An Evolutionary Formulation of the Crossing Number Problem

<table>The coding convention for genome <svg height="16.35" id="M61" style="vertical-align:-4.15506pt" version="1.1" viewbox="0 0 57.950001 16.35" width="57.950001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M546 430q-14 -33 -37.5 -140t-38.5 -209q-29 -195 -156 -289q-29 -21 -70 -37t-78 -16q-62 0 -102.5 37t-40.5 69q0 27 26 48q19 15 25 -3q13 -43 46.5 -69t87.5 -26q55 0 83 21q29 22 50 65t43 140q14 64 29 163h-2q-55 -81 -149 -152q-59 -44 -102 -44q-23 0 -43.5 30
t-20.5 85q0 74 32.5 145.5t84.5 117.5q40 35 100 58.5t117 23.5q21 0 65 -8q30 -6 44 -6zM456 386q-40 16 -81 16q-29 0 -64 -13q-42 -17 -85 -97q-19 -36 -31.5 -86t-12.5 -86q0 -64 30 -64q20 0 52.5 21t60.5 52q74 81 95 123q26 62 36 134z" id="x1D454"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.738,16.138)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" id="x33"></path></g>
<g transform="matrix(.017,-0,0,-.017,20.788,11.113)"><path d="M448 1h-83q-118 0 -201.5 74.5t-83.5 179.5t83.5 179.5t201.5 74.5h83v-50h-84q-87 0 -150.5 -51.5t-73.5 -127.5h308v-50h-308q10 -76 73.5 -127.5t150.5 -51.5h84v-50z" id="x2208"></path></g><g transform="matrix(.017,-0,0,-.017,34.471,11.113)"><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,-.012,47.675,15.188)"><path d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8
l-26 -65h-233z" id="x35"></path></g>
<g transform="matrix(.017,-0,0,-.017,54.012,11.113)"><path d="M114 412q23 0 39 -16t16 -41t-16 -41.5t-40 -16.5t-39.5 16.5t-15.5 41.5t15.5 41t40.5 16zM95 130q31 0 61 -30t30 -78q0 -53 -38 -88t-92 -52l-11 30q76 32 76 85q0 26 -17.5 43.5t-44.5 23.5q-13 3 -13 24q0 19 14.5 30.5t34.5 11.5z" id="x3B"></path></g>
</svg> coding conventions, R: right-going edge, L: left-going edge, for the case where  <svg height="16.4125" id="M62" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 43.637501 16.4125" width="43.637501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path d="M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74
t114 146l32 145l74 26h11q-52 -199 -80 -347q-8 -39 6 -39q7 0 32 17.5t47 39.5z" id="x1D462"></path></g>
<g transform="matrix(.012,-0,0,-.012,15.088,16.212)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
<g transform="matrix(.017,-0,0,-.017,18.963,12.138)"><path d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" id="x2C"></path></g><g transform="matrix(.017,-0,0,-.017,25.66,12.138)"><path d="M457 332q0 -81 -41 -161.5t-105.5 -131.5t-129.5 -51q-46 0 -79 30.5t-33 86.5q0 18 7 51q18 95 46 187q6 19 6 33q0 7 -7 7q-24 0 -78 -64l-20 23q32 48 73 77t78 29q33 0 33 -46q0 -38 -10 -70q-28 -92 -39 -152q-7 -38 -7 -57q0 -78 69 -78q68 0 118 75.5t50 194.5
q0 43 -12 57q-11 12 -11 24q0 19 15 35.5t34 16.5q18 0 30.5 -34.5t12.5 -81.5z" id="x1D463"></path></g>
<g transform="matrix(.012,-0,0,-.012,33.825,16.212)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
<g transform="matrix(.017,-0,0,-.017,37.7,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg> forms a vertical line, either R or L can be used. </table>

Discrete Dynamics in Nature and Society

An Evolutionary Formulation of the Crossing Number Problem

Figure 5