{"id":233,"date":"2020-04-16T03:54:51","date_gmt":"2020-04-16T03:54:51","guid":{"rendered":"http:\/\/blogs.rollins.edu\/graphembeddings\/?page_id=233"},"modified":"2020-05-05T10:06:24","modified_gmt":"2020-05-05T10:06:24","slug":"k33","status":"publish","type":"page","link":"https:\/\/blogs.rollins.edu\/graphembeddings\/embedding-complete-bipartite-graphs-with-cayley-maps\/k33\/","title":{"rendered":"K3,3"},"content":{"rendered":"\n\n\n

The optimal genus for \"K_{3,3}\" is \"g=1\". We were able to obtain the same genus using a Cayley Map with group \"Z_6\" and rotation \"\rho=(1\" \"3\" \"5)\".<\/p>\n\n\n\n

\"This
\"C_M(Z_6,(\"1 3 5\"))\"
The face set is \"\mathcal{F}=\{(\"3 5\"),(\"1\")\}\", where \"F_0\" and \"F_1\" are both 6-gons, and there are two \"F_0\" faces. Using the Euler Characteristic Formula, \"\chi=0\", so g=1.<\/figcaption><\/figure><\/div>\n\n\n\n
\"\"
Cayley Map \"C_M(K_6,(1\" \"3\" \"5))\" embedded into a torus (\"g=1\"). When the vertices are connected, the three hexagons (right)<\/em> form a torus (left)<\/em>. The green and yellow faces use the permutation \"\lambda_0=(3\" \"5)\", and the blue face uses the permutation \"\lambda_1=(1)\".<\/figcaption><\/figure><\/div>\n\n\n\n


Because the rotation for \"Z_6\" only has three elements, there is only one other possible rotation, \"\rho=(3\" 1 \"5)\". This rotation produces the same genus as \"\rho=(1\" \"3\" \"5)\", but with different faces: 

*** QuickLaTeX cannot compile formula:\n\mathcal{F}={(3\n\n*** Error message:\nMissing } inserted.\r\nleading text: $\mathcal{F}={(3$\r\n\n<\/pre> 
*** QuickLaTeX cannot compile formula:\n1),(5)}\n\n*** Error message:\nExtra }, or forgotten $.\r\nleading text: $1),(5)}\r\n\n<\/pre>. Using \"D_3\" instead of \"Z_6\" produces the same genus as well, but \"\mathcal{F}={\rho}\".<\/p>\n\n\n\n
\"This
\"C_M(D_3,(\"frfr^2f))\\mathcal{F}=\\{(f
*** QuickLaTeX cannot compile formula:\n\hspace{1}\n\n*** Error message:\nIllegal unit of measure (pt inserted).\r\nleading text: $\hspace{1}\r\n\n<\/pre>rf
*** QuickLaTeX cannot compile formula:\n\hspace{1}\n\n*** Error message:\nIllegal unit of measure (pt inserted).\r\nleading text: $\hspace{1}\r\n\n<\/pre>r^2f)\\}\", and\"\\lambda_0\"produces three 6-gons. Using the Euler Characteristic Formula,\"\\chi=0\", so\"g=1$.<\/figcaption><\/figure><\/div>\n","protected":false},"excerpt":{"rendered":"
The optimal genus for $K_{3,3}$ is $g=1$. We were able to obtain the same genus using a Cayley Map with group $Z_6$ and rotation $\\rho=(1$ $3$ $5)$. Because the rotation for $Z_6$ only has three elements, there is only one other possible rotation, $\\rho=(3$ 1 $5)$. This rotation produces the same genus as $\\rho=(1$ $3$ […]<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":97,"menu_order":7,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-233","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/233","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=233"}],"version-history":[{"count":25,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/233\/revisions"}],"predecessor-version":[{"id":424,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/233\/revisions\/424"}],"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=233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}