Pub Date : 2023-11-14DOI: 10.4153/s0008414x23000755
Adam S. Sikora
We study systems of $2$-tangle equations which play an important role in the analysis of enzyme actions on DNA strands. We show that every system of framed tangle equations has at most one framed rational solution. Furthermore, we show that the Jones Unknot conjecture implies that if a system of tangle equations has a rational solution then that solution is unique among all $2$-tangles. This result potentially opens a door to a purely topological disproof of the Jones Unknot conjecture. We introduce the notion of the Kauffman bracket ratio ${T}_qin mathbb Q(q)$ of any $2$-tangle $T$ and we conjecture that for $q=1$ it is the slope of meridionally incompressible surfaces in $D^3-T$. We prove that conjecture for algebraic $T$. We also prove that for rational $T$, the brackets ${T}_q$ coincide with the $q$-rationals of Morier-Genoud-Ovsienko. Additionally, we relate systems of tangle equations to the Cosmetic Surgery Conjecture and the Nugatory Crossing Conjecture.
{"title":"Tangle Equations, the Jones conjecture, slopes of surfaces in tangle complements, and <i>q</i>-deformed rationals","authors":"Adam S. Sikora","doi":"10.4153/s0008414x23000755","DOIUrl":"https://doi.org/10.4153/s0008414x23000755","url":null,"abstract":"We study systems of $2$-tangle equations which play an important role in the analysis of enzyme actions on DNA strands. We show that every system of framed tangle equations has at most one framed rational solution. Furthermore, we show that the Jones Unknot conjecture implies that if a system of tangle equations has a rational solution then that solution is unique among all $2$-tangles. This result potentially opens a door to a purely topological disproof of the Jones Unknot conjecture. We introduce the notion of the Kauffman bracket ratio ${T}_qin mathbb Q(q)$ of any $2$-tangle $T$ and we conjecture that for $q=1$ it is the slope of meridionally incompressible surfaces in $D^3-T$. We prove that conjecture for algebraic $T$. We also prove that for rational $T$, the brackets ${T}_q$ coincide with the $q$-rationals of Morier-Genoud-Ovsienko. Additionally, we relate systems of tangle equations to the Cosmetic Surgery Conjecture and the Nugatory Crossing Conjecture.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"32 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134953566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-13DOI: 10.4153/s0008414x23000743
JUNYAN ZHAO
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{"title":"COMPACTIFICATIONS OF MODULI OF DEL PEZZO SURFACES VIA LINE ARRANGEMENT AND K-STABILITY","authors":"JUNYAN ZHAO","doi":"10.4153/s0008414x23000743","DOIUrl":"https://doi.org/10.4153/s0008414x23000743","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"4 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136347547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-09DOI: 10.4153/s0008414x23000731
Projesh Nath Choudhury, Apoorva Khare
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{"title":"The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra","authors":"Projesh Nath Choudhury, Apoorva Khare","doi":"10.4153/s0008414x23000731","DOIUrl":"https://doi.org/10.4153/s0008414x23000731","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":" 20","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-31DOI: 10.4153/s0008414x2300072x
ARNAUD PLESSIS
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{"title":"A NEW WAY TO TACKLE A CONJECTURE OF RÉMOND","authors":"ARNAUD PLESSIS","doi":"10.4153/s0008414x2300072x","DOIUrl":"https://doi.org/10.4153/s0008414x2300072x","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"49 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135813586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.4153/s0008414x23000718
Dmitri I. Panyushev
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{"title":"The Frobenius semiradical, generic stabilisers, and Poisson centre for nilradicals","authors":"Dmitri I. Panyushev","doi":"10.4153/s0008414x23000718","DOIUrl":"https://doi.org/10.4153/s0008414x23000718","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"47 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136261894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.4153/s0008414x2300069x
Matthew Badger, Alyssa Genschaw
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{"title":"Lower bounds on Bourgain’s constant for harmonic measure","authors":"Matthew Badger, Alyssa Genschaw","doi":"10.4153/s0008414x2300069x","DOIUrl":"https://doi.org/10.4153/s0008414x2300069x","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"41 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136262052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.4153/s0008414x23000706
Leon Barth, Deniz Kus
We use the dual functional realization of loop algebras to study the prime irreducible objects in the Hernandez-Leclerc category for the quantum affine algebra associated to $mathfrak{sl}_{n+1}$. When the HL category is realized as a monoidal categorification of a cluster algebra cite{HL10,HL13}, these representations correspond precisely to the cluster variables and the frozen variables are minimal affinizations. For any height function, we determine the classical decomposition of these representations with respect to the Hopf subalgebra $mathbf{U}_q(mathfrak{sl}_{n+1})$ and describe the graded multiplicities of their graded limits in terms of lattice points of convex polytopes. Combined with cite{BCMo15} we obtain the graded decomposition of stable prime Demazure modules in level two integrable highest weight representations of the corresponding affine Lie algebra.
{"title":"PRIME REPRESENTATIONS IN THE HERNANDEZ-LECLERC CATEGORY: CLASSICAL DECOMPOSITIONS","authors":"Leon Barth, Deniz Kus","doi":"10.4153/s0008414x23000706","DOIUrl":"https://doi.org/10.4153/s0008414x23000706","url":null,"abstract":"We use the dual functional realization of loop algebras to study the prime irreducible objects in the Hernandez-Leclerc category for the quantum affine algebra associated to $mathfrak{sl}_{n+1}$. When the HL category is realized as a monoidal categorification of a cluster algebra cite{HL10,HL13}, these representations correspond precisely to the cluster variables and the frozen variables are minimal affinizations. For any height function, we determine the classical decomposition of these representations with respect to the Hopf subalgebra $mathbf{U}_q(mathfrak{sl}_{n+1})$ and describe the graded multiplicities of their graded limits in terms of lattice points of convex polytopes. Combined with cite{BCMo15} we obtain the graded decomposition of stable prime Demazure modules in level two integrable highest weight representations of the corresponding affine Lie algebra.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136233441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.4153/s0008414x23000688
NICHOLAS DAVIDSON, JONATHAN R. KUJAWA, ROBERT MUTH
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"WEBS OF TYPE P","authors":"NICHOLAS DAVIDSON, JONATHAN R. KUJAWA, ROBERT MUTH","doi":"10.4153/s0008414x23000688","DOIUrl":"https://doi.org/10.4153/s0008414x23000688","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"135 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136262225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.4153/s0008414x23000676
YUSUKE AIKAWA, RYOKICHI TANAKA, TAKUYA YAMAUCHI
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"ISOGENY GRAPHS ON SUPERSPECIAL ABELIAN VARIETIES: EIGENVALUES AND CONNECTION TO BRUHAT-TITS BUILDINGS","authors":"YUSUKE AIKAWA, RYOKICHI TANAKA, TAKUYA YAMAUCHI","doi":"10.4153/s0008414x23000676","DOIUrl":"https://doi.org/10.4153/s0008414x23000676","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135616818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.4153/s0008414x23000639
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"CJM volume 75 issue 6 Cover and Front matter","authors":"","doi":"10.4153/s0008414x23000639","DOIUrl":"https://doi.org/10.4153/s0008414x23000639","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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