10 59: Difference between revisions

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{{Template:Basic Knot Invariants|name=10_59}}
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{{Rolfsen Knot Page|
n = 10 |
k = 59 |
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,2,-10,5,-3,4,-2,6,-9,10,-5,7,-8,9,-6,8,-7/goTop.html |
braid_table = <table cellspacing=0 cellpadding=0 border=0>
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]]</td></tr>
</table> |
braid_crossings = 10 |
braid_width = 5 |
braid_index = 5 |
same_alexander = [[9_40]], [[K11n66]], |
same_jones = [[10_106]], |
khovanov_table = <table border=1>
<tr align=center>
<td width=13.3333%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
</table></td>
<td width=6.66667%>-4</td ><td width=6.66667%>-3</td ><td width=6.66667%>-2</td ><td width=6.66667%>-1</td ><td width=6.66667%>0</td ><td width=6.66667%>1</td ><td width=6.66667%>2</td ><td width=6.66667%>3</td ><td width=6.66667%>4</td ><td width=6.66667%>5</td ><td width=6.66667%>6</td ><td width=13.3333%>&chi;</td></tr>
<tr align=center><td>15</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>
<tr align=center><td>13</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>11</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>3</td></tr>
<tr align=center><td>9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>6</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>-4</td></tr>
<tr align=center><td>7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>6</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
<tr align=center><td>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>6</td><td bgcolor=yellow>6</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>6</td><td bgcolor=yellow>6</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>3</td></tr>
<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-3</td></tr>
<tr align=center><td>-3</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>3</td></tr>
<tr align=center><td>-5</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>-7</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
</table> |
coloured_jones_2 = <math>q^{20}-3 q^{19}+2 q^{18}+6 q^{17}-17 q^{16}+12 q^{15}+22 q^{14}-53 q^{13}+26 q^{12}+56 q^{11}-100 q^{10}+29 q^9+95 q^8-127 q^7+16 q^6+117 q^5-119 q^4-7 q^3+112 q^2-83 q-25+81 q^{-1} -39 q^{-2} -27 q^{-3} +40 q^{-4} -9 q^{-5} -14 q^{-6} +11 q^{-7} -3 q^{-9} + q^{-10} </math> |
coloured_jones_3 = <math>q^{39}-3 q^{38}+2 q^{37}+2 q^{36}-q^{35}-8 q^{34}+9 q^{33}+14 q^{32}-23 q^{31}-27 q^{30}+48 q^{29}+53 q^{28}-85 q^{27}-101 q^{26}+132 q^{25}+175 q^{24}-183 q^{23}-267 q^{22}+211 q^{21}+391 q^{20}-232 q^{19}-504 q^{18}+217 q^{17}+610 q^{16}-178 q^{15}-691 q^{14}+122 q^{13}+730 q^{12}-42 q^{11}-748 q^{10}-27 q^9+711 q^8+118 q^7-665 q^6-182 q^5+573 q^4+255 q^3-481 q^2-286 q+358+307 q^{-1} -245 q^{-2} -291 q^{-3} +138 q^{-4} +251 q^{-5} -56 q^{-6} -191 q^{-7} - q^{-8} +133 q^{-9} +24 q^{-10} -77 q^{-11} -30 q^{-12} +39 q^{-13} +23 q^{-14} -16 q^{-15} -14 q^{-16} +6 q^{-17} +5 q^{-18} -3 q^{-20} + q^{-21} </math> |
coloured_jones_4 = <math>q^{64}-3 q^{63}+2 q^{62}+2 q^{61}-5 q^{60}+8 q^{59}-11 q^{58}+11 q^{57}+4 q^{56}-33 q^{55}+29 q^{54}-13 q^{53}+53 q^{52}-2 q^{51}-150 q^{50}+47 q^{49}+44 q^{48}+229 q^{47}-q^{46}-494 q^{45}-71 q^{44}+180 q^{43}+757 q^{42}+186 q^{41}-1157 q^{40}-612 q^{39}+204 q^{38}+1758 q^{37}+886 q^{36}-1891 q^{35}-1692 q^{34}-274 q^{33}+2903 q^{32}+2192 q^{31}-2188 q^{30}-2929 q^{29}-1351 q^{28}+3594 q^{27}+3638 q^{26}-1811 q^{25}-3696 q^{24}-2626 q^{23}+3524 q^{22}+4612 q^{21}-991 q^{20}-3719 q^{19}-3609 q^{18}+2852 q^{17}+4866 q^{16}-56 q^{15}-3125 q^{14}-4127 q^{13}+1833 q^{12}+4499 q^{11}+836 q^{10}-2120 q^9-4177 q^8+630 q^7+3613 q^6+1555 q^5-855 q^4-3700 q^3-508 q^2+2306 q+1817+367 q^{-1} -2651 q^{-2} -1163 q^{-3} +887 q^{-4} +1437 q^{-5} +1091 q^{-6} -1343 q^{-7} -1095 q^{-8} -105 q^{-9} +677 q^{-10} +1074 q^{-11} -347 q^{-12} -575 q^{-13} -397 q^{-14} +77 q^{-15} +618 q^{-16} +50 q^{-17} -129 q^{-18} -250 q^{-19} -115 q^{-20} +215 q^{-21} +66 q^{-22} +26 q^{-23} -75 q^{-24} -75 q^{-25} +45 q^{-26} +15 q^{-27} +23 q^{-28} -9 q^{-29} -21 q^{-30} +6 q^{-31} +5 q^{-33} -3 q^{-35} + q^{-36} </math> |
coloured_jones_5 = <math>q^{95}-3 q^{94}+2 q^{93}+2 q^{92}-5 q^{91}+4 q^{90}+5 q^{89}-9 q^{88}+q^{87}+4 q^{86}-17 q^{85}+11 q^{84}+32 q^{83}-4 q^{82}-22 q^{81}-41 q^{80}-49 q^{79}+50 q^{78}+155 q^{77}+101 q^{76}-115 q^{75}-315 q^{74}-273 q^{73}+163 q^{72}+669 q^{71}+638 q^{70}-185 q^{69}-1235 q^{68}-1341 q^{67}+41 q^{66}+2021 q^{65}+2581 q^{64}+489 q^{63}-2989 q^{62}-4499 q^{61}-1653 q^{60}+3912 q^{59}+7115 q^{58}+3765 q^{57}-4486 q^{56}-10276 q^{55}-6985 q^{54}+4288 q^{53}+13723 q^{52}+11201 q^{51}-3061 q^{50}-16799 q^{49}-16208 q^{48}+544 q^{47}+19255 q^{46}+21420 q^{45}+2918 q^{44}-20420 q^{43}-26301 q^{42}-7214 q^{41}+20423 q^{40}+30315 q^{39}+11642 q^{38}-19211 q^{37}-33076 q^{36}-15851 q^{35}+17079 q^{34}+34581 q^{33}+19447 q^{32}-14466 q^{31}-34836 q^{30}-22171 q^{29}+11463 q^{28}+34118 q^{27}+24228 q^{26}-8535 q^{25}-32600 q^{24}-25448 q^{23}+5395 q^{22}+30482 q^{21}+26291 q^{20}-2395 q^{19}-27787 q^{18}-26495 q^{17}-905 q^{16}+24567 q^{15}+26408 q^{14}+4043 q^{13}-20727 q^{12}-25537 q^{11}-7347 q^{10}+16367 q^9+24111 q^8+10123 q^7-11590 q^6-21606 q^5-12422 q^4+6642 q^3+18415 q^2+13587 q-2015-14338 q^{-1} -13643 q^{-2} -1920 q^{-3} +10000 q^{-4} +12425 q^{-5} +4685 q^{-6} -5718 q^{-7} -10237 q^{-8} -6122 q^{-9} +2092 q^{-10} +7441 q^{-11} +6285 q^{-12} +556 q^{-13} -4635 q^{-14} -5428 q^{-15} -2035 q^{-16} +2191 q^{-17} +4023 q^{-18} +2550 q^{-19} -507 q^{-20} -2531 q^{-21} -2278 q^{-22} -464 q^{-23} +1262 q^{-24} +1700 q^{-25} +798 q^{-26} -431 q^{-27} -1037 q^{-28} -752 q^{-29} -12 q^{-30} +520 q^{-31} +535 q^{-32} +178 q^{-33} -201 q^{-34} -325 q^{-35} -164 q^{-36} +49 q^{-37} +141 q^{-38} +122 q^{-39} +19 q^{-40} -71 q^{-41} -64 q^{-42} -9 q^{-43} +12 q^{-44} +24 q^{-45} +23 q^{-46} -9 q^{-47} -14 q^{-48} - q^{-49} +5 q^{-52} -3 q^{-54} + q^{-55} </math> |
coloured_jones_6 = <math>q^{132}-3 q^{131}+2 q^{130}+2 q^{129}-5 q^{128}+4 q^{127}+q^{126}+7 q^{125}-19 q^{124}+q^{123}+20 q^{122}-25 q^{121}+16 q^{120}+18 q^{119}+21 q^{118}-73 q^{117}-31 q^{116}+64 q^{115}-44 q^{114}+82 q^{113}+112 q^{112}+42 q^{111}-292 q^{110}-247 q^{109}+90 q^{108}+21 q^{107}+497 q^{106}+606 q^{105}+152 q^{104}-1059 q^{103}-1376 q^{102}-425 q^{101}+220 q^{100}+2204 q^{99}+2916 q^{98}+1258 q^{97}-2865 q^{96}-5443 q^{95}-3951 q^{94}-726 q^{93}+6525 q^{92}+10723 q^{91}+7429 q^{90}-4175 q^{89}-15120 q^{88}-16463 q^{87}-8829 q^{86}+11686 q^{85}+28398 q^{84}+27511 q^{83}+3240 q^{82}-28283 q^{81}-44153 q^{80}-35885 q^{79}+6963 q^{78}+53013 q^{77}+69038 q^{76}+33878 q^{75}-31299 q^{74}-82485 q^{73}-89733 q^{72}-24463 q^{71}+67566 q^{70}+124650 q^{69}+94827 q^{68}-5378 q^{67}-110736 q^{66}-159124 q^{65}-88087 q^{64}+51617 q^{63}+169514 q^{62}+170878 q^{61}+53259 q^{60}-107320 q^{59}-215770 q^{58}-165370 q^{57}+2969 q^{56}+181264 q^{55}+232043 q^{54}+124295 q^{53}-71352 q^{52}-237847 q^{51}-226639 q^{50}-57865 q^{49}+159895 q^{48}+258830 q^{47}+180526 q^{46}-22299 q^{45}-226671 q^{44}-255929 q^{43}-107466 q^{42}+122916 q^{41}+254329 q^{40}+210290 q^{39}+20838 q^{38}-197919 q^{37}-258084 q^{36}-138340 q^{35}+85001 q^{34}+232503 q^{33}+219810 q^{32}+54199 q^{31}-162683 q^{30}-245235 q^{29}-157373 q^{28}+47338 q^{27}+201336 q^{26}+218957 q^{25}+84517 q^{24}-120303 q^{23}-222151 q^{22}-171612 q^{21}+4097 q^{20}+158404 q^{19}+208727 q^{18}+115433 q^{17}-65976 q^{16}-183941 q^{15}-177651 q^{14}-44691 q^{13}+99222 q^{12}+180816 q^{11}+138738 q^{10}-2953 q^9-125398 q^8-163229 q^7-86074 q^6+29317 q^5+128622 q^4+138288 q^3+51737 q^2-53665 q-120386-100518 q^{-1} -31211 q^{-2} +60715 q^{-3} +105510 q^{-4} +76269 q^{-5} +8646 q^{-6} -59455 q^{-7} -79642 q^{-8} -59587 q^{-9} +1761 q^{-10} +53025 q^{-11} +63614 q^{-12} +38668 q^{-13} -6747 q^{-14} -38361 q^{-15} -51131 q^{-16} -26219 q^{-17} +8138 q^{-18} +31017 q^{-19} +34299 q^{-20} +17373 q^{-21} -3803 q^{-22} -24902 q^{-23} -23742 q^{-24} -11308 q^{-25} +4420 q^{-26} +15214 q^{-27} +15812 q^{-28} +9456 q^{-29} -4507 q^{-30} -9950 q^{-31} -10160 q^{-32} -5012 q^{-33} +1515 q^{-34} +6152 q^{-35} +7515 q^{-36} +2307 q^{-37} -919 q^{-38} -3667 q^{-39} -3731 q^{-40} -2189 q^{-41} +469 q^{-42} +2720 q^{-43} +1729 q^{-44} +1151 q^{-45} -279 q^{-46} -1025 q^{-47} -1360 q^{-48} -625 q^{-49} +458 q^{-50} +384 q^{-51} +612 q^{-52} +269 q^{-53} -387 q^{-55} -305 q^{-56} +13 q^{-57} -27 q^{-58} +134 q^{-59} +108 q^{-60} +88 q^{-61} -67 q^{-62} -71 q^{-63} +2 q^{-64} -33 q^{-65} +12 q^{-66} +15 q^{-67} +32 q^{-68} -9 q^{-69} -14 q^{-70} +6 q^{-71} -7 q^{-72} +5 q^{-75} -3 q^{-77} + q^{-78} </math> |
coloured_jones_7 = <math>q^{175}-3 q^{174}+2 q^{173}+2 q^{172}-5 q^{171}+4 q^{170}+q^{169}+3 q^{168}-3 q^{167}-19 q^{166}+17 q^{165}+12 q^{164}-20 q^{163}+12 q^{162}+3 q^{161}+13 q^{160}-18 q^{159}-78 q^{158}+56 q^{157}+67 q^{156}-14 q^{155}+39 q^{154}-35 q^{153}-21 q^{152}-111 q^{151}-247 q^{150}+152 q^{149}+325 q^{148}+266 q^{147}+262 q^{146}-232 q^{145}-506 q^{144}-785 q^{143}-920 q^{142}+363 q^{141}+1475 q^{140}+2037 q^{139}+1839 q^{138}-467 q^{137}-2745 q^{136}-4522 q^{135}-4567 q^{134}-245 q^{133}+5220 q^{132}+9665 q^{131}+10101 q^{130}+2561 q^{129}-8323 q^{128}-18334 q^{127}-21216 q^{126}-9528 q^{125}+10914 q^{124}+32221 q^{123}+41491 q^{122}+25211 q^{121}-10127 q^{120}-51064 q^{119}-74442 q^{118}-56212 q^{117}-608 q^{116}+72262 q^{115}+123499 q^{114}+110369 q^{113}+30341 q^{112}-89468 q^{111}-187950 q^{110}-194369 q^{109}-91202 q^{108}+90580 q^{107}+262071 q^{106}+312183 q^{105}+195053 q^{104}-60419 q^{103}-333639 q^{102}-459859 q^{101}-349449 q^{100}-17771 q^{99}+383829 q^{98}+625054 q^{97}+554478 q^{96}+156798 q^{95}-391708 q^{94}-787693 q^{93}-798981 q^{92}-360028 q^{91}+339456 q^{90}+921759 q^{89}+1060754 q^{88}+620370 q^{87}-216558 q^{86}-1004231 q^{85}-1313097 q^{84}-916642 q^{83}+27372 q^{82}+1017968 q^{81}+1525433 q^{80}+1221199 q^{79}+214803 q^{78}-959249 q^{77}-1677609 q^{76}-1503543 q^{75}-482798 q^{74}+836629 q^{73}+1756491 q^{72}+1737910 q^{71}+750054 q^{70}-668316 q^{69}-1765174 q^{68}-1909362 q^{67}-990367 q^{66}+478970 q^{65}+1715130 q^{64}+2013147 q^{63}+1186804 q^{62}-290062 q^{61}-1624327 q^{60}-2057064 q^{59}-1333116 q^{58}+119079 q^{57}+1512667 q^{56}+2053620 q^{55}+1430994 q^{54}+26504 q^{53}-1393841 q^{52}-2018872 q^{51}-1491657 q^{50}-146548 q^{49}+1278259 q^{48}+1966185 q^{47}+1525924 q^{46}+247682 q^{45}-1165838 q^{44}-1904384 q^{43}-1547603 q^{42}-339934 q^{41}+1054432 q^{40}+1836659 q^{39}+1563270 q^{38}+433405 q^{37}-934402 q^{36}-1760650 q^{35}-1578328 q^{34}-535662 q^{33}+799050 q^{32}+1669468 q^{31}+1588648 q^{30}+650127 q^{29}-638978 q^{28}-1554650 q^{27}-1589201 q^{26}-773601 q^{25}+452900 q^{24}+1406920 q^{23}+1566532 q^{22}+898023 q^{21}-240689 q^{20}-1220620 q^{19}-1510509 q^{18}-1009004 q^{17}+13846 q^{16}+994613 q^{15}+1407490 q^{14}+1089286 q^{13}+214501 q^{12}-735297 q^{11}-1253458 q^{10}-1122016 q^9-421402 q^8+458102 q^7+1047594 q^6+1093289 q^5+586349 q^4-183526 q^3-802844 q^2-999366 q-688534-61652 q^{-1} +538408 q^{-2} +844839 q^{-3} +717559 q^{-4} +254425 q^{-5} -281280 q^{-6} -648043 q^{-7} -673235 q^{-8} -376793 q^{-9} +58547 q^{-10} +434006 q^{-11} +568182 q^{-12} +423484 q^{-13} +108117 q^{-14} -231142 q^{-15} -425527 q^{-16} -402092 q^{-17} -207529 q^{-18} +64525 q^{-19} +272883 q^{-20} +330682 q^{-21} +241488 q^{-22} +51003 q^{-23} -135388 q^{-24} -234540 q^{-25} -223883 q^{-26} -111853 q^{-27} +31530 q^{-28} +137589 q^{-29} +174030 q^{-30} +126180 q^{-31} +32687 q^{-32} -57695 q^{-33} -113998 q^{-34} -109320 q^{-35} -59734 q^{-36} +4273 q^{-37} +59446 q^{-38} +77364 q^{-39} +60427 q^{-40} +23421 q^{-41} -19957 q^{-42} -44848 q^{-43} -46742 q^{-44} -30772 q^{-45} -2624 q^{-46} +19304 q^{-47} +29060 q^{-48} +26577 q^{-49} +11643 q^{-50} -3586 q^{-51} -14239 q^{-52} -18098 q^{-53} -12038 q^{-54} -3445 q^{-55} +4502 q^{-56} +9813 q^{-57} +8779 q^{-58} +5232 q^{-59} +428 q^{-60} -4316 q^{-61} -5140 q^{-62} -4126 q^{-63} -1847 q^{-64} +1118 q^{-65} +2202 q^{-66} +2606 q^{-67} +1934 q^{-68} +94 q^{-69} -822 q^{-70} -1286 q^{-71} -1165 q^{-72} -354 q^{-73} -12 q^{-74} +464 q^{-75} +739 q^{-76} +340 q^{-77} +92 q^{-78} -184 q^{-79} -290 q^{-80} -119 q^{-81} -142 q^{-82} -34 q^{-83} +142 q^{-84} +103 q^{-85} +74 q^{-86} -9 q^{-87} -56 q^{-88} +6 q^{-89} -33 q^{-90} -33 q^{-91} +12 q^{-92} +15 q^{-93} +23 q^{-94} -14 q^{-96} +6 q^{-97} -7 q^{-99} +5 q^{-102} -3 q^{-104} + q^{-105} </math> |
computer_talk =
<table>
<tr valign=top>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[10, 59]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10],
X[14, 8, 15, 7], X[18, 12, 19, 11], X[20, 15, 1, 16],
X[16, 19, 17, 20], X[12, 18, 13, 17], X[6, 14, 7, 13]]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[10, 59]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -9, 10, -5, 7, -8, 9,
-6, 8, -7]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[10, 59]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 8, 10, 14, 2, 18, 6, 20, 12, 16]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[10, 59]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[5, {-1, 2, -1, 2, -3, 2, 2, 4, -3, 4}]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{5, 10}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[10, 59]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>5</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[10, 59]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:10_59_ML.gif]]</td></tr><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[10, 59]]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 1, 3, 3, NotAvailable, 1}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[10, 59]][t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -3 7 18 2 3
-23 + t - -- + -- + 18 t - 7 t + t
2 t
t</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[10, 59]][z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 6
1 - z - z + z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[9, 40], Knot[10, 59], Knot[11, NonAlternating, 66]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[10, 59]], KnotSignature[Knot[10, 59]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{75, 2}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[10, 59]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -3 3 6 2 3 4 5 6 7
-9 + q - -- + - + 12 q - 12 q + 12 q - 10 q + 6 q - 3 q + q
2 q
q</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 59], Knot[10, 106]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[10, 59]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -10 -6 2 2 2 4 6 8 10 12 14
q - q + -- - -- + 2 q - q + 4 q - q + q - q - 3 q +
4 2
q q
16 18 22
2 q - q + q</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[10, 59]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 2 2
-6 3 4 2 2 z 4 z 5 z 2 2 4
-2 + a - -- + -- + a - 4 z + -- - ---- + ---- + a z - 2 z -
4 2 6 4 2
a a a a a
4 4 6
2 z 3 z z
---- + ---- + --
4 2 2
a a a</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[10, 59]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 2
-6 3 4 2 z 4 z 5 z 2 z 3 z
-2 - a - -- - -- - a + -- - --- - --- - 2 a z + 8 z - -- + ---- +
4 2 7 3 a 8 6
a a a a a a
2 2 3 3 3 3
10 z 11 z 2 2 3 z 4 z 20 z 21 z 3
----- + ----- + 3 a z - ---- + ---- + ----- + ----- + 8 a z -
4 2 7 5 3 a
a a a a a
4 4 4 4 5 5 5
4 z 5 z 11 z 8 z 2 4 3 z 7 z 28 z
6 z + -- - ---- - ----- - ---- - 3 a z + ---- - ---- - ----- -
8 6 4 2 7 5 3
a a a a a a a
5 6 6 6 7 7
27 z 5 6 5 z z 9 z 2 6 6 z 11 z
----- - 9 a z - 4 z + ---- + -- - ---- + a z + ---- + ----- +
a 6 4 2 5 3
a a a a a
7 8 8 9 9
8 z 7 8 4 z 7 z z z
---- + 3 a z + 3 z + ---- + ---- + -- + --
a 4 2 3 a
a a a</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[10, 59]], Vassiliev[3][Knot[10, 59]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{-1, -1}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[10, 59]][q, t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 3 1 2 1 4 2 5 4 q
7 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- +
7 4 5 3 3 3 3 2 2 q t t
q t q t q t q t q t
3 5 5 2 7 2 7 3 9 3 9 4
6 q t + 6 q t + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t +
11 4 11 5 13 5 15 6
4 q t + q t + 2 q t + q t</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[10, 59], 2][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -10 3 11 14 9 40 27 39 81 2
-25 + q - -- + -- - -- - -- + -- - -- - -- + -- - 83 q + 112 q -
9 7 6 5 4 3 2 q
q q q q q q q
3 4 5 6 7 8 9 10
7 q - 119 q + 117 q + 16 q - 127 q + 95 q + 29 q - 100 q +
11 12 13 14 15 16 17 18
56 q + 26 q - 53 q + 22 q + 12 q - 17 q + 6 q + 2 q -
19 20
3 q + q</nowiki></code></td></tr>
</table> }}

Latest revision as of 17:01, 1 September 2005

10 58.gif

10_58

10 60.gif

10_60

10 59.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 59's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 59 at Knotilus!


Two figure 8 knots on a loop, interlinked.

Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13
Gauss code 1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code 4 8 10 14 2 18 6 20 12 16
Conway Notation [22,211,2]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif

Length is 10, width is 5,

Braid index is 5

10 59 ML.gif 10 59 AP.gif
[{9, 12}, {11, 3}, {12, 10}, {7, 11}, {6, 8}, {5, 7}, {4, 2}, {3, 6}, {1, 4}, {2, 9}, {8, 1}, {10, 5}]

[edit Notes on presentations of 10 59]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-3][-9]
Hyperbolic Volume 13.3899
A-Polynomial See Data:10 59/A-polynomial

[edit Notes for 10 59's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant -2

[edit Notes for 10 59's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 75, 2 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant
The G2 invariant

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_40, K11n66,}

Same Jones Polynomial (up to mirroring, ): {10_106,}

Vassiliev invariants

V2 and V3: (-1, -1)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 2 is the signature of 10 59. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-10123456χ
15          11
13         2 -2
11        41 3
9       62  -4
7      64   2
5     66    0
3    66     0
1   47      3
-1  25       -3
-3 14        3
-5 2         -2
-71          1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials