Pub Date : 2024-11-20DOI: 10.1016/j.aim.2024.110021
Borys Kadets , Isabel Vogt
This paper is devoted to understanding curves X over a number field k that possess infinitely many solutions in extensions of k of degree at most d; such solutions are the titular low degree points. For it is known ([9], [2]) that such curves, after a base change to , admit a map of degree at most d onto or an elliptic curve. For the analogous statement was shown to be false [3]. We prove that once the genus of X is high enough, the low degree points still have geometric origin: they can be obtained as pullbacks of low degree points from a lower genus curve. We introduce a discrete-geometric invariant attached to such curves: a family of subspace configurations, with many interesting properties. This structure gives a natural alternative construction of curves from [3]. As an application of our methods, we obtain a classification of such curves over k for , and a classification over for .
本文致力于理解数域 k 上的曲线 X，它在度数至多为 d 的 k 的广延中有无穷多个解；这些解就是所谓的低度点。对于 d=2,3 已知（[9], [2]），这类曲线在基数变为 k‾ 之后，可以有一个至多 d 度的映射到 P1 或椭圆曲线上。对于 d⩾4，类似的说法被证明是错误的[3]。我们证明，一旦 X 的种属足够高，低度点仍然具有几何起源：它们可以作为低度点从较低种属曲线的回拉而得到。我们引入了附加于此类曲线的离散几何不变量：具有许多有趣性质的子空间配置族。这种结构为 [3] 中的曲线提供了一种自然的替代构造。作为我们方法的应用，我们得到了 d=2,3 时 k 上的此类曲线分类，以及 d=4,5 时 k¯ 上的分类。
{"title":"Subspace configurations and low degree points on curves","authors":"Borys Kadets , Isabel Vogt","doi":"10.1016/j.aim.2024.110021","DOIUrl":"10.1016/j.aim.2024.110021","url":null,"abstract":"<div><div>This paper is devoted to understanding curves <em>X</em> over a number field <em>k</em> that possess infinitely many solutions in extensions of <em>k</em> of degree at most <em>d</em>; such solutions are the titular low degree points. For <span><math><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span> it is known (<span><span>[9]</span></span>, <span><span>[2]</span></span>) that such curves, after a base change to <span><math><mover><mrow><mi>k</mi></mrow><mo>‾</mo></mover></math></span>, admit a map of degree at most <em>d</em> onto <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> or an elliptic curve. For <span><math><mi>d</mi><mo>⩾</mo><mn>4</mn></math></span> the analogous statement was shown to be false <span><span>[3]</span></span>. We prove that once the genus of <em>X</em> is high enough, the low degree points still have geometric origin: they can be obtained as pullbacks of low degree points from a lower genus curve. We introduce a discrete-geometric invariant attached to such curves: a family of subspace configurations, with many interesting properties. This structure gives a natural alternative construction of curves from <span><span>[3]</span></span>. As an application of our methods, we obtain a classification of such curves over <em>k</em> for <span><math><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span>, and a classification over <span><math><mover><mrow><mi>k</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span> for <span><math><mi>d</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110021"},"PeriodicalIF":1.5,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.aim.2024.110023
Xuezhang Chen , Wei Wei , Nan Wu
We establish three families of Sobolev trace inequalities of orders two and four in the unit ball under higher order moments constraint, and are able to construct smooth test functions to show all such inequalities are almost optimal. Some distinct feature in almost sharp examples between the fourth order and second order Sobolev trace inequalities is discovered. This has been neglected in higher order Sobolev inequality case in [21]. As a byproduct, the method of our construction can be used to show the sharpness of the generalized Lebedev-Milin inequality under constraints.
{"title":"Almost sharp Sobolev trace inequalities in the unit ball under constraints","authors":"Xuezhang Chen , Wei Wei , Nan Wu","doi":"10.1016/j.aim.2024.110023","DOIUrl":"10.1016/j.aim.2024.110023","url":null,"abstract":"<div><div>We establish three families of Sobolev trace inequalities of orders two and four in the unit ball under higher order moments constraint, and are able to construct <em>smooth</em> test functions to show all such inequalities are <em>almost optimal</em>. Some distinct feature in <em>almost sharp</em> examples between the fourth order and second order Sobolev trace inequalities is discovered. This has been neglected in higher order Sobolev inequality case in <span><span>[21]</span></span>. As a byproduct, the method of our construction can be used to show the sharpness of the generalized Lebedev-Milin inequality under constraints.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110023"},"PeriodicalIF":1.5,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.aim.2024.110025
Raf Cluckers , Georges Comte , Jean-Philippe Rolin , Tamara Servi
We consider several systems of algebras of real- and complex-valued functions, which appear in o-minimal geometry and related geometrically tame contexts. For each such system, we prove its stability under parametric integration and we study the asymptotics of the functions as well as the nature of their parametric Mellin transforms.
{"title":"Mellin transforms of power-constructible functions","authors":"Raf Cluckers , Georges Comte , Jean-Philippe Rolin , Tamara Servi","doi":"10.1016/j.aim.2024.110025","DOIUrl":"10.1016/j.aim.2024.110025","url":null,"abstract":"<div><div>We consider several systems of algebras of real- and complex-valued functions, which appear in o-minimal geometry and related geometrically tame contexts. For each such system, we prove its stability under parametric integration and we study the asymptotics of the functions as well as the nature of their parametric Mellin transforms.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110025"},"PeriodicalIF":1.5,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.aim.2024.110017
Sangjib Kim , Soo Teck Lee
We study the algebras of polynomials invariant under the action of the complex classical groups , and . We show that each of these algebras is a flat deformation of the semigroup algebra on an explicitly defined semigroup.
我们研究了在复经典群 GLn、Sp2n 和 On 作用下不变的多项式代数。我们证明，这些代数的每一个都是一个明确定义的半群上的半群代数的平面变形。
{"title":"Toric degeneration of algebras of invariants","authors":"Sangjib Kim , Soo Teck Lee","doi":"10.1016/j.aim.2024.110017","DOIUrl":"10.1016/j.aim.2024.110017","url":null,"abstract":"<div><div>We study the algebras of polynomials invariant under the action of the complex classical groups <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We show that each of these algebras is a flat deformation of the semigroup algebra on an explicitly defined semigroup.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110017"},"PeriodicalIF":1.5,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.aim.2024.110031
Nefton Pali, Bruno Salvy
This article deals with an explicit canonical construction of a maximal totally real embedding for real analytic manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent bundle or of its complexified tangent bundle. The existence of maximal totally real embeddings for real analytic manifolds is known from previous celebrated works by Bruhat-Whitney [1] and Grauert [4]. Their construction is based on the use of analytic continuation of local frames and local coordinates that are far from being canonical or explicit. As a consequence, the form of the corresponding complex structure has been a mystery since the very beginning. A quite simple recursive expression for such complex structures has been provided in the first author's work “On maximal totally real embeddings” [12]. In our series of articles we focus on the case of torsion free connections. In the present article we give a fiberwise Taylor expansion of the canonical complex structure which is expressed in terms of symmetrization of curvature monomials and a rather simple and explicit expression of the coefficients of the expansion. We explain also a rather simple geometric characterization of such canonical complex structures. Our main result and argument can be useful for the study of open questions in the theory of the embeddings in consideration such as their moduli space.
{"title":"Explicit maximal totally real embeddings","authors":"Nefton Pali, Bruno Salvy","doi":"10.1016/j.aim.2024.110031","DOIUrl":"10.1016/j.aim.2024.110031","url":null,"abstract":"<div><div>This article deals with an explicit canonical construction of a maximal totally real embedding for real analytic manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent bundle or of its complexified tangent bundle. The existence of maximal totally real embeddings for real analytic manifolds is known from previous celebrated works by Bruhat-Whitney <span><span>[1]</span></span> and Grauert <span><span>[4]</span></span>. Their construction is based on the use of analytic continuation of local frames and local coordinates that are far from being canonical or explicit. As a consequence, the form of the corresponding complex structure has been a mystery since the very beginning. A quite simple recursive expression for such complex structures has been provided in the first author's work “On maximal totally real embeddings” <span><span>[12]</span></span>. In our series of articles we focus on the case of torsion free connections. In the present article we give a fiberwise Taylor expansion of the canonical complex structure which is expressed in terms of symmetrization of curvature monomials and a rather simple and explicit expression of the coefficients of the expansion. We explain also a rather simple geometric characterization of such canonical complex structures. Our main result and argument can be useful for the study of open questions in the theory of the embeddings in consideration such as their moduli space.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110031"},"PeriodicalIF":1.5,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1016/j.aim.2024.110029
Matías Menni
To each simplicial set X we naturally assign an étendue whose internal logic captures information about the geometry of X. In particular, we show that, for ‘non-singular’ objects X and Y, the étendues and are equivalent if, and only if, X and Y have the same dimension. Many of the results apply to presheaf toposes over ‘well-founded’ sites.
对于每个简单集合 X，我们自然会分配一个 "E´X"，其内部逻辑捕捉了 X 的几何信息。我们特别指出，对于 "非星形 "对象 X 和 Y，如果且仅当 X 和 Y 具有相同维度时，"E´X "和 "E´Y "是等价的。许多结果都适用于 "有根据 "站点上的预叶拓扑。
{"title":"The étendue of a combinatorial space and its dimension","authors":"Matías Menni","doi":"10.1016/j.aim.2024.110029","DOIUrl":"10.1016/j.aim.2024.110029","url":null,"abstract":"<div><div>To each simplicial set <em>X</em> we naturally assign an étendue <span><math><mover><mrow><mi>E</mi></mrow><mrow><mo>´</mo></mrow></mover><mi>X</mi></math></span> whose internal logic captures information about the geometry of <em>X</em>. In particular, we show that, for ‘non-singular’ objects <em>X</em> and <em>Y</em>, the étendues <span><math><mover><mrow><mi>E</mi></mrow><mrow><mo>´</mo></mrow></mover><mi>X</mi></math></span> and <span><math><mover><mrow><mi>E</mi></mrow><mrow><mo>´</mo></mrow></mover><mi>Y</mi></math></span> are equivalent if, and only if, <em>X</em> and <em>Y</em> have the same dimension. Many of the results apply to presheaf toposes over ‘well-founded’ sites.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110029"},"PeriodicalIF":1.5,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1016/j.aim.2024.110030
Moritz Kassmann , Marvin Weidner
We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest equation in this theory, the fractional Kolmogorov equation. Our result reflects a purely nonlocal phenomenon since the Harnack inequality holds true for local kinetic equations like the Kolmogorov equation.
{"title":"The Harnack inequality fails for nonlocal kinetic equations","authors":"Moritz Kassmann , Marvin Weidner","doi":"10.1016/j.aim.2024.110030","DOIUrl":"10.1016/j.aim.2024.110030","url":null,"abstract":"<div><div>We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest equation in this theory, the fractional Kolmogorov equation. Our result reflects a purely nonlocal phenomenon since the Harnack inequality holds true for local kinetic equations like the Kolmogorov equation.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110030"},"PeriodicalIF":1.5,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.aim.2024.109999
Dilip Raghavan , Juris Steprāns
It will be shown to be consistent that there are at least two non-isomorphic selective ultrafilters, but no stable ordered-union ultrafilters. This answers a question of Blass from his 1987 paper [6].
{"title":"Stable ordered-union versus selective ultrafilters","authors":"Dilip Raghavan , Juris Steprāns","doi":"10.1016/j.aim.2024.109999","DOIUrl":"10.1016/j.aim.2024.109999","url":null,"abstract":"<div><div>It will be shown to be consistent that there are at least two non-isomorphic selective ultrafilters, but no stable ordered-union ultrafilters. This answers a question of Blass from his 1987 paper <span><span>[6]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 109999"},"PeriodicalIF":1.5,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142650798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-07DOI: 10.1016/j.aim.2024.110000
Xiang-Ke Chang , Xiao-Min Chen
The Camassa–Holm hierarchy can be regarded as isospectral flows of the inhomogeneous string. This paper is devoted to the exploration of the second flow in the Camassa–Holm hierarchy (2ndCH) together with its peakon dynamical system as well as their nonisospectral generalizations. It is shown that a reduction of the peakon dynamical system of the 2ndCH equation results in the two-component modified Camassa–Holm (2mCH) interlacing peakon dynamical system. This reduction result is then extended to the nonisospectral case. More precisely, a nonisospectral extension of the 2ndCH is proposed together with its multipeakons based on classical determinant technique. It is also demonstrated that the corresponding peakon dynamical system can be reduced to the generalized nonisospectral 2mCH interlacing peakon dynamical system. Moreover, a special case of the proposed equation is investigated and a new phenomenon of 2-peakon is observed.
{"title":"On the peakon dynamical system of the second flow in the Camassa–Holm hierarchy","authors":"Xiang-Ke Chang , Xiao-Min Chen","doi":"10.1016/j.aim.2024.110000","DOIUrl":"10.1016/j.aim.2024.110000","url":null,"abstract":"<div><div>The Camassa–Holm hierarchy can be regarded as isospectral flows of the inhomogeneous string. This paper is devoted to the exploration of the second flow in the Camassa–Holm hierarchy (2ndCH) together with its peakon dynamical system as well as their nonisospectral generalizations. It is shown that a reduction of the peakon dynamical system of the 2ndCH equation results in the two-component modified Camassa–Holm (2mCH) interlacing peakon dynamical system. This reduction result is then extended to the nonisospectral case. More precisely, a nonisospectral extension of the 2ndCH is proposed together with its multipeakons based on classical determinant technique. It is also demonstrated that the corresponding peakon dynamical system can be reduced to the generalized nonisospectral 2mCH interlacing peakon dynamical system. Moreover, a special case of the proposed equation is investigated and a new phenomenon of 2-peakon is observed.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 110000"},"PeriodicalIF":1.5,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142650796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-06DOI: 10.1016/j.aim.2024.109993
Cristina Ana-Maria Anghel
Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by immersed Lagrangian submanifolds. The main result proves that the coloured Jones polynomial and coloured Alexander polynomial come as different specialisations of an intersection pairing of the same homology classes over two variables, with extra framing corrections in each case. This model can be evaluated at roots of unity. The first corollary explains Bigelow's picture for the Jones polynomial with noodles and forks from the quantum point of view. Secondly, we conclude that the coloured Alexander polynomial is a graded intersection pairing in a -covering of the configuration space in the punctured disc. This paper comes with a sequel article where, based on this result, we construct another topological model with homology classes given by explicit embedded Lagrangians, which are more suitable for computations. As a corollary of these two papers, we provide two intersection models (an immersed one and an embedded one) each of which leads to the Jones polynomial and Alexander polynomial by suitable specialisations.
彩色琼斯和亚历山大多项式是恢复琼斯和亚历山大多项式首项的量子不变量序列。我们用拓扑工具证明，它们在概念上可以被看作是浸没拉格朗日子实体给出的显式同调类之间覆盖空间的交集配对。主要结果证明，第 N 次彩色琼斯多项式和第 N 次彩色亚历山大多项式是两个变量上相同同构类的交集配对的不同特化，在每种情况下都有额外的框架修正。这个模型可以在统一根上进行评估。第一个推论从量子的角度解释了毕格罗关于琼斯多项式与面条和叉子的图景。其次，我们得出结论：N 次彩色亚历山大多项式是穿刺圆盘配置空间 Z⊕Z2N 覆盖中的分级交对。本文还附有一篇续文，在此基础上，我们构建了另一个拓扑模型，其同调类由显式嵌入拉格朗日给出，更适合计算。作为这两篇论文的推论，我们提供了两个交集模型（一个浸入模型和一个嵌入模型），每个模型都能通过适当的特殊化得出琼斯多项式和亚历山大多项式。
{"title":"Coloured Jones and Alexander polynomials as topological intersections of cycles in configuration spaces","authors":"Cristina Ana-Maria Anghel","doi":"10.1016/j.aim.2024.109993","DOIUrl":"10.1016/j.aim.2024.109993","url":null,"abstract":"<div><div>Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by immersed Lagrangian submanifolds. The main result proves that the <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math></span> coloured Jones polynomial and <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math></span> coloured Alexander polynomial come as different specialisations of an intersection pairing of the same homology classes over two variables, with extra framing corrections in each case. This model can be evaluated at roots of unity. The first corollary explains Bigelow's picture for the Jones polynomial with noodles and forks from the quantum point of view. Secondly, we conclude that the <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math></span> coloured Alexander polynomial is a graded intersection pairing in a <span><math><mi>Z</mi><mo>⊕</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn><mi>N</mi></mrow></msub></math></span>-covering of the configuration space in the punctured disc. This paper comes with a sequel article where, based on this result, we construct another topological model with homology classes given by explicit embedded Lagrangians, which are more suitable for computations. As a corollary of these two papers, we provide two intersection models (an immersed one and an embedded one) each of which leads to the Jones polynomial and Alexander polynomial by suitable specialisations.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"459 ","pages":"Article 109993"},"PeriodicalIF":1.5,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142650797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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