{"id":165,"date":"2020-04-16T03:35:03","date_gmt":"2020-04-16T03:35:03","guid":{"rendered":"http:\/\/blogs.rollins.edu\/graphembeddings\/?page_id=165"},"modified":"2020-05-05T10:21:29","modified_gmt":"2020-05-05T10:21:29","slug":"k22-k33","status":"publish","type":"page","link":"https:\/\/blogs.rollins.edu\/graphembeddings\/embedding-complete-bipartite-graphs-with-cayley-maps\/k22-k33\/","title":{"rendered":"K2,2"},"content":{"rendered":"\n\n\n

According to Ringel, the optimal genus for \"K_{2,2}\" is \"g=0\". We were able to obtain the same genus using a Cayley Map with group \"Z_4\" and rotation \"\rho=(1\" \"3).\" <\/p>\n\n\n\n

\"\"
\"C_M(Z_4,(1\" \"3))\"
The face set is \"\mathcal{F}=\{(3),(1)\}\", where \"F_0\" and \"F_1\" are both 4-gons. Using the Euler Characteristic Formula, \"\chi=2\", so \"g=0\".<\/figcaption><\/figure><\/div>\n\n\n\n
\"\"
Cayley Map \"C_M(Z_4,(1\" \"3))\" embedded into a sphere (\"g=0\"). The blue square uses permutation \"\lambda_0=(3)\" and forms the top half of the sphere, and the yellow square uses permutation \"\lambda_1=(1)\" and forms the bottom half.<\/figcaption><\/figure><\/div>\n\n\n\n
<\/div>\n\n\n\n

Using the Dihedral Group \"D_2\" yielded the same genus, but with one distinct face appearing twice. The reason for this difference is that each element in the rotation using \"Z_4\" is followed by its inverse, whereas in the rotation for \"D_2\", each element is its own inverse.<\/p>\n\n\n\n

\"\"
Cayley Map \"C_M(D_2,(f\" \\hspace{1} \"rf))\".
The face set is \"\mathcal{F}=\{(f\" \\hspace{1}\"rf)\}\", where \"F_0\" is two 4-gons. By the Euler Characteristic Formula, \"\chi=2\", so \"g=0\".<\/figcaption><\/figure><\/div>\n\n\n\n
\"\"
Cayley Map \"C_M(D_2,(f\" \"rf))\" embedded into a sphere (\"g=0\"). The blue and green squares use permutation \"(f\" \"rf)\", and each face forms half the sphere.<\/figcaption><\/figure><\/div>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

According to Ringel, the optimal genus for $K_{2,2}$ is $g=0$. We were able to obtain the same genus using a Cayley Map with group $Z_4$ and rotation $\\rho=(1$ $3).$ Using the Dihedral Group $D_2$ yielded the same genus, but with one distinct face appearing twice. The reason for this difference is that each element in […]<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":97,"menu_order":6,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-165","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/165","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/comments?post=165"}],"version-history":[{"count":30,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/165\/revisions"}],"predecessor-version":[{"id":425,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/165\/revisions\/425"}],"up":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/97"}],"wp:attachment":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/media?parent=165"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}