{"id":10,"date":"2010-05-12T16:14:18","date_gmt":"2010-05-12T16:14:18","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=10"},"modified":"2010-08-25T08:59:24","modified_gmt":"2010-08-25T08:59:24","slug":"a-new-ansatz-for-extremal-laurent-polynomials","status":"publish","type":"post","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=10","title":{"rendered":"A new ansatz for extremal Laurent polynomials"},"content":{"rendered":"<p>This post describes a new method for generating Laurent polynomials in 3 variables.\u00a0 Many of these Laurent polynomials are extremal or of low ramification, and they include the extremal Laurent polynomials mirror to 15 of the 17 minimal Fano 3-folds. We call this method the <em>Minkowksi ansatz<\/em>.<\/p>\n<p>Let <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/> be a 3-dimensional reflexive polytope.\u00a0 We will construct a Laurent polynomial with Newton polytope equal to <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/>, or in other words we will explain how to assign a coefficient to each integer point in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/>.\u00a0 This goes as follows.<\/p>\n<p><strong>Lattice Minkowski sums<\/strong><\/p>\n<p>We say that a polygon <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/f09\/f09564c9ca56850d4cd6b3319e541aee-ffffff-000000-0.png' alt='Q' title='Q' class='latex' \/> is the <em>lattice Minkowski sum<\/em> of polygons <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/e1e\/e1e1d3d40573127e9ee0480caf1283d6-ffffff-000000-0.png' alt='R' title='R' class='latex' \/> and <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/5db\/5dbc98dcc983a70728bd082d1a47546e-ffffff-000000-0.png' alt='S' title='S' class='latex' \/> if and only if both:<\/p>\n<ul>\n<li><img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/4b2\/4b27423e8e9f7cabf50b65e40ae79d9a-ffffff-000000-0.png' alt='Q = R + S' title='Q = R + S' class='latex' \/>, so that <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/f09\/f09564c9ca56850d4cd6b3319e541aee-ffffff-000000-0.png' alt='Q' title='Q' class='latex' \/> is the Minkowski sum of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/e1e\/e1e1d3d40573127e9ee0480caf1283d6-ffffff-000000-0.png' alt='R' title='R' class='latex' \/> and <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/5db\/5dbc98dcc983a70728bd082d1a47546e-ffffff-000000-0.png' alt='S' title='S' class='latex' \/> as usual<\/li>\n<li>the integer lattice in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/f09\/f09564c9ca56850d4cd6b3319e541aee-ffffff-000000-0.png' alt='Q' title='Q' class='latex' \/> is the sum of the integer lattices in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/e1e\/e1e1d3d40573127e9ee0480caf1283d6-ffffff-000000-0.png' alt='R' title='R' class='latex' \/> and <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/5db\/5dbc98dcc983a70728bd082d1a47546e-ffffff-000000-0.png' alt='S' title='S' class='latex' \/>.<\/li>\n<\/ul>\n<p>Note that any of the the polygons <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/eb7\/eb7e942dd4efdc3bbdd39a592ec2c8ba-ffffff-000000-0.png' alt='Q, R, S' title='Q, R, S' class='latex' \/> here are allowed to be  degenerate.<\/p>\n<p>Examples:\u00a0 here are two lattice Minkowksi decompositions <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/c3d\/c3dedb2e64eb7dfaf4a0536876ef7fe9-ffffff-000000-0.png' alt='P = Q+R' title='P = Q+R' class='latex' \/> of a hexagon:<\/p>\n<div id=\"attachment_17\" style=\"width: 310px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/hexagondecomp1.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-17\" class=\"size-medium wp-image-17\" title=\"hexagondecomp1\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/hexagondecomp1-300x108.png\" alt=\"\" width=\"300\" height=\"108\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/hexagondecomp1-300x108.png 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/hexagondecomp1.png 484w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><p id=\"caption-attachment-17\" class=\"wp-caption-text\">A lattice Minkowksi decomposition<\/p><\/div>\n<div id=\"attachment_18\" style=\"width: 310px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/hexagondecomp2.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-18\" class=\"size-medium wp-image-18\" title=\"hexagondecomp2\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/hexagondecomp2-300x108.png\" alt=\"\" width=\"300\" height=\"108\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/hexagondecomp2-300x108.png 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/hexagondecomp2.png 484w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><p id=\"caption-attachment-18\" class=\"wp-caption-text\">Another lattice Minkowski decomposition<\/p><\/div>\n<p>Note that the same lattice polygon can have more than one lattice Minkowski decomposition.\u00a0 Note also that the first decomposition here is not a complete decomposition into lattice-Minkowksi-irreducible pieces, because the square <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/f09\/f09564c9ca56850d4cd6b3319e541aee-ffffff-000000-0.png' alt='Q' title='Q' class='latex' \/> can be further decomposed as the sum of a vertical and a horizontal line.<\/p>\n<div id=\"attachment_23\" style=\"width: 310px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/notadecomp.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-23\" class=\"size-medium wp-image-23\" title=\"notadecomp\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/notadecomp-300x108.png\" alt=\"\" width=\"300\" height=\"108\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/notadecomp-300x108.png 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/notadecomp.png 484w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><p id=\"caption-attachment-23\" class=\"wp-caption-text\">This is not a lattice Minkowski decomposition<\/p><\/div>\n<p>The example above is not a lattice Minkowski decomposition, because the lattice in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/> is not the sum of the lattices in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/f09\/f09564c9ca56850d4cd6b3319e541aee-ffffff-000000-0.png' alt='Q' title='Q' class='latex' \/> and <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/e1e\/e1e1d3d40573127e9ee0480caf1283d6-ffffff-000000-0.png' alt='R' title='R' class='latex' \/>.\u00a0 In fact <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/> is lattice Minkowski irreducible.<\/p>\n<p><strong>Decompose the facets into irreducible pieces<\/strong><\/p>\n<p>There are 4319 3-dimensional reflexive polytopes.\u00a0 These polytopes contain a total of 344 distinct facets, where we regard two facets as the same if and only if they differ by a lattice-preserving automorphism.\u00a0 Of these facets, 79 are lattice Minkowski irreducible.\u00a0 These 79 facets are also the non-degenerate polygons which occur when the 344 total facets are decomposed into lattice Minkowksi irreducible pieces.\u00a0 Of those 79 facets, exactly 8 contain no interior lattice points.\u00a0 Those 8 triangles, which we call <em>admissible triangles<\/em> are all of type <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/f93\/f93809ae14fb28ef6dbe11c99529c51b-ffffff-000000-0.png' alt='A_n' title='A_n' class='latex' \/>:<\/p>\n<div id=\"attachment_31\" style=\"width: 310px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/Antriangles1.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-31\" class=\"size-medium wp-image-31\" title=\"Antriangles\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/Antriangles1-300x117.png\" alt=\"\" width=\"300\" height=\"117\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/Antriangles1-300x117.png 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/Antriangles1.png 484w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><p id=\"caption-attachment-31\" class=\"wp-caption-text\">The eight admissible triangles<\/p><\/div>\n<p>In other words, the cones over these triangles give affine toric varieties that are transverse <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/f93\/f93809ae14fb28ef6dbe11c99529c51b-ffffff-000000-0.png' alt='A_n' title='A_n' class='latex' \/> singularities, for <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/c87\/c8794dc135e76032c3f1572f966d1604-ffffff-000000-0.png' alt='1 \\leq n \\leq 8' title='1 \\leq n \\leq 8' class='latex' \/>.<\/p>\n<p><strong>The ansatz<\/strong><\/p>\n<p>Given a 3-dimensional reflexive polytope <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/>, we construct a possibly-empty list of Laurent polynomials as follows.\u00a0 For each facet <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/800\/800618943025315f869e4e1f09471012-ffffff-000000-0.png' alt='F' title='F' class='latex' \/> of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/>, decompose <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/800\/800618943025315f869e4e1f09471012-ffffff-000000-0.png' alt='F' title='F' class='latex' \/> into lattice-Minkowksi-irreducible pieces in all possible ways.\u00a0 Discard any such decomposition of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/800\/800618943025315f869e4e1f09471012-ffffff-000000-0.png' alt='F' title='F' class='latex' \/> which contains a non-degenerate polygon that is not an admissible triangle.\u00a0 Any remaining decomposition of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/800\/800618943025315f869e4e1f09471012-ffffff-000000-0.png' alt='F' title='F' class='latex' \/> will consist of line segments and admissible triangles.\u00a0 To this decomposition we associate a Laurent polynomial which is the product of certain basic Laurent polynomials corresponding to line segments and\u00a0 to admissible triangles.\u00a0 The basic Laurent polynomials for admissible triangles are:<\/p>\n<div id=\"attachment_34\" style=\"width: 310px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/basicansatz.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-34\" class=\"size-medium wp-image-34\" title=\"basicansatz\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/basicansatz-300x122.png\" alt=\"\" width=\"300\" height=\"122\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/basicansatz-300x122.png 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/basicansatz.png 484w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><p id=\"caption-attachment-34\" class=\"wp-caption-text\">The coefficients of the basic Laurent polynomials  for admissible triangles.<\/p><\/div>\n<p>and so on for the remaining admissible triangles.\u00a0 The basic Laurent polynomials for line segments are:<\/p>\n<div id=\"attachment_38\" style=\"width: 198px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/basicansatz2.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-38\" class=\"size-medium wp-image-38    \" title=\"basicansatz2\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/basicansatz2-235x300.png\" alt=\"\" width=\"188\" height=\"240\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/basicansatz2-235x300.png 235w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/05\/basicansatz2.png 380w\" sizes=\"auto, (max-width: 188px) 100vw, 188px\" \/><\/a><p id=\"caption-attachment-38\" class=\"wp-caption-text\">The coefficients of the basic Laurent polynomials for line segments<\/p><\/div>\n<p>and so on for other line segments.<\/p>\n<p>So now, for each facet <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/800\/800618943025315f869e4e1f09471012-ffffff-000000-0.png' alt='F' title='F' class='latex' \/> of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/>, we have a list <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/2ab\/2aba37b4d5b91d7475afd4c48dc6cd83-ffffff-000000-0.png' alt='L_F' title='L_F' class='latex' \/> of Laurent polynomials; this list will be empty if <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/800\/800618943025315f869e4e1f09471012-ffffff-000000-0.png' alt='F' title='F' class='latex' \/> cannot be written as a lattice Minkowksi sum of line segments and admissible triangles. In other words for each facet <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/800\/800618943025315f869e4e1f09471012-ffffff-000000-0.png' alt='F' title='F' class='latex' \/> we have list of ways of assigning coefficients to each integer point in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/800\/800618943025315f869e4e1f09471012-ffffff-000000-0.png' alt='F' title='F' class='latex' \/>.\u00a0 We seek a list of Laurent polynomials with Newton polytope equal to <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/>, or in other words a list of ways of assigning coefficients to each integer point in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/>.\u00a0 This is produced by assigning the coefficient zero to the origin (which is the only interior point of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/>) and then assigning coefficients to the integer points on facets of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/> as specified in the facet lists (but amalgamated in all possible ways, so if there are <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/d66\/d6667ffd4c60ec1fc999e4d4f98ca139-ffffff-000000-0.png' alt='n_F' title='n_F' class='latex' \/> elements in the list for facet <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/800\/800618943025315f869e4e1f09471012-ffffff-000000-0.png' alt='F' title='F' class='latex' \/> then the number of elements in the list for <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/> is <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/913\/913b237afa99ef99aa65471dd32dac37-ffffff-000000-0.png' alt='\\prod_{\\text{facets F}} n_F' title='\\prod_{\\text{facets F}} n_F' class='latex' \/>).<\/p>\n<p><strong>Points to Note<\/strong><\/p>\n<ul>\n<li>This ansatz almost generalizes the earlier recipes given by Pryjzalkowski and Galkin, but differs a little because of the difference between Minkowski decomposition and lattice Minkowksi decomposition.<\/li>\n<li>Altman has studied the deformation theory of affine toric varieties and discovered a close connection with Minkowski decompositions.\u00a0 Since we expect to find the local system associated to an extremal Laurent polynomial <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/8fa\/8fa14cdd754f91cc6554c9e71929cce7-ffffff-000000-0.png' alt='f' title='f' class='latex' \/> as a piece of the quantum cohomology local system associated to a smoothing of the Newton polytope of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/8fa\/8fa14cdd754f91cc6554c9e71929cce7-ffffff-000000-0.png' alt='f' title='f' class='latex' \/>, this is encouraging.\u00a0 But note that Minkowski decomposition and lattice Minkowksi decomposition are not the same.<\/li>\n<li>We suspect that if <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/> is a 3-dimensional reflexive polytope containing a facet with no admissible lattice Minkowski decompositions then the toric variety corresponding to <em><img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/44c\/44c29edb103a2872f519ad0c9a0fdaaa-ffffff-000000-0.png' alt='P' title='P' class='latex' \/> does not smooth<\/em>.\u00a0 More on this later.<\/li>\n<li>This ansatz also fits well with\u00a0 <a href=\"http:\/\/www.ams.org\/mathscinet\/search\/publications.html?pg1=IID&amp;s1=225645\">Kouchnirenko&#8217;s criterion for a Laurent polynomial to be degenerate.<\/a><\/li>\n<\/ul>\n<p>(I learned this last point from Hiroshi.)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post describes a new method for generating Laurent polynomials in 3 variables.\u00a0 Many of these Laurent polynomials are extremal or of low ramification, and they include the extremal Laurent polynomials mirror to 15 of the 17 minimal Fano 3-folds. We call this method the Minkowksi ansatz. Let be a 3-dimensional reflexive polytope.\u00a0 We will [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[3],"class_list":["post-10","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-theory"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/10","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10"}],"version-history":[{"count":39,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/10\/revisions"}],"predecessor-version":[{"id":90,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/10\/revisions\/90"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}