{"title":"结理论和聚类代数 III：Posets","authors":"Véronique Bazier-Matte, Ralf Schiffler","doi":"arxiv-2409.11287","DOIUrl":null,"url":null,"abstract":"In previous work, we associated a module $T(i)$ to every segment $i$ of a\nlink diagram $K$ and showed that there is a poset isomorphism between the\nsubmodules of $T(i)$ and the Kauffman states of $K$ relative to $i$. In this\npaper, we show that the posets are distributive lattices and give explicit\ndescriptions of the join irreducibles in both posets. We also prove that the\nsubposet of join irreducible Kauffman states is isomorphic to the poset of the\ncoefficient quiver of $T(i)$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Knot theory and cluster algebra III: Posets\",\"authors\":\"Véronique Bazier-Matte, Ralf Schiffler\",\"doi\":\"arxiv-2409.11287\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In previous work, we associated a module $T(i)$ to every segment $i$ of a\\nlink diagram $K$ and showed that there is a poset isomorphism between the\\nsubmodules of $T(i)$ and the Kauffman states of $K$ relative to $i$. In this\\npaper, we show that the posets are distributive lattices and give explicit\\ndescriptions of the join irreducibles in both posets. We also prove that the\\nsubposet of join irreducible Kauffman states is isomorphic to the poset of the\\ncoefficient quiver of $T(i)$.\",\"PeriodicalId\":501038,\"journal\":{\"name\":\"arXiv - MATH - Representation Theory\",\"volume\":\"13 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Representation Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.11287\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Representation Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11287","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
In previous work, we associated a module $T(i)$ to every segment $i$ of a
link diagram $K$ and showed that there is a poset isomorphism between the
submodules of $T(i)$ and the Kauffman states of $K$ relative to $i$. In this
paper, we show that the posets are distributive lattices and give explicit
descriptions of the join irreducibles in both posets. We also prove that the
subposet of join irreducible Kauffman states is isomorphic to the poset of the
coefficient quiver of $T(i)$.
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