{"id":958,"date":"2025-09-24T21:50:58","date_gmt":"2025-09-24T19:50:58","guid":{"rendered":"http:\/\/blog.mathyuan.com\/?p=958"},"modified":"2025-10-04T10:02:26","modified_gmt":"2025-10-04T08:02:26","slug":"%e6%89%ad%e7%bb%93floer%e5%90%8c%e8%b0%83%e5%8c%ba%e5%88%86%e3%80%8c%e5%88%87%e7%89%87%e5%9c%86%e7%9b%98%e3%80%8d","status":"publish","type":"post","link":"https:\/\/blog.mathyuan.com\/?p=958","title":{"rendered":"\u626d\u7ed3Floer\u540c\u8c03\u533a\u5206\u300c\u5207\u7247\u5706\u76d8\u300d"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u672c\u7bc7\u4e3b\u8981\u6284\u4e00\u4e0bJuhasz\u548cZemke\u57282020\u5e74\u7684\u6587\u7ae0<a href=\"https:\/\/arxiv.org\/abs\/1804.09589\">Distinguishing Slice Disks Using Knot Floer Homology<\/a>\u3002\u73b0\u6b63\u6301\u7eed\u66f4\u65b0\u4e2d\uff01<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u7b80\u4ecb<\/h3>\n\n\n\n<p>\u6211\u4eec\u5e0c\u671b\u505a\u5230\u5982\u4e0b\u4e8b\u60c5\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5206\u7c7b$D^4$\u4e2d\u7684\u626d\u7ed3$K$\u7684\u5149\u6ed1\u5207\u7247\u5706\u76d8\uff1b<\/li>\n\n\n\n<li>\u8003\u8651Fox\u63d0\u51fa\u7684\u300c\u5207\u7247\u5706\u76d8\u731c\u60f3\u300d\uff0c\u5373\u662f\u300cEvery slice knot is ribbon\u300d\uff0c\u5176\u53cd\u5411\u7531Fox\u5df2\u7ecf\u8bc1\u660e\uff0c\u5373\u300cEvery ribbon knot is slice\u300d\uff1b<\/li>\n<\/ul>\n\n\n\n<p>\u5728\u6587\u7ae0\u4e2d\uff0c\u6709\u4e00\u4e9b\u7b49\u4ef7\u5173\u7cfb\u9700\u8981\u6ce8\u610f\uff0c\u4ed6\u4eec\u7684\u5f3a\u5ea6\u4e5f\u8bb8\u4e0d\u540c\uff08\uff1f\uff09<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u300c\u4fdd\u6301$B^4$\u8fb9\u754c\u6052\u540c\u7684\u5916\u56f4\u540c\u75d5\u300d<\/li>\n\n\n\n<li>\u300c\u5fae\u5206\u540c\u80da\u300d$B^4\\rightarrow B^4$<\/li>\n\n\n\n<li>\u300c\u4fdd\u6301\u8fb9\u754c\u6052\u540c\u7684<strong>\u7a33\u5b9a<\/strong>\u5916\u56f4\u540c\u75d5\u300d<\/li>\n\n\n\n<li>\u300c<strong>\u7a33\u5b9a<\/strong>\u5fae\u5206\u540c\u80da\u300d<\/li>\n<\/ol>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u300c\u7a33\u5b9a\u5fae\u5206\u540c\u80da\u300d\u6307\u7684\u662f\uff1a\u82e5\u901a\u8fc7\u4e0e\u540c\u6837\u6570\u91cf\u7684$S^2\\times S^2$\u8fde\u901a\u548c\uff0c\u4e24\u4e2a\u6d41\u5f62\u5982\u679c\u662f\u5fae\u5206\u540c\u80da\u7684\uff0c\u90a3\u4e48\u5c31\u662f\u7a33\u5b9a\u5fae\u5206\u540c\u80da\u7684\u3002<a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/016686418490004X\" data-type=\"link\" data-id=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/016686418490004X\">Gompf\u57281984\u5e74<\/a>\u8bc1\u660e\u4e86\uff1a\u5bf9\u4e8e\u7a33\u5b9a\u62d3\u6251\u540c\u80da\u76842\u4e2a\u95ed\u5149\u6ed14\u7ef4\u6d41\u5f62$M$\u548c$M&#8217;$\uff0c\u90a3\u4e48<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>$M\\# S^2\\tilde{\\times} S^2$\u4e0e$M&#8217;\\# S^2\\tilde{\\times} S^2$\u662f\u7a33\u5b9a\u5fae\u5206\u540c\u80da\u7684\uff1b<\/li>\n\n\n\n<li>\u5982\u679cM\u548cM&#8217;\u90fd\u662f\u53ef\u5b9a\u5411\u7684\uff0c\u90a3\u4e48\u4ed6\u4eec\u672c\u8eab\u5c31\u4e92\u76f8\u7a33\u5b9a\u5fae\u5206\u540c\u80da\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u6240\u4ee5\u300c\u7a33\u5b9a\u5fae\u5206\u540c\u80da\u300d\u5f31\u4e8e\u5fae\u5206\u540c\u80da\uff0c\u4e14\u4e0e\u300c\u7a33\u5b9a\u62d3\u6251\u540c\u80da\u300d\u76f8\u5dee\u4e0d\u5927\u3002<\/p>\n<\/blockquote>\n\n\n\n<p>\u6700\u7ec8\u53ef\u4ee5\u5b66\u4e60\u5230\u4ee5\u4e0b\u7406\u8bba\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u7528knot Floer homology\u6784\u9020slice disk\u7684\u4e0d\u53d8\u91cf [Marengon, Juhasz, 2016]\uff1b<\/li>\n\n\n\n<li>\u5229\u7528link Floer TQFT\u6784\u9020trace\/cotrace\u6620\u5c04\uff1b<\/li>\n\n\n\n<li>\u5728\u4e00\u4e2a\u65e0\u7a77\u591aslice disks\u7684\u65cf\u4e0a\uff0c\u5199\u4e00\u4e9b\u4e0d\u53d8\u91cf\u7684\u516c\u5f0f\u3002\u5229\u7528\u6784\u9020\u7684\u4e0d\u53d8\u91cfand\u516c\u5f0f\uff0c\u53ef\u4ee5\u5728\u300cstable isotopy\u548cstable diffeomorphism\u300d\u7684\u610f\u4e49\u4e0b\uff0c\u533a\u5206\u8fd9\u4e9b\u5207\u7247\u5706\u76d8\u3002\u4e8b\u5b9e\u4e0a\uff0c\u5229\u7528\u67d0\u4e9b\u57fa\u672c\u7fa4\u5c31\u53ef\u4ee5\u5728\u300cstable isotopy\u300d\u7684\u610f\u4e49\u4e0b\u533a\u5206\u4e00\u4e9b\u5207\u7247\u5706\u76d8\uff0c\u4f46\u662f\u8fd9\u4e2a\u7ed3\u679c\u4f3c\u4e4e\u4e0d\u592a\u80fd\u62d3\u5c55\u5230\u300cstable diffeomorphism\u300d\u4e2d\uff1b<\/li>\n\n\n\n<li>\u5bf9\u6bd4\u4e0a\u9762\u6240\u8bf4\u7684\u300c\u57fa\u672c\u7fa4\u65b9\u6cd5\u300d\u548c\u300cHeegaard Floer\u65b9\u6cd5\u300d\uff0c\u8bd5\u56fe\u5bfb\u627e\u57fa\u672c\u7fa4\u548clink cobordism\u7684\u5173\u7cfb\uff1f<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>\u5177\u4f53\u6765\u8bf4\uff0c\u7ed9\u5b9a\u626d\u7ed3$K\\subset S^3$<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$B$\uff1a\u626d\u7ed3$K$\u4e0a\u67d0\u70b9\u7684\u90bb\u57df\uff1b<\/li>\n\n\n\n<li>$d:(S^3,K)\\rightarrow (S^3,K)$\u81ea\u540c\u80da\uff0c\u4f7f\u5f97$d|_B=\\mathrm{id}$\u200b;<\/li>\n<\/ul>\n\n\n\n<p>\u5229\u7528<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>deform-spinning (Litherland, 1979);<\/li>\n\n\n\n<li>twist-spinning (Zeeman, 1965);<\/li>\n\n\n\n<li>roll-spinning (Fox, 1966);<\/li>\n<\/ol>\n\n\n\n<p>\u53ef\u4ee5\u5f97\u5230$-K\\# K$\u7684\u5207\u7247\u5706\u76d8$D_{K,d}$\uff0c\u5e76\u4e14<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u5207\u7247\u5706\u76d8$D_{K,d}$\u7684diffeomorphism type\u90fd\u76f8\u540c\uff0c\u65e0\u8bba$B$\u5982\u4f55\u9009\u53d6 [Prop 3.10];<\/li>\n\n\n\n<li>\u4f46\u662f$D_{K,d}$\u7684isotopy class\u5374\u7531d\u5728\u6620\u5c04\u7c7b\u7fa4$\\mathrm{Diff}(S^3,K,B)$\u7684\u7c7b\u552f\u4e00\u51b3\u5b9a\uff0c\u8fd9\u5c31\u548c$B$\u7684\u9009\u53d6\u6709\u5173\u3002<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>\u540c\u65f6\uff0c\u6211\u4eec\u8003\u8651decorated knot $(K,P)$\u7684\u5207\u7247\u5706\u76d8$D=D_{K,P}$\u3002\u5229\u7528\u8be5\u5207\u7247\u5706\u76d8\uff0c\u53ef\u4ee5\u5f97\u5230\u300c\u5e3d\u5b50\u7248\u672c\u300dHeegaard Floer\u540c\u8c03\u4e2d\u7684\u4e00\u4e2a\u540c\u8c03\u7c7b<br>$$<br>t_{D,P}\\neq 0\\in \\widehat{\\mathrm{HFK}}(K,P)<br>$$<br>\u7136\u540e\u65ad\u8a00\uff1a$t_{D,P}$\u662f\u5207\u7247\u5706\u76d8$D$\u7684\u540c\u75d5\u4e0d\u53d8\u91cf\uff0c\u5e76\u4e14t\u5728D\u548c2-\u626d\u7ed3\u8fde\u901a\u548c\u65f6\u4fdd\u6301\u4e0d\u53d8\u3002\u5982\u679c\u8bb0$V:=\\widehat{\\mathrm{HFK}}(K,P)$\uff0c\u90a3\u4e48\u6709\u5b9a\u74065.1\u8868\u793a\uff1a<br>$$<br>\\widehat{\\mathrm{HFK}}(-K\\# K,P)\\cong V^*\\otimes V\\cong\\mathrm{Hom}(V,V) <br>$$<br>\u5e76\u4e14\u5728\u8fd9\u4e2a\u540c\u6784\u4e0b\uff0c$t_{D,P}$\u6253\u5230\u81ea\u540c\u80da$d$\u8bf1\u5bfc\u7684\u6620\u5c04$d^*\\in\\mathrm{Hom}(V,V)$\u3002\u7279\u522b\u5730\uff0c\u5982\u679c\u81ea\u540c\u80da$d$\u6070\u597d\u662f\u8bf1\u5bfcroll-spinning\u7684\u90a3\u4e2a\u5fae\u5206\u540c\u80da\uff0c\u90a3\u4e48<br>$$<br>\\LARGE{t_{D_{k,r^l},P}}<br>$$<br>\u80fd\u591f\u5728<strong>\u7a33\u5b9a\u540c\u75d5<\/strong>\u4e0b\u533a\u5206\u8fd9\u4e00\u65cf\u5207\u7247\u5706\u76d8${D_{k,r^l}:l\\in\\mathbb{Z}}$\u3002\u8fd9\u4e00\u7ed3\u679c\u5f53\u7136\u662f\u5f3a\u5927\u7684\uff0c\u56e0\u4e3a\u5bf9\u65e0\u7a77\u591a\u4e2a\u626d\u7ed3$K$\uff0c\u8fd9\u4e9b\u5207\u7247\u5706\u76d8\u7684\u8865\u7a7a\u95f4\u65f6\u5fae\u5206\u540c\u80da\u7684\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u4e5f\u7279\u522b\u5730\u5f97\u5230\u4e86decorated knot $(K,P)$\uff0c\u4f7f\u5f97\u5b83\u7684\u67d0\u4e24\u4e2a\u5207\u7247\u5706\u76d8\u5177\u6709\u4e0d\u540c\u7684\u4e0d\u53d8\u91cft\u3002<\/p>\n\n\n\n<p>\u4e00\u822c\u5730\uff0c\u8fd8\u53ef\u4ee5\u8003\u8651\u4e24\u4e2a\u626d\u7ed3$K$, $K&#8217;$\u786e\u5b9a\u7684\u5207\u7247\u5706\u76d8\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$\\mathcal{C}:(K,P)\\rightarrow (K&#8217;,P&#8217;)$\uff1a\u4e00\u4e2adecorated concordance\uff1b<\/li>\n\n\n\n<li>$D_\\mathcal{C}$\uff1a$-K\\# K$\u7684\u5207\u7247\u5706\u76d8\uff1b<\/li>\n<\/ul>\n\n\n\n<p>\u90a3\u4e48\u7c7b\u4f3c\u4e0e\u4e0a\u9762\u7684\u5b9a\u7406\uff0c\u6211\u4eec\u4f1a\u6709<br>$$<br>\\widehat{\\mathrm{HFK}}(-K\\# K&#8217;,P)\\cong V^*\\otimes V&#8217;\\cong\\mathrm{Hom}(V,V&#8217;)<br>$$<br>\u4f46\u662f\uff0c\u8fd9\u4e2a\u540c\u6784\u5e76\u975e\u81ea\u7136\u540c\u6784\uff1a\u5b83\u7531\u4e00\u4e2a\u300c\u8fde\u901a\u548c\u7403\u9762$\u300dS$\u51b3\u5b9a\u3002\u65e0\u8bba\u5982\u4f55\uff0c\u9009\u53d6\u4e00\u4e2a\u8fd9\u6837\u7684\u540c\u6784\uff0c$t_{D_\\mathcal{C}}$\u5c31\u4f1a\u6253\u5230$F_\\mathcal{C}\\in\\mathrm{Hom}(V,V&#8217;)$\u4e2d\uff0c\u8fd9\u4e2a\u6620\u5c04\u7684rank\u4f1a\u7ed9\u6211\u4eec\u5e26\u6765\u5207\u7247\u5706\u76d8$D$\u7684\u4e0d\u53d8\u91cf\uff08\u4e0e\u8fde\u901a\u548c\u7403\u9762$S$\u6709\u5173\uff09\uff0c\u53ef\u4ee5\u8bb0\u4e3a<br>$$<br>\\mathrm{rk}_S(D):=\\mathrm{rk}(F_\\mathcal{C}).<br>$$<br>\u6709\u4e00\u4e9b\u7ed3\u8bba<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5982\u82e5$D_\\mathcal{C}=D_K\\natural D_{K&#8217;}$\uff0c\u5219$\\mathrm{rk}_S(D)=1$\uff1b<\/li>\n\n\n\n<li>\u5982\u82e5$K\\neq K&#8217;$\u5747\u4e0d\u662f\u7d20\u7684\uff0c\u5219\u300c\u8fde\u901a\u548c\u7403\u9762\u300d$S$\u552f\u4e00\uff0c\u6b64\u65f6$\\mathrm{rk}(t_{D,P})$\u6210\u4e3aD\u7684<strong>\u7a33\u5b9a\u5fae\u5206\u540c\u80da<\/strong>\u4e0d\u53d8\u91cf\uff1b<\/li>\n<\/ul>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u4e0b\u9762\u4e00\u4e9b\u7ed3\u8bba\u53ef\u80fd\u66f4\u76f4\u63a5\u5730\u548cHeegaard Floer\u540c\u8c03\u6709\u5173\u3002<\/p>\n<\/blockquote>\n\n\n\n<p>\u5982\u679c$D$\u662f$-K\\# K&#8217;$\u7684\u5207\u7247\u5706\u76d8\uff0c\u8bb0$\\mathrm{rk}(D):=\\mathrm{t_{D,P}}$\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>t\u662f\u4fdd\u6301Alexander\u548cMaslov\u5206\u6b21\u7684\uff0c\u4e8e\u662frank(D)\u53ef\u4ee5\u62c6\u6210$\\mathrm{rk}_j(D,i)$\u548c$\\mathrm{rk}(D,i)$\u7684\u5206\u6b21\u7248\u672c\u3002<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$H\\subset D^4$\uff1a\u4e0e\u5207\u7247\u5706\u76d8$D$\u6a2a\u622a\u76f8\u4ea4\u7684properly embedded\u4e09\u7ef4\u7403\uff0c\u6ee1\u8db3$\\partial H=S$\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u90a3\u4e48<\/p>\n\n\n\n<p><strong>Theorem 6.7<\/strong> \u7528trivial tangles $(D^3,D^1)$\u300c\u76d6\u4e0a (cap off)\u300d$(H,D\\cap H)$\uff0c\u53ef\u4ee5\u5f97\u5230\u94fe\u73af$L\\subset S^3$\uff0c\u8fd9\u4e2a\u94fe\u73af\u6ee1\u8db3$\\mathrm{rk}_S(D)\\leq\\mathrm{rk}(\\widehat{HFK}(L))$\u3002\u7279\u522b\u5730\uff0c\u5982\u679c\u8fd9\u4e2a\u94fe\u73af\u662f\u4e00\u4e2a\u626d\u7ed3\uff0c\u90a3\u4e48 <br>$$ <br>\\max_{i\\in\\mathbb{Z}}{\\mathrm{rk}_S(D,i)\\neq 0}\\leq g(L).<br>$$<\/p>\n\n\n\n<p>\u5728$t_{D,P}$\u8fd9\u4e2a\u4e0d\u53d8\u91cf\u4e0a\uff0c\u8fd8\u6709\u4e00\u4e2a\u7ed3\u8bba\uff1a<\/p>\n\n\n\n<p><strong>Theorem<\/strong> (Kim, 2010). \u5bf9\u4e8e\u4efb\u610f\u626d\u7ed3$K$\uff0c\u5b58\u5728\u4ece$K$\u5230$K&#8217;$\u7684concordance $C$\uff0c\u7531\u67d0\u79cd$K$\u7684satellite\u5f97\u5230\u3002\u5229\u7528\u8fd9\u4e2aconcordance\uff0c\u53ef\u4ee5\u6709<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$K$, $K&#8217;$, $C:K\\rightarrow K&#8217;$ concordance\uff1b<\/li>\n\n\n\n<li>$P$, $P&#8217;$: \u5206\u522b\u662f\u4e24\u4e2a\u626d\u7ed3\u7684decoration\uff1b<\/li>\n\n\n\n<li>$\\sigma$\uff1aconcordance $C$\u4e0a\u4e0eP, P&#8217;\u5747\u76f8\u5bb9\u7684decoration\uff1b<\/li>\n\n\n\n<li>$\\mathcal{C}:=(C,\\sigma)$\uff1b<\/li>\n<\/ul>\n\n\n\n<p>\u82e5$D$\u548c$D&#8217;$\u90fd\u5206\u522b\u662f$K$\u548c$K&#8217;$\u7684\u5207\u7247\u5706\u76d8\uff0c\u5e76\u4e14\u4ed6\u4eec\u7684$t$\u4e0d\u540c\uff0c\u90a3\u4e48<br>$$<br>t_{C\\cup D,P&#8217;}\\neq t_{C\\cup D&#8217;,P&#8217;}<br>$$<\/p>\n\n\n\n<p>\u8fd9\u662f\u56e0\u4e3a$\\mathrm{LHS}=F_\\mathcal{C}(t_{D,P})$\uff0c\u800c$\\mathrm{RHS}=F_\\mathcal{C}(t_{D&#8217;,P&#8217;})$\uff0c\u4e14concordance\u6620\u5c04\u662f\u5355\u5c04\u3002\u8fd9\u4e2a\u7ed3\u8bba\u4e8b\u5b9e\u4e0a\u544a\u8bc9\u6211\u4eec\uff1a\u5982\u679c$t$\u80fd\u533a\u5206\u300c\u53ef\u80fdcomposite\u626d\u7ed3\u300d$K$\u7684\u5207\u7247\u5706\u76d8$(D,P)$\u548c$(D&#8217;,P&#8217;)$\uff0c\u90a3\u4e48\u5b83\u5c31\u80fd\u5728<strong>\u7a33\u5b9a\u540c\u75d5<\/strong>\u610f\u4e49\u4e0b\uff0c\u533a\u5206\u300c\u7d20\u626d\u7ed3$K&#8217;$\u7684\u5207\u7247\u5706\u76d8\u300d$C\\cup D$\u4e0e$C\\cup D&#8217;$\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n \n","protected":false},"excerpt":{"rendered":"<p>Knot Floer\u540c\u8c03\u4f55\u4ee5\u533a\u5206\u300c\u5207\u7247\u5706\u76d8\u300d <a class=\"more-link\" href=\"https:\/\/blog.mathyuan.com\/?p=958\">\u7ee7\u7eed\u9605\u8bfb <span class=\"screen-reader-text\">  \u626d\u7ed3Floer\u540c\u8c03\u533a\u5206\u300c\u5207\u7247\u5706\u76d8\u300d<\/span><span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[17,8],"class_list":["post-958","post","type-post","status-publish","format-standard","hentry","category-notes","tag-17","tag-8"],"_links":{"self":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/958","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=958"}],"version-history":[{"count":30,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/958\/revisions"}],"predecessor-version":[{"id":1025,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/958\/revisions\/1025"}],"wp:attachment":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=958"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=958"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=958"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}