{"title":"论{\\it量子}的几何基础结的A-多项式","authors":"Dmitry Galakhov, Alexei Morozov","doi":"arxiv-2408.08181","DOIUrl":null,"url":null,"abstract":"A simple geometric way is suggested to derive the Ward identities in the\nChern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots.\nIn quasi-classical limit it is closely related to the well publicized\naugmentation theory and contact geometry. Quantization allows to present it in\nmuch simpler terms, what could make these techniques available to a broader\naudience. To avoid overloading of the presentation, only the case of the\ncolored Jones polynomial for the trefoil knot is considered, though various\ngeneralizations are straightforward. Restriction to solely Jones polynomials\n(rather than full HOMFLY-PT) is related to a serious simplification, provided\nby the use of Kauffman calculus. Going beyond looks realistic, however it\nremains a problem, both challenging and promising.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"3 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On geometric bases for {\\\\it quantum} A-polynomials of knots\",\"authors\":\"Dmitry Galakhov, Alexei Morozov\",\"doi\":\"arxiv-2408.08181\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A simple geometric way is suggested to derive the Ward identities in the\\nChern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots.\\nIn quasi-classical limit it is closely related to the well publicized\\naugmentation theory and contact geometry. Quantization allows to present it in\\nmuch simpler terms, what could make these techniques available to a broader\\naudience. To avoid overloading of the presentation, only the case of the\\ncolored Jones polynomial for the trefoil knot is considered, though various\\ngeneralizations are straightforward. Restriction to solely Jones polynomials\\n(rather than full HOMFLY-PT) is related to a serious simplification, provided\\nby the use of Kauffman calculus. Going beyond looks realistic, however it\\nremains a problem, both challenging and promising.\",\"PeriodicalId\":501317,\"journal\":{\"name\":\"arXiv - MATH - Quantum Algebra\",\"volume\":\"3 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Quantum Algebra\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.08181\",\"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 - Quantum Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.08181","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On geometric bases for {\it quantum} A-polynomials of knots
A simple geometric way is suggested to derive the Ward identities in the
Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots.
In quasi-classical limit it is closely related to the well publicized
augmentation theory and contact geometry. Quantization allows to present it in
much simpler terms, what could make these techniques available to a broader
audience. To avoid overloading of the presentation, only the case of the
colored Jones polynomial for the trefoil knot is considered, though various
generalizations are straightforward. Restriction to solely Jones polynomials
(rather than full HOMFLY-PT) is related to a serious simplification, provided
by the use of Kauffman calculus. Going beyond looks realistic, however it
remains a problem, both challenging and promising.