Pub Date : 2020-10-16DOI: 10.1142/S021821652150036X
Subhankar Dey, Hakan Doga
In this paper, we give a combinatorial description of the concordance invariant $varepsilon$ defined by Hom in cite{hom2011knot}, prove some properties of this invariant using grid homology techniques. We also compute $varepsilon$ of $(p,q)$ torus knots and prove that $varepsilon(mathbb{G}_+)=1$ if $mathbb{G}_+$ is a grid diagram for a positive braid. Furthermore, we show how $varepsilon$ behaves under $(p,q)$-cabling of negative torus knots.
{"title":"A Combinatorial Description of the Knot Concordance Invariant Epsilon","authors":"Subhankar Dey, Hakan Doga","doi":"10.1142/S021821652150036X","DOIUrl":"https://doi.org/10.1142/S021821652150036X","url":null,"abstract":"In this paper, we give a combinatorial description of the concordance invariant $varepsilon$ defined by Hom in cite{hom2011knot}, prove some properties of this invariant using grid homology techniques. We also compute $varepsilon$ of $(p,q)$ torus knots and prove that $varepsilon(mathbb{G}_+)=1$ if $mathbb{G}_+$ is a grid diagram for a positive braid. Furthermore, we show how $varepsilon$ behaves under $(p,q)$-cabling of negative torus knots.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90941794","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 : 2020-10-10DOI: 10.4310/PAMQ.2021.V17.N1.A14
A. Savini
Let $N$ be a compact manifold with a foliation $mathscr{F}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation $mathscr{F}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose to have a foliation-preserving homeomorphism $f:(N,mathscr{F}_N) rightarrow (M,mathscr{F}_M)$ which is $C^1$-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies $h(N,mathscr{F}_N)$ and $h(M,mathscr{F}_M)$ and it holds $h(M,mathscr{F}_M) leq h(N,mathscr{F}_N)$. Additionally, if equality holds, then the leaves must be homothetic.
{"title":"Entropy rigidity for foliations by strictly convex projective manifolds","authors":"A. Savini","doi":"10.4310/PAMQ.2021.V17.N1.A14","DOIUrl":"https://doi.org/10.4310/PAMQ.2021.V17.N1.A14","url":null,"abstract":"Let $N$ be a compact manifold with a foliation $mathscr{F}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation $mathscr{F}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose to have a foliation-preserving homeomorphism $f:(N,mathscr{F}_N) rightarrow (M,mathscr{F}_M)$ which is $C^1$-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies $h(N,mathscr{F}_N)$ and $h(M,mathscr{F}_M)$ and it holds $h(M,mathscr{F}_M) leq h(N,mathscr{F}_N)$. Additionally, if equality holds, then the leaves must be homothetic.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77240407","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 the differences between various concordance invariants of knots, including Rasmussen's $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are nontrivial for some quasi-alternating knots, we show the additivity of Turaev genus for a certain class of knots. This leads us to the first example of an infinite family of quasi-alternating knots with Turaev genus exactly $g$ for any fixed positive integer $g$, solving a question of Champanerkar-Kofman.
{"title":"Concordance Invariants and the Turaev Genus","authors":"H. Jung, Sungkyung Kang, Seungwon Kim","doi":"10.1093/IMRN/RNAB055","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB055","url":null,"abstract":"We show that the differences between various concordance invariants of knots, including Rasmussen's $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are nontrivial for some quasi-alternating knots, we show the additivity of Turaev genus for a certain class of knots. This leads us to the first example of an infinite family of quasi-alternating knots with Turaev genus exactly $g$ for any fixed positive integer $g$, solving a question of Champanerkar-Kofman.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81541132","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 : 2020-09-30DOI: 10.1142/S0129167X21500233
Jose Ceniceros, Sam Nelson
We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions. As an application we define new single-variable and two-variable polynomial invariants of oriented pseudoknots and singular knots and links. We provide examples to show that the new invariants are proper enhancements of the counting invariant are are not determined by the Jablan polynomial.
{"title":"Cocycle enhancements of psyquandle counting invariants","authors":"Jose Ceniceros, Sam Nelson","doi":"10.1142/S0129167X21500233","DOIUrl":"https://doi.org/10.1142/S0129167X21500233","url":null,"abstract":"We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions. As an application we define new single-variable and two-variable polynomial invariants of oriented pseudoknots and singular knots and links. We provide examples to show that the new invariants are proper enhancements of the counting invariant are are not determined by the Jablan polynomial.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72689881","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 give an explicit formula for the adjoint Reidemeister torsion of two-bridge knots and prove that the adjoint Reidemeister torsion satisfies a certain type of vanishing identities.
{"title":"Adjoint Reidemeister torsions of two-bridge knots","authors":"Seokbeom Yoon","doi":"10.1090/proc/15981","DOIUrl":"https://doi.org/10.1090/proc/15981","url":null,"abstract":"We give an explicit formula for the adjoint Reidemeister torsion of two-bridge knots and prove that the adjoint Reidemeister torsion satisfies a certain type of vanishing identities.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76213369","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 paper, we establish a version of the adjunction inequality for closed symplectic 4-manifolds. As in a previous paper on the Thom conjecture, we use contact geometry and trisections of 4-manifolds to reduce this inequality to the slice-Bennequin inequality for knots in the 4-ball. As this latter result can be proved using Khovanov homology, we completely avoid gauge theoretic techniques. This inequality can be used to give gauge-theory-free proofs of several landmark results in 4-manifold topology, such as detecting exotic smooth structures, the symplectic Thom conjecture, and exluding connected sum decompositions of certain symplectic 4-manifolds.
{"title":"Symplectic trisections and the adjunction inequality","authors":"Peter Lambert-Cole","doi":"10.14288/1.0395253","DOIUrl":"https://doi.org/10.14288/1.0395253","url":null,"abstract":"In this paper, we establish a version of the adjunction inequality for closed symplectic 4-manifolds. As in a previous paper on the Thom conjecture, we use contact geometry and trisections of 4-manifolds to reduce this inequality to the slice-Bennequin inequality for knots in the 4-ball. As this latter result can be proved using Khovanov homology, we completely avoid gauge theoretic techniques. This inequality can be used to give gauge-theory-free proofs of several landmark results in 4-manifold topology, such as detecting exotic smooth structures, the symplectic Thom conjecture, and exluding connected sum decompositions of certain symplectic 4-manifolds.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"1963 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91334998","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 : 2020-09-15DOI: 10.1142/S021821652150005X
Xifeng Jin
We show that, for any integers, $g geq 3$ and $n geq 2$, there exists a link in $S^3$ such that its complement has a genus $g$ Heegaard splitting with distance $n$.
{"title":"Heegaard distance of the link complements in S3","authors":"Xifeng Jin","doi":"10.1142/S021821652150005X","DOIUrl":"https://doi.org/10.1142/S021821652150005X","url":null,"abstract":"We show that, for any integers, $g geq 3$ and $n geq 2$, there exists a link in $S^3$ such that its complement has a genus $g$ Heegaard splitting with distance $n$.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"230 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79694916","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 slice disks for knots in the boundary of a smooth compact 4-manifold $X^{4}$. We call a knot $K subset partial X$ deep slice in $X$ if there is a smooth properly embedded 2-disk in $X$ with boundary $K$, but $K$ is not concordant to the unknot in a collar neighborhood $partial X times I$ of the boundary. We point out how this concept relates to various well-known conjectures and give some criteria for the nonexistence of such deep slice knots. Then we show, using the Wall self-intersection invariant and a result of Rohlin, that every 4-manifold consisting of just one 0- and a nonzero number of 2-handles always has a deep slice knot in the boundary. We end by considering 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. A generalization of the Murasugi-Tristram inequality is used to show that there does not exist a compact, oriented 4-manifold $V$ with spherical boundary such that every knot $K subset S^3 = partial V$ is slice in $V$ via a null-homologous disk.
{"title":"Deep and shallow slice knots in 4-manifolds","authors":"M. Klug, Benjamin Matthias Ruppik","doi":"10.1090/bproc/89","DOIUrl":"https://doi.org/10.1090/bproc/89","url":null,"abstract":"We consider slice disks for knots in the boundary of a smooth compact 4-manifold $X^{4}$. We call a knot $K subset partial X$ deep slice in $X$ if there is a smooth properly embedded 2-disk in $X$ with boundary $K$, but $K$ is not concordant to the unknot in a collar neighborhood $partial X times I$ of the boundary. We point out how this concept relates to various well-known conjectures and give some criteria for the nonexistence of such deep slice knots. Then we show, using the Wall self-intersection invariant and a result of Rohlin, that every 4-manifold consisting of just one 0- and a nonzero number of 2-handles always has a deep slice knot in the boundary. We end by considering 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. A generalization of the Murasugi-Tristram inequality is used to show that there does not exist a compact, oriented 4-manifold $V$ with spherical boundary such that every knot $K subset S^3 = partial V$ is slice in $V$ via a null-homologous disk.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76428920","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 : 2020-09-06DOI: 10.1142/S0218216521500401
Ph. G. Korablev, V. Tarkaev
Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such an arcs disjoint from crossings. In the paper we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.
{"title":"A relation between the crossing number and the height of a knotoid","authors":"Ph. G. Korablev, V. Tarkaev","doi":"10.1142/S0218216521500401","DOIUrl":"https://doi.org/10.1142/S0218216521500401","url":null,"abstract":"Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such an arcs disjoint from crossings. In the paper we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"159 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77281207","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 : 2020-09-04DOI: 10.1142/S0218216521500383
I. Dynnikov, V. Sokolova
We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of `simpler' moves not increasing the complexity of the diagram along the way.
{"title":"Multiflypes of rectangular diagrams of links","authors":"I. Dynnikov, V. Sokolova","doi":"10.1142/S0218216521500383","DOIUrl":"https://doi.org/10.1142/S0218216521500383","url":null,"abstract":"We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of `simpler' moves not increasing the complexity of the diagram along the way.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"147 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83241794","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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