{"id":3469,"date":"2024-11-03T10:53:00","date_gmt":"2024-11-03T02:53:00","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=3469"},"modified":"2025-02-26T11:33:05","modified_gmt":"2025-02-26T03:33:05","slug":"alexander_polynomial_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2024\/11\/03\/alexander_polynomial_mark\/","title":{"rendered":"\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u6ce8\u8bb0"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u6700\u8fd1\u5728\u5b66\u4e60TopoXu\u8001\u5e08\u7684\u62d3\u6251\u6570\u636e\u5206\u6790\u8bfe\u7a0b, \u8bb8\u591a\u8bfe\u7a0b\u5185\u5bb9\u53c2\u8003\u4e86\u848b\u4f2f\u9a79\u8001\u5e08\u7684\u300a\u7ef3\u5708\u7684\u6570\u5b66\u300b\u4e00\u4e66. \u4f46\u9057\u61be\u7684\u662f, \u4e66\u4e0a\u5bf9\u4e8e\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u53ca\u5eb7\u5a01\u65b9\u6cd5\u4ecb\u7ecd\u7684\u7bc7\u5e45\u5f88\u5c11, \u800c\u8fd9\u4e00\u90e8\u5206\u5374\u53c8\u662f\u7ebd\u7ed3\u7406\u8bba\u91cc\u6781\u5176\u91cd\u8981\u7684\u4e00\u90e8\u5206, \u56e0\u6b64\u7279\u4ee5\u672c\u6587\u8bb0\u5f55\u4e00\u4e9b\u5173\u4e8e\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u53ca\u5eb7\u5a01\u65b9\u6cd5\u7684\u8981\u70b9.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/www.bilibili.com\/video\/BV1HaigeJE2q\/?spm_id_from=333.999.0.0&#038;vd_source=71042933bac98e100d097571a3b97b3a\">\u62d3\u6251\u6570\u636e\u5206\u6790lecture4_part1<\/a><br \/>\n2. <a href=\"https:\/\/www.bilibili.com\/video\/BV1CbigenEjg?spm_id_from=333.788.videopod.sections&#038;vd_source=71042933bac98e100d097571a3b97b3a\">\u62d3\u6251\u6570\u636e\u5206\u6790lecture4_part2<\/a><br \/>\n3. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/662505840\">\u7ebd\u7ed3\u7406\u8bba\u548c\u743c\u65af\u591a\u9879\u5f0f<\/a><br \/>\n4. \u59dc\u4f2f\u9a79. \u7ef3\u5708\u7684\u6570\u5b66[M]. \u6e56\u5357\u654e\u80b2\u51fa\u7248\u793e, 1991.<\/p>\n<p>\u5386\u53f2\u4e0a\u7b2c\u4e00\u4e2a\u7ebd\u7ed3\u7684\u62d3\u6251\u4e0d\u53d8\u91cf\u662f\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0fAlexander Polynomial, \u5b83\u662f\u7531James Alexander\u4e8e1928\u5e74\u5254\u9664. \u5b83\u53ef\u4ee5\u7528\u4e8e\u533a\u5206\u4e00\u90e8\u5206\u7ebd\u7ed3. \u53ea\u8981\u4e24\u4e2a\u7ebd\u7ed3$L$\u548c$L&#8217;$\u7684\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u4e0d\u76f8\u540c, \u5219\u5b83\u4eec\u4e00\u5b9a\u662f\u5f7c\u6b64\u4e0d\u540c\u75d5\u7684. \u4f46\u5982\u679c\u4e24\u4e2a\u7ebd\u7ed3\u7684\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u8ba1\u7b97\u7ed3\u679c\u76f8\u540c\u7684\u8bdd, \u6211\u4eec\u5374\u5e76\u4e0d\u80fd\u8bf4\u5b83\u4eec\u5c31\u662f\u5f7c\u6b64\u540c\u75d5\u7684\u7ebd\u7ed3. \u6211\u4eec\u5c06\u5728\u672c\u6587\u7684\u672b\u5c3e, \u7ed9\u51fa\u4e00\u4e2a\u4e0d\u5e73\u51e1\u7684\u4f8b\u5b50, \u5b83\u7684\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u4e0e\u5e73\u51e1\u7ebd\u7ed3\u7684\u76f8\u540c, \u4f46\u5b83\u4e00\u5b9a\u4e0d\u662f\u5e73\u51e1\u7ebd\u7ed3.<\/p>\n<p><strong>1. \u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u7684\u5b9a\u4e49<\/strong><\/p>\n<p>\u7ebd\u7ed3\u7684\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u5176\u672c\u8d28\u662f\u5355\u53d8\u91cf\u6d1b\u6717\u591a\u9879\u5f0fLaurant Polynominal. \u6d1b\u6717\u591a\u9879\u5f0f\u4e0e\u666e\u901a\u591a\u9879\u5f0f\u7684\u4e0d\u540c\u70b9\u5728\u4e8e\u5b83\u5141\u8bb8\u53d8\u91cf\u7684\u8d1f\u6574\u6570\u6b21\u5e42\u7684\u51fa\u73b0. \u5373$L(t) = t + 1 + t^{-1}$\u662f\u6d1b\u6717\u591a\u9879\u5f0f\u7684\u4e00\u4e2a\u4f8b\u5b50. \u4e8b\u5b9e\u4e0a, \u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u4e0e\u743c\u65af\u591a\u9879\u5f0f\u7684\u672c\u8d28\u90fd\u662f\u5355\u53d8\u91cf\u6d1b\u6717\u591a\u9879\u5f0f, \u800cHOMFLY\u591a\u9879\u5f0f\u5219\u662f\u53cc\u53d8\u91cf\u6d1b\u6717\u591a\u9879\u5f0f.<\/p>\n<p>\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u7684\u8ba1\u7b97\u65b9\u5f0f\u6709\u4e24\u79cd, \u7b2c\u4e00\u79cd\u662f\u7531James Alexander\u672c\u4eba\u5728\u539f\u59cb\u8bba\u6587\u63d0\u51fa\u7684, \u7528\u7684\u65b9\u6cd5\u662f\u8ba1\u7b97\u6295\u5f71\u56fe\u7d22\u51b3\u5b9a\u7684\u77e9\u9635\u7684\u884c\u5217\u5f0f. \u7b2c\u4e8c\u79cd\u662f\u7531John H.Conway\u4e8e1969\u5e74\u63d0\u51fa, \u4ed6\u4f7f\u7528\u4e00\u79cd\u88ab\u79f0\u4f5c\u662fSkein Relation\u7684\u65b9\u5f0f, \u8fd9\u79cd\u65b9\u5f0f\u63a8\u5e7f\u6027\u5f88\u5f3a, \u53ef\u4ee5\u7528\u4e8e\u743c\u65af\u591a\u9879\u5f0f\u53ca\u5176HOMFLY\u591a\u9879\u5f0f\u7684\u8ba1\u7b97.<\/p>\n<p>\u4e3a\u4e86\u540e\u9762\u7684\u8868\u8ff0\u65b9\u4fbf, \u6211\u4eec\u628a\u7b2c\u4e00\u79cd\u65b9\u6cd5\u79f0\u4e3a\u4e9a\u5386\u5c71\u5927\u65b9\u6cd5, \u7b2c\u4e8c\u79cd\u65b9\u6cd5\u79f0\u4e3a\u5eb7\u5a01\u65b9\u6cd5. \u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u6709\u7684\u65f6\u5019\u4e5f\u88ab\u79f0\u4f5c\u662f\u4e9a\u5386\u5c71\u5927-\u5eb7\u5a01\u591a\u9879\u5f0fAlexander-Conway Polynominal.<\/p>\n<p><strong>2. \u5eb7\u5a01\u65b9\u6cd5<\/strong><\/p>\n<p>\u5eb7\u5a01\u65b9\u6cd5\u7684\u8ba1\u7b97\u6cd5\u5219\u4e3b\u8981\u9075\u5faa\u4ee5\u4e0b\u4e24\u6761\u65b9\u7a0b:<br \/>\n$\\\\$ Rule 1. $\\bigtriangleup (\\bigcirc) = 1$.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2024\/11\/skein_relation.jpg\" alt=\"\" width=\"177\" height=\"96\" class=\"aligncenter size-full wp-image-3490\" \/><\/p>\n<p>Rule 2. $\\bigtriangleup(L_+) &#8211; \\bigtriangleup(L_-) + (t^{\\frac{1}{2}} &#8211; t^{-\\frac{1}{2}})\\bigtriangleup(L_0) = 0$.<br \/>\n$\\\\$ $\\cdot$ \u6cd5\u5219\u4e00: \u5728\u65b9\u7a0b\u4e2d, \u6240\u6709\u51fa\u73b0\u7684\u5e73\u51e1\u7ebd\u7ed3\u53ca\u5176\u7b49\u4ef7\u6295\u5f71\u56fe\u7684\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u90fd\u662f1.<br \/>\n$\\\\$ $\\cdot$ \u6cd5\u5219\u4e8c: \u62c6\u63a5\u5173\u7cfb\u5f0fSkein Relation, \u4e3b\u8981\u9488\u5bf9\u7ebd\u7ed3\u7684\u5176\u4e2d\u4e00\u4e2a\u4ea4\u53c9\u70b9, \u6b64\u65f6\u7ebd\u7ed3\u53ef\u80fd\u662f$L_+$\u6216\u8005$L_-$\u4e4b\u4e00(\u6709\u7684\u65f6\u5019\u4e5f\u53ef\u80fd\u662f$L_0$), \u7136\u540e\u5199\u51fa\u8be5\u4ea4\u53c9\u70b9\u7684\u53e6\u5916\u4e00\u79cd\u6295\u5f71\u56fe, \u4ee5\u53ca\u53bb\u9664\u8be5\u4ea4\u53c9\u70b9\u7684\u6295\u5f71\u56fe, \u5e76\u4e58\u4ee5\u76f8\u5173\u7684\u7cfb\u6570(\u53ef\u80fd\u5305\u542b\u672a\u77e5\u6570$t$) \u7528\u4ee5\u7ec4\u6210\u65b9\u7a0b. \u5373$$\\bigtriangleup(L_+) &#8211; \\bigtriangleup(L_-) + (t^{\\frac{1}{2}} &#8211; t^{-\\frac{1}{2}})\\bigtriangleup(L_0) = 0.$$\u5eb7\u5a01\u65b9\u6cd5\u5c31\u662f\u53cd\u590d\u5229\u7528\u4ee5\u4e0a\u4e24\u4e2a\u6cd5\u5219, \u9010\u4e2a\u89e3\u5f00\u4ea4\u53c9\u70b9, \u5e76\u4e14\u5229\u7528R-Moves\u548c\u5df2\u77e5\u7ebd\u7ed3\u6216\u94fe\u73af\u7684\u8ba1\u7b97\u7ed3\u679c, \u8fed\u4ee3\u5f97\u51fa\u76ee\u6807\u7ebd\u7ed3\u6216\u94fe\u73af\u7684\u8ba1\u7b97\u7ed3\u679c. \u6ce8\u610f: \u6b64\u5904\u6211\u4eec\u5229\u7528\u4e86\u4e00\u4e2a\u4e8b\u5b9e, \u5c31\u662f\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u5728R-Moves\u4e0b\u4fdd\u6301\u4e0d\u53d8, \u5176\u8bc1\u660e\u53ef\u4ee5\u53c2\u770b\u4e9a\u5386\u5c71\u5927\u7684\u539f\u6587.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u5728\u5b66\u4e60TopoXu\u8001\u5e08\u7684\u62d3\u6251\u6570\u636e\u5206\u6790\u8bfe\u7a0b, \u8bb8\u591a\u8bfe\u7a0b\u5185\u5bb9\u53c2\u8003\u4e86\u848b\u4f2f\u9a79\u8001\u5e08\u7684\u300a\u7ef3\u5708\u7684\u6570\u5b66\u300b\u4e00\u4e66. \u4f46\u9057\u61be\u7684\u662f &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2024\/11\/03\/alexander_polynomial_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u4e9a\u5386\u5c71\u5927\u591a\u9879\u5f0f\u6ce8\u8bb0<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[23,28],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3469"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=3469"}],"version-history":[{"count":21,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3469\/revisions"}],"predecessor-version":[{"id":3641,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3469\/revisions\/3641"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=3469"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=3469"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=3469"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}