Pub Date : 2021-10-13DOI: 10.26493/1855-3974.2724.83d
A. Herman, Roghayeh Maleki
In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative association scheme corresponds to a standard integral table algebra with integral multiplicities that satisfies all of the parameter restrictions known to hold for association schemes. For each rank and involution type, we generate an algebraic variety for which any suitable integral solution corresponds to a standard integral table algebra with integral multiplicities, and then try to find the smallest suitable solution. Our main results show the eigenvalues of commutative association schemes of rank 4 and nonsymmetric commutative association schemes of rank 5 will always be cyclotomic. In the rank 5 cases these our conclusions rely on calculations done by computer for Gr"obner bases or for bases of rational vector spaces spanned by polynomials. We give several examples of feasible parameter sets for small symmetric association schemes of rank 5 that have noncyclotomic eigenvalues.
{"title":"The search for small association schemes with noncyclotomic eigenvalues","authors":"A. Herman, Roghayeh Maleki","doi":"10.26493/1855-3974.2724.83d","DOIUrl":"https://doi.org/10.26493/1855-3974.2724.83d","url":null,"abstract":"In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative association scheme corresponds to a standard integral table algebra with integral multiplicities that satisfies all of the parameter restrictions known to hold for association schemes. For each rank and involution type, we generate an algebraic variety for which any suitable integral solution corresponds to a standard integral table algebra with integral multiplicities, and then try to find the smallest suitable solution. Our main results show the eigenvalues of commutative association schemes of rank 4 and nonsymmetric commutative association schemes of rank 5 will always be cyclotomic. In the rank 5 cases these our conclusions rely on calculations done by computer for Gr\"obner bases or for bases of rational vector spaces spanned by polynomials. We give several examples of feasible parameter sets for small symmetric association schemes of rank 5 that have noncyclotomic eigenvalues.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76719788","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 : 2021-09-23DOI: 10.26493/1855-3974.2730.6ac
H. Boden, Matthew Shimoda
Given a knot, we develop methods for finding the braid representative that minimizes the number of simple walks. Such braids lead to an efficient method for computing the colored Jones polynomial of $K$, following an approach developed by Armond and implemented by Hajij and Levitt. We use this method to compute the colored Jones polynomial in closed form for the knots $5_2, 6_1,$ and $7_2$. The set of simple walks can change under reflection, rotation, and cyclic permutation of the braid, and we prove an invariance property which relates the simple walks of a braid to those of its reflection under cyclic permutation. We study the growth rate of the number of simple walks for families of torus knots. Finally, we present a table of braid words that minimize the number of simple walks for knots up to 13 crossings.
{"title":"Braid representatives minimizing the number of simple walks","authors":"H. Boden, Matthew Shimoda","doi":"10.26493/1855-3974.2730.6ac","DOIUrl":"https://doi.org/10.26493/1855-3974.2730.6ac","url":null,"abstract":"Given a knot, we develop methods for finding the braid representative that minimizes the number of simple walks. Such braids lead to an efficient method for computing the colored Jones polynomial of $K$, following an approach developed by Armond and implemented by Hajij and Levitt. We use this method to compute the colored Jones polynomial in closed form for the knots $5_2, 6_1,$ and $7_2$. The set of simple walks can change under reflection, rotation, and cyclic permutation of the braid, and we prove an invariance property which relates the simple walks of a braid to those of its reflection under cyclic permutation. We study the growth rate of the number of simple walks for families of torus knots. Finally, we present a table of braid words that minimize the number of simple walks for knots up to 13 crossings.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91467630","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 : 2021-09-13DOI: 10.26493/1855-3974.2638.d0b
B. Alspach, Aditya Joshi
We examine the chromatic index of generalized truncations of graphs and multigraphs.
研究了图和多图的广义截断的色指数。
{"title":"On the chromatic index of generalized truncations","authors":"B. Alspach, Aditya Joshi","doi":"10.26493/1855-3974.2638.d0b","DOIUrl":"https://doi.org/10.26493/1855-3974.2638.d0b","url":null,"abstract":"We examine the chromatic index of generalized truncations of graphs and multigraphs.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86866133","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 : 2021-09-11DOI: 10.26493/1855-3974.2694.56a
F. Volta, A. Lucchini
The M"{o}bius function of the subgroup lettice of a finite group $G$ has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let $A$ be a subgroup of the automorphism group $rm{Aut}(G)$ of a finite group $G$ and denote by $mathcal C_A(G)$ the set of $A$-conjugacy classes of subgroups of $G.$ For $Hleq G$ let $[H]_A~=~{~H^a ~mid ~ain ~A}$ be the element of $mathcal C_A(G)$ containing $H.$ We may define an ordering in $mathcal C_A(G)$ in the following way: $[H]_Aleq [K]_A$ if $H^aleq K$ for some $ain A$. We consider the M"{o}bius function $mu_A$ of the corresponding poset and analyse its properties and possible applications.
{"title":"The A-Möbius function of a finite group","authors":"F. Volta, A. Lucchini","doi":"10.26493/1855-3974.2694.56a","DOIUrl":"https://doi.org/10.26493/1855-3974.2694.56a","url":null,"abstract":"The M\"{o}bius function of the subgroup lettice of a finite group $G$ has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let $A$ be a subgroup of the automorphism group $rm{Aut}(G)$ of a finite group $G$ and denote by $mathcal C_A(G)$ the set of $A$-conjugacy classes of subgroups of $G.$ For $Hleq G$ let $[H]_A~=~{~H^a ~mid ~ain ~A}$ be the element of $mathcal C_A(G)$ containing $H.$ We may define an ordering in $mathcal C_A(G)$ in the following way: $[H]_Aleq [K]_A$ if $H^aleq K$ for some $ain A$. We consider the M\"{o}bius function $mu_A$ of the corresponding poset and analyse its properties and possible applications.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89030317","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 : 2021-09-07DOI: 10.26493/1855-3974.2433.ba6
Lowell Abrams, Daniel C. Slilaty
A spherical quadrangulation is an embedding of a graph G in the sphere in which each facial boundary walk has length four. Vertices that are not of degree four in G are called curvature vertices . In this paper we classify all spherical quadrangulations with n -fold rotational symmetry ( n ≥ 3 ) that have minimum degree 3 and the least possible number of curvature vertices, and describe all such spherical quadrangulations in terms of nets of quadrilaterals. The description reveals that such rotationally symmetric quadrangulations necessarily also have dihedral symmetry.
{"title":"Characterization of a family of rotationally symmetric spherical quadrangulations","authors":"Lowell Abrams, Daniel C. Slilaty","doi":"10.26493/1855-3974.2433.ba6","DOIUrl":"https://doi.org/10.26493/1855-3974.2433.ba6","url":null,"abstract":"A spherical quadrangulation is an embedding of a graph G in the sphere in which each facial boundary walk has length four. Vertices that are not of degree four in G are called curvature vertices . In this paper we classify all spherical quadrangulations with n -fold rotational symmetry ( n ≥ 3 ) that have minimum degree 3 and the least possible number of curvature vertices, and describe all such spherical quadrangulations in terms of nets of quadrilaterals. The description reveals that such rotationally symmetric quadrangulations necessarily also have dihedral symmetry.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90240426","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 : 2021-09-01DOI: 10.26493/1855-3974.2669.58c
Paul M. Terwilliger
This paper concerns the positive part U q + of the quantum group U q ( sl ^ 2 ) . The algebra U q + has a presentation involving two generators that satisfy the cubic q -Serre relations. We recently introduced an algebra U q + called the alternating central extension of U q + . We presented U q + by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of U q + that involves a small subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of U q + .
{"title":"A compact presentation for the alternating central extension of the positive part of Uq(sl^2)","authors":"Paul M. Terwilliger","doi":"10.26493/1855-3974.2669.58c","DOIUrl":"https://doi.org/10.26493/1855-3974.2669.58c","url":null,"abstract":"This paper concerns the positive part U q + of the quantum group U q ( sl ^ 2 ) . The algebra U q + has a presentation involving two generators that satisfy the cubic q -Serre relations. We recently introduced an algebra U q + called the alternating central extension of U q + . We presented U q + by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of U q + that involves a small subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of U q + .","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80441184","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 : 2021-08-19DOI: 10.26493/1855-3974.2584.68d
Ian Gleason, I. Hubard
{"title":"The antiprism of an abstract polytope","authors":"Ian Gleason, I. Hubard","doi":"10.26493/1855-3974.2584.68d","DOIUrl":"https://doi.org/10.26493/1855-3974.2584.68d","url":null,"abstract":"","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84586447","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 : 2021-08-17DOI: 10.26493/1855-3974.2053.c7b
Q. Yan, Xian'an Jin
Kotzig showed that every connected 4 -regular plane graph has an A -trail—an Eulerian circuit that turns either left or right at each vertex. However, this statement is not true for Eulerian plane graphs and determining if an Eulerian plane graph has an A -trail is NP-hard. The aim of this paper is to give a characterization of Eulerian embedded graphs having an A -trail. Andersen et al. showed the existence of orthogonal pairs of A -trails in checkerboard colourable 4 -regular graphs embedded on the plane, torus and projective plane. A problem posed in their paper is to characterize Eulerian embedded graphs (not necessarily checkerboard colourable) which contain two orthogonal A -trails. In this article, we solve this problem in terms of twisted duals. Several related results are also obtained.
{"title":"A-trails of embedded graphs and twisted duals","authors":"Q. Yan, Xian'an Jin","doi":"10.26493/1855-3974.2053.c7b","DOIUrl":"https://doi.org/10.26493/1855-3974.2053.c7b","url":null,"abstract":"Kotzig showed that every connected 4 -regular plane graph has an A -trail—an Eulerian circuit that turns either left or right at each vertex. However, this statement is not true for Eulerian plane graphs and determining if an Eulerian plane graph has an A -trail is NP-hard. The aim of this paper is to give a characterization of Eulerian embedded graphs having an A -trail. Andersen et al. showed the existence of orthogonal pairs of A -trails in checkerboard colourable 4 -regular graphs embedded on the plane, torus and projective plane. A problem posed in their paper is to characterize Eulerian embedded graphs (not necessarily checkerboard colourable) which contain two orthogonal A -trails. In this article, we solve this problem in terms of twisted duals. Several related results are also obtained.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85232957","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 : 2021-08-17DOI: 10.26493/1855-3974.2698.6fe
Karen Meagher, M. N. Shirazi, B. Stevens
A $(k,ell)$-partition is a set partition which has $ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $left| P_{i} cap Q_{j} right|geq t$. In this paper we prove a version of the ErdH{o}s-Ko-Rado theorem for partially $2$-intersecting $(k,ell)$-partitions. In particular, we show for $ell$ sufficiently large, the set of all $(k,ell)$-partitions in which a block contains a fixed pair is the largest set of 2-partially intersecting $(k,ell)$-partitions. For for $k=3$, we show this result holds for all $ell$.
{"title":"An extension of the Erdős-Ko-Rado theorem to uniform set partitions","authors":"Karen Meagher, M. N. Shirazi, B. Stevens","doi":"10.26493/1855-3974.2698.6fe","DOIUrl":"https://doi.org/10.26493/1855-3974.2698.6fe","url":null,"abstract":"A $(k,ell)$-partition is a set partition which has $ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $left| P_{i} cap Q_{j} right|geq t$. In this paper we prove a version of the ErdH{o}s-Ko-Rado theorem for partially $2$-intersecting $(k,ell)$-partitions. In particular, we show for $ell$ sufficiently large, the set of all $(k,ell)$-partitions in which a block contains a fixed pair is the largest set of 2-partially intersecting $(k,ell)$-partitions. For for $k=3$, we show this result holds for all $ell$.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86189656","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 : 2021-08-17DOI: 10.26493/1855-3974.2257.6de
Jay Zimmerman, Coy L. May
{"title":"Maximal order group actions on Riemann surfaces","authors":"Jay Zimmerman, Coy L. May","doi":"10.26493/1855-3974.2257.6de","DOIUrl":"https://doi.org/10.26493/1855-3974.2257.6de","url":null,"abstract":"","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86942481","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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