a free action of\u0000$H$ such that the geometric manifold $N(P,H)$ has a hyperelliptic involution in\u0000$mathbb Z_2^m/H$, and for $mathbb X=mathbb R^4$, $mathbb L^4$, $mathbb\u0000L^3times mathbb R$ and $mathbb L^2times mathbb R^2$ there are no such\u0000polytopes.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863690","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}
{"title":"Positive scalar curvature with point singularities","authors":"Simone Cecchini, Georg Frenck, Rudolf Zeidler","doi":"arxiv-2407.20163","DOIUrl":"https://doi.org/arxiv-2407.20163","url":null,"abstract":"We show that in every dimension $n geq 8$, there exists a smooth closed\u0000manifold $M^n$ which does not admit a smooth positive scalar curvature (\"psc\")\u0000metric, but $M$ admits an $mathrm{L}^infty$-metric which is smooth and has\u0000psc outside a singular set of codimension $geq 8$. This provides\u0000counterexamples to a conjecture of Schoen. In fact, there are such examples of\u0000arbitrarily high dimension with only single point singularities. In addition,\u0000we provide examples of $mathrm{L}^infty$-metrics on $mathbb{R}^n$ for\u0000certain $n geq 8$ which are smooth and have psc outside the origin, but cannot\u0000be smoothly approximated away from the origin by everywhere smooth Riemannian\u0000metrics of non-negative scalar curvature. This stands in precise contrast to\u0000established smoothing results via Ricci--DeTurck flow for singular metrics with\u0000stronger regularity assumptions. Finally, as a positive result, we describe a\u0000$mathrm{KO}$-theoretic condition which obstructs the existence of\u0000$mathrm{L}^infty$-metrics that are smooth and of psc outside a finite subset.\u0000This shows that closed enlargeable spin manifolds do not carry such metrics.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863665","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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