A Rabbit Hole between Topology and Geometry

<table>The toroidal graph <svg height="15.25" id="M467" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 52.375 15.25" width="52.375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M764 650l-7 -26q-61 -10 -91.5 -22.5t-76.5 -48.5l-235 -184q104 -152 208 -276q31 -37 54.5 -48.5t68.5 -16.5l-7 -28h-156q-138 170 -213 284q-19 29 -33.5 35t-33.5 -7l-29 -173q-15 -75 -5.5 -90.5t74.5 -20.5l-5 -28h-260l7 28q58 5 74 21.5t30 89.5l70 378
q12 68 2 83t-72 22l6 28h257l-7 -28q-62 -7 -76 -21.5t-27 -83.5l-32 -172q29 11 78 46q112 84 217 179q18 16 23.5 25.5t-1 14.5t-26.5 8l-30 4l5 28h249z" id="x1D43E"></path></g>
<g transform="matrix(.012,-0,0,-.012,13.25,15.187)"><path d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" id="x34"></path></g>
<g transform="matrix(.017,-0,0,-.017,23.362,11.112)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g><g transform="matrix(.017,-0,0,-.017,37.114,11.112)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g><g transform="matrix(.017,-0,0,-.017,45.274,11.112)"><path d="M391 364q0 -28 -20.5 -53.5t-50 -43.5t-70 -34t-73.5 -25t-68 -17v-29q0 -48 20 -81t65 -33q73 0 157 76l16 -23q-107 -113 -221 -113q-21 0 -40.5 6.5t-39 22t-31.5 47t-12 75.5q0 69 33.5 139.5t95.5 114.5q77 55 143 55q42 0 69 -24.5t27 -59.5zM313 350
q0 26 -14.5 40.5t-37.5 14.5q-32 0 -44 -7q-82 -46 -104 -175q200 49 200 127z" id="x1D452"></path></g>
</svg> and its dual <svg height="15.25" id="M468" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 52.375 15.25" width="52.375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M764 650l-7 -26q-61 -10 -91.5 -22.5t-76.5 -48.5l-235 -184q104 -152 208 -276q31 -37 54.5 -48.5t68.5 -16.5l-7 -28h-156q-138 170 -213 284q-19 29 -33.5 35t-33.5 -7l-29 -173q-15 -75 -5.5 -90.5t74.5 -20.5l-5 -28h-260l7 28q58 5 74 21.5t30 89.5l70 378
q12 68 2 83t-72 22l6 28h257l-7 -28q-62 -7 -76 -21.5t-27 -83.5l-32 -172q29 11 78 46q112 84 217 179q18 16 23.5 25.5t-1 14.5t-26.5 8l-30 4l5 28h249z" id="x1D43E"></path></g>
<g transform="matrix(.012,-0,0,-.012,13.25,15.187)"><path d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" id="x34"></path></g>
<g transform="matrix(.017,-0,0,-.017,23.362,11.112)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g><g transform="matrix(.017,-0,0,-.017,37.114,11.112)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g><g transform="matrix(.017,-0,0,-.017,45.274,11.112)"><path d="M391 364q0 -28 -20.5 -53.5t-50 -43.5t-70 -34t-73.5 -25t-68 -17v-29q0 -48 20 -81t65 -33q73 0 157 76l16 -23q-107 -113 -221 -113q-21 0 -40.5 6.5t-39 22t-31.5 47t-12 75.5q0 69 33.5 139.5t95.5 114.5q77 55 143 55q42 0 69 -24.5t27 -59.5zM313 350
q0 26 -14.5 40.5t-37.5 14.5q-32 0 -44 -7q-82 -46 -104 -175q200 49 200 127z" id="x1D452"></path></g>
</svg>.</table>

International Scholarly Research Notices

A Rabbit Hole between Topology and Geometry

Figure 8