10 91

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10 90.gif

10_90

10 92.gif

10_92

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10 91 Quick Notes


10 91 Further Notes and Views

Knot presentations

Planar diagram presentation X6271 X20,6,1,5 X16,9,17,10 X10,3,11,4 X2,18,3,17 X14,7,15,8 X8,15,9,16 X12,20,13,19 X18,12,19,11 X4,13,5,14
Gauss code 1, -5, 4, -10, 2, -1, 6, -7, 3, -4, 9, -8, 10, -6, 7, -3, 5, -9, 8, -2
Dowker-Thistlethwaite code 6 10 20 14 16 18 4 8 2 12
Conway Notation [.3.2.20]

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 13.487
A-Polynomial See Data:10 91/A-polynomial

[edit Notes for 10 91's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ 4 }[/math]
Rasmussen s-Invariant 0

[edit Notes for 10 91's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-4 t^3+9 t^2-14 t+17-14 t^{-1} +9 t^{-2} -4 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+4 z^6+5 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 73, 0 }
Jones polynomial [math]\displaystyle{ -q^5+3 q^4-6 q^3+9 q^2-11 q+13-11 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +6 z^6-4 a^2 z^4-4 z^4 a^{-2} +13 z^4-5 a^2 z^2-5 z^2 a^{-2} +12 z^2-2 a^2-2 a^{-2} +5 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+5 z^8 a^{-2} +9 z^8+4 a^3 z^7+a z^7+2 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-7 a^2 z^6-13 z^6 a^{-2} +3 z^6 a^{-4} -26 z^6+a^5 z^5-6 a^3 z^5-7 a z^5-13 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+7 a^2 z^4+16 z^4 a^{-2} -6 z^4 a^{-4} +35 z^4-2 a^5 z^3+9 a z^3+18 z^3 a^{-1} +9 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2-7 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -19 z^2+a^5 z-4 a z-6 z a^{-1} -3 z a^{-3} +2 a^2+2 a^{-2} +5 }[/math]
The A2 invariant [math]\displaystyle{ -q^{14}+q^{12}-2 q^{10}+q^8-q^4+4 q^2-1+4 q^{-2} - q^{-4} + q^{-8} -2 q^{-10} + q^{-12} - q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-5 q^{70}-3 q^{68}+16 q^{66}-27 q^{64}+36 q^{62}-38 q^{60}+23 q^{58}-q^{56}-34 q^{54}+70 q^{52}-92 q^{50}+95 q^{48}-66 q^{46}+4 q^{44}+68 q^{42}-130 q^{40}+151 q^{38}-122 q^{36}+46 q^{34}+44 q^{32}-116 q^{30}+138 q^{28}-93 q^{26}+9 q^{24}+77 q^{22}-122 q^{20}+97 q^{18}-19 q^{16}-82 q^{14}+161 q^{12}-175 q^{10}+132 q^8-27 q^6-94 q^4+190 q^2-223+188 q^{-2} -92 q^{-4} -27 q^{-6} +132 q^{-8} -176 q^{-10} +165 q^{-12} -89 q^{-14} -10 q^{-16} +93 q^{-18} -126 q^{-20} +87 q^{-22} -3 q^{-24} -84 q^{-26} +137 q^{-28} -124 q^{-30} +54 q^{-32} +40 q^{-34} -124 q^{-36} +160 q^{-38} -142 q^{-40} +74 q^{-42} +7 q^{-44} -77 q^{-46} +108 q^{-48} -102 q^{-50} +72 q^{-52} -28 q^{-54} -10 q^{-56} +32 q^{-58} -42 q^{-60} +36 q^{-62} -23 q^{-64} +12 q^{-66} -6 q^{-70} +7 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_91.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{11}+2 q^9-3 q^7+3 q^5-2 q^3+2 q+2 q^{-1} -2 q^{-3} +3 q^{-5} -3 q^{-7} +2 q^{-9} - q^{-11} }[/math]
2 [math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+6 q^{26}-6 q^{24}-5 q^{22}+17 q^{20}-8 q^{18}-17 q^{16}+24 q^{14}-25 q^{10}+17 q^8+10 q^6-18 q^4+q^2+14+ q^{-2} -17 q^{-4} +10 q^{-6} +17 q^{-8} -25 q^{-10} + q^{-12} +24 q^{-14} -18 q^{-16} -8 q^{-18} +18 q^{-20} -5 q^{-22} -8 q^{-24} +6 q^{-26} -2 q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ -q^{63}+2 q^{61}+q^{59}-2 q^{57}-3 q^{55}+3 q^{53}+5 q^{51}-9 q^{49}-7 q^{47}+18 q^{45}+18 q^{43}-28 q^{41}-43 q^{39}+32 q^{37}+77 q^{35}-19 q^{33}-111 q^{31}-16 q^{29}+135 q^{27}+61 q^{25}-134 q^{23}-107 q^{21}+113 q^{19}+136 q^{17}-74 q^{15}-147 q^{13}+33 q^{11}+134 q^9+11 q^7-114 q^5-43 q^3+82 q+78 q^{-1} -47 q^{-3} -108 q^{-5} +18 q^{-7} +130 q^{-9} +22 q^{-11} -147 q^{-13} -60 q^{-15} +144 q^{-17} +100 q^{-19} -123 q^{-21} -127 q^{-23} +81 q^{-25} +138 q^{-27} -33 q^{-29} -122 q^{-31} -12 q^{-33} +92 q^{-35} +34 q^{-37} -55 q^{-39} -38 q^{-41} +23 q^{-43} +30 q^{-45} -7 q^{-47} -17 q^{-49} +2 q^{-51} +7 q^{-53} -3 q^{-57} +2 q^{-61} - q^{-63} }[/math]
4 [math]\displaystyle{ q^{104}-2 q^{102}-q^{100}+2 q^{98}-q^{96}+6 q^{94}-3 q^{92}-q^{90}+3 q^{88}-15 q^{86}+4 q^{84}-9 q^{82}+19 q^{80}+48 q^{78}-16 q^{76}-39 q^{74}-104 q^{72}-6 q^{70}+168 q^{68}+139 q^{66}+21 q^{64}-307 q^{62}-297 q^{60}+114 q^{58}+432 q^{56}+462 q^{54}-242 q^{52}-737 q^{50}-435 q^{48}+355 q^{46}+1046 q^{44}+398 q^{42}-712 q^{40}-1097 q^{38}-314 q^{36}+1080 q^{34}+1080 q^{32}-83 q^{30}-1165 q^{28}-957 q^{26}+511 q^{24}+1151 q^{22}+518 q^{20}-669 q^{18}-1032 q^{16}-79 q^{14}+741 q^{12}+699 q^{10}-144 q^8-745 q^6-410 q^4+316 q^2+684+232 q^{-2} -476 q^{-4} -699 q^{-6} -32 q^{-8} +739 q^{-10} +644 q^{-12} -221 q^{-14} -1039 q^{-16} -505 q^{-18} +672 q^{-20} +1106 q^{-22} +288 q^{-24} -1105 q^{-26} -1051 q^{-28} +173 q^{-30} +1194 q^{-32} +915 q^{-34} -570 q^{-36} -1158 q^{-38} -528 q^{-40} +619 q^{-42} +1068 q^{-44} +183 q^{-46} -600 q^{-48} -745 q^{-50} -112 q^{-52} +579 q^{-54} +439 q^{-56} +35 q^{-58} -384 q^{-60} -324 q^{-62} +69 q^{-64} +201 q^{-66} +194 q^{-68} -39 q^{-70} -150 q^{-72} -56 q^{-74} +6 q^{-76} +81 q^{-78} +24 q^{-80} -26 q^{-82} -13 q^{-84} -15 q^{-86} +15 q^{-88} +6 q^{-90} -5 q^{-92} + q^{-94} -3 q^{-96} +3 q^{-98} -2 q^{-102} + q^{-104} }[/math]
5 [math]\displaystyle{ -q^{155}+2 q^{153}+q^{151}-2 q^{149}+q^{147}-2 q^{145}-6 q^{143}-q^{141}+7 q^{139}+7 q^{137}+14 q^{135}+9 q^{133}-21 q^{131}-42 q^{129}-34 q^{127}+6 q^{125}+67 q^{123}+114 q^{121}+68 q^{119}-86 q^{117}-230 q^{115}-229 q^{113}-10 q^{111}+337 q^{109}+541 q^{107}+312 q^{105}-327 q^{103}-914 q^{101}-900 q^{99}-60 q^{97}+1158 q^{95}+1774 q^{93}+986 q^{91}-942 q^{89}-2613 q^{87}-2479 q^{85}-132 q^{83}+2956 q^{81}+4253 q^{79}+2154 q^{77}-2281 q^{75}-5656 q^{73}-4835 q^{71}+270 q^{69}+6021 q^{67}+7522 q^{65}+2799 q^{63}-4910 q^{61}-9312 q^{59}-6264 q^{57}+2355 q^{55}+9646 q^{53}+9218 q^{51}+970 q^{49}-8359 q^{47}-10865 q^{45}-4266 q^{43}+5890 q^{41}+10936 q^{39}+6697 q^{37}-2954 q^{35}-9629 q^{33}-7863 q^{31}+307 q^{29}+7487 q^{27}+7789 q^{25}+1633 q^{23}-5202 q^{21}-6905 q^{19}-2664 q^{17}+3225 q^{15}+5673 q^{13}+3104 q^{11}-1790 q^9-4664 q^7-3236 q^5+875 q^3+4027 q+3531 q^{-1} -167 q^{-3} -3868 q^{-5} -4244 q^{-7} -626 q^{-9} +3982 q^{-11} +5379 q^{-13} +1868 q^{-15} -3940 q^{-17} -6868 q^{-19} -3681 q^{-21} +3417 q^{-23} +8205 q^{-25} +5965 q^{-27} -1999 q^{-29} -8937 q^{-31} -8380 q^{-33} -285 q^{-35} +8536 q^{-37} +10289 q^{-39} +3193 q^{-41} -6785 q^{-43} -11078 q^{-45} -6071 q^{-47} +3866 q^{-49} +10342 q^{-51} +8163 q^{-53} -435 q^{-55} -8081 q^{-57} -8823 q^{-59} -2710 q^{-61} +4870 q^{-63} +7921 q^{-65} +4688 q^{-67} -1549 q^{-69} -5752 q^{-71} -5199 q^{-73} -1054 q^{-75} +3147 q^{-77} +4391 q^{-79} +2384 q^{-81} -845 q^{-83} -2851 q^{-85} -2536 q^{-87} -611 q^{-89} +1307 q^{-91} +1917 q^{-93} +1124 q^{-95} -203 q^{-97} -1060 q^{-99} -1016 q^{-101} -324 q^{-103} +405 q^{-105} +645 q^{-107} +391 q^{-109} -35 q^{-111} -300 q^{-113} -280 q^{-115} -85 q^{-117} +108 q^{-119} +145 q^{-121} +67 q^{-123} -18 q^{-125} -54 q^{-127} -42 q^{-129} -4 q^{-131} +25 q^{-133} +14 q^{-135} -2 q^{-137} -3 q^{-139} -3 q^{-141} -3 q^{-143} +2 q^{-145} +3 q^{-147} -3 q^{-149} +2 q^{-153} - q^{-155} }[/math]
6 [math]\displaystyle{ q^{216}-2 q^{214}-q^{212}+2 q^{210}-q^{208}+2 q^{206}+2 q^{204}+10 q^{202}-5 q^{200}-17 q^{198}-6 q^{196}-15 q^{194}+22 q^{190}+66 q^{188}+35 q^{186}-26 q^{184}-53 q^{182}-116 q^{180}-105 q^{178}-10 q^{176}+211 q^{174}+268 q^{172}+173 q^{170}+q^{168}-370 q^{166}-611 q^{164}-514 q^{162}+164 q^{160}+824 q^{158}+1177 q^{156}+1011 q^{154}-149 q^{152}-1661 q^{150}-2616 q^{148}-1812 q^{146}+264 q^{144}+2928 q^{142}+4822 q^{140}+3798 q^{138}-158 q^{136}-5401 q^{134}-8227 q^{132}-6897 q^{130}-564 q^{128}+8415 q^{126}+14040 q^{124}+12017 q^{122}+1420 q^{120}-12120 q^{118}-21811 q^{116}-19675 q^{114}-3864 q^{112}+17532 q^{110}+32243 q^{108}+29010 q^{106}+7550 q^{104}-23624 q^{102}-45160 q^{100}-41132 q^{98}-10684 q^{96}+31190 q^{94}+58901 q^{92}+54223 q^{90}+13506 q^{88}-40305 q^{86}-74157 q^{84}-64849 q^{82}-13776 q^{80}+49834 q^{78}+88138 q^{76}+72211 q^{74}+10264 q^{72}-61061 q^{70}-97033 q^{68}-73330 q^{66}-3373 q^{64}+71176 q^{62}+100362 q^{60}+67117 q^{58}-7919 q^{56}-76638 q^{54}-95921 q^{52}-54917 q^{50}+19905 q^{48}+77417 q^{46}+83879 q^{44}+37456 q^{42}-28664 q^{40}-71967 q^{38}-66850 q^{36}-19669 q^{34}+33992 q^{32}+61163 q^{30}+46783 q^{28}+5305 q^{26}-34721 q^{24}-47784 q^{22}-28341 q^{20}+5852 q^{18}+32118 q^{16}+33749 q^{14}+13526 q^{12}-13663 q^{10}-28520 q^8-22157 q^6-1010 q^4+20049 q^2+25575+12761 q^{-2} -10407 q^{-4} -26842 q^{-6} -24716 q^{-8} -3554 q^{-10} +23004 q^{-12} +35485 q^{-14} +23930 q^{-16} -7415 q^{-18} -37620 q^{-20} -45703 q^{-22} -21294 q^{-24} +21842 q^{-26} +54957 q^{-28} +54079 q^{-30} +14814 q^{-32} -38933 q^{-34} -72716 q^{-36} -58950 q^{-38} -3884 q^{-40} +58180 q^{-42} +86227 q^{-44} +58431 q^{-46} -9267 q^{-48} -75600 q^{-50} -94134 q^{-52} -52644 q^{-54} +23575 q^{-56} +86248 q^{-58} +94811 q^{-60} +44229 q^{-62} -34894 q^{-64} -89869 q^{-66} -88817 q^{-68} -33787 q^{-70} +39509 q^{-72} +85832 q^{-74} +78917 q^{-76} +24663 q^{-78} -38975 q^{-80} -75574 q^{-82} -65579 q^{-84} -19254 q^{-86} +33738 q^{-88} +62390 q^{-90} +52059 q^{-92} +15336 q^{-94} -25610 q^{-96} -47167 q^{-98} -40662 q^{-100} -12962 q^{-102} +17501 q^{-104} +33234 q^{-106} +29980 q^{-108} +11451 q^{-110} -9670 q^{-112} -22494 q^{-114} -21164 q^{-116} -9504 q^{-118} +4152 q^{-120} +13708 q^{-122} +14400 q^{-124} +7972 q^{-126} -1388 q^{-128} -7710 q^{-130} -9017 q^{-132} -6002 q^{-134} -348 q^{-136} +4107 q^{-138} +5613 q^{-140} +3770 q^{-142} +821 q^{-144} -1881 q^{-146} -3184 q^{-148} -2387 q^{-150} -647 q^{-152} +988 q^{-154} +1497 q^{-156} +1301 q^{-158} +468 q^{-160} -464 q^{-162} -779 q^{-164} -598 q^{-166} -136 q^{-168} +137 q^{-170} +336 q^{-172} +284 q^{-174} +40 q^{-176} -102 q^{-178} -129 q^{-180} -57 q^{-182} -31 q^{-184} +38 q^{-186} +60 q^{-188} +14 q^{-190} -11 q^{-192} -17 q^{-194} +2 q^{-196} -10 q^{-198} +11 q^{-202} -2 q^{-206} -3 q^{-208} +3 q^{-210} -2 q^{-214} + q^{-216} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{14}+q^{12}-2 q^{10}+q^8-q^4+4 q^2-1+4 q^{-2} - q^{-4} + q^{-8} -2 q^{-10} + q^{-12} - q^{-14} }[/math]
1,1 [math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+52 q^{36}-86 q^{34}+134 q^{32}-200 q^{30}+273 q^{28}-356 q^{26}+454 q^{24}-534 q^{22}+577 q^{20}-576 q^{18}+508 q^{16}-358 q^{14}+124 q^{12}+156 q^{10}-458 q^8+750 q^6-996 q^4+1162 q^2-1226+1198 q^{-2} -1050 q^{-4} +834 q^{-6} -548 q^{-8} +246 q^{-10} +58 q^{-12} -320 q^{-14} +494 q^{-16} -598 q^{-18} +623 q^{-20} -584 q^{-22} +494 q^{-24} -390 q^{-26} +293 q^{-28} -202 q^{-30} +128 q^{-32} -80 q^{-34} +46 q^{-36} -22 q^{-38} +10 q^{-40} -4 q^{-42} + q^{-44} }[/math]
2,0 [math]\displaystyle{ q^{38}-q^{36}-q^{34}+2 q^{32}-2 q^{30}-3 q^{28}+4 q^{26}+3 q^{24}-4 q^{22}-3 q^{20}+7 q^{18}+3 q^{16}-12 q^{14}+q^{12}+8 q^{10}-7 q^8-6 q^6+8 q^4+4 q^2-4+5 q^{-2} +9 q^{-4} -3 q^{-6} -5 q^{-8} +10 q^{-10} + q^{-12} -11 q^{-14} +4 q^{-16} +6 q^{-18} -5 q^{-20} -6 q^{-22} +3 q^{-24} +3 q^{-26} -4 q^{-28} -2 q^{-30} +3 q^{-32} - q^{-36} + q^{-38} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{34}-2 q^{32}+q^{30}+4 q^{28}-7 q^{26}+2 q^{24}+9 q^{22}-15 q^{20}+2 q^{18}+15 q^{16}-21 q^{14}-q^{12}+16 q^{10}-14 q^8-2 q^6+13 q^4+3 q^2+2 q^{-2} +13 q^{-4} -3 q^{-6} -14 q^{-8} +15 q^{-10} - q^{-12} -21 q^{-14} +15 q^{-16} +2 q^{-18} -15 q^{-20} +10 q^{-22} +2 q^{-24} -6 q^{-26} +4 q^{-28} -2 q^{-32} + q^{-34} }[/math]
1,0,0 [math]\displaystyle{ -q^{17}+q^{15}-3 q^{13}+2 q^{11}-3 q^9+2 q^7-q^5+3 q^3+2 q+2 q^{-1} +3 q^{-3} - q^{-5} +2 q^{-7} -3 q^{-9} +2 q^{-11} -3 q^{-13} + q^{-15} - q^{-17} }[/math]
1,0,1 [math]\displaystyle{ q^{56}-4 q^{54}+10 q^{52}-14 q^{50}+7 q^{48}+16 q^{46}-49 q^{44}+71 q^{42}-56 q^{40}-13 q^{38}+107 q^{36}-182 q^{34}+175 q^{32}-43 q^{30}-161 q^{28}+356 q^{26}-413 q^{24}+264 q^{22}+51 q^{20}-409 q^{18}+607 q^{16}-567 q^{14}+273 q^{12}+91 q^{10}-359 q^8+410 q^6-263 q^4+82 q^2+28+54 q^{-2} -201 q^{-4} +340 q^{-6} -272 q^{-8} +25 q^{-10} +331 q^{-12} -591 q^{-14} +606 q^{-16} -387 q^{-18} +6 q^{-20} +308 q^{-22} -460 q^{-24} +390 q^{-26} -188 q^{-28} -33 q^{-30} +176 q^{-32} -192 q^{-34} +126 q^{-36} -29 q^{-38} -42 q^{-40} +61 q^{-42} -46 q^{-44} +18 q^{-46} +4 q^{-48} -10 q^{-50} +8 q^{-52} -4 q^{-54} + q^{-56} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{40}-q^{38}+4 q^{34}-2 q^{32}-3 q^{30}+8 q^{28}-2 q^{26}-11 q^{24}+5 q^{22}+6 q^{20}-16 q^{18}-9 q^{16}+9 q^{14}-2 q^{12}-17 q^{10}+7 q^8+21 q^6-8 q^4+5 q^2+29+4 q^{-2} -11 q^{-4} +18 q^{-6} +4 q^{-8} -19 q^{-10} -5 q^{-12} +8 q^{-14} -8 q^{-16} -15 q^{-18} +7 q^{-20} +7 q^{-22} -8 q^{-24} - q^{-26} +8 q^{-28} -3 q^{-30} -2 q^{-32} +3 q^{-34} - q^{-36} - q^{-38} + q^{-40} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{20}+q^{18}-3 q^{16}+q^{14}-2 q^{12}-q^{10}+q^8-q^6+4 q^4+q^2+5+ q^{-2} +4 q^{-4} - q^{-6} + q^{-8} - q^{-10} -2 q^{-12} + q^{-14} -3 q^{-16} + q^{-18} - q^{-20} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+8 q^{28}-11 q^{26}+16 q^{24}-21 q^{22}+23 q^{20}-24 q^{18}+21 q^{16}-15 q^{14}+7 q^{12}+4 q^{10}-16 q^8+28 q^6-37 q^4+45 q^2-46+46 q^{-2} -37 q^{-4} +29 q^{-6} -16 q^{-8} +5 q^{-10} +7 q^{-12} -15 q^{-14} +21 q^{-16} -24 q^{-18} +23 q^{-20} -22 q^{-22} +16 q^{-24} -12 q^{-26} +8 q^{-28} -4 q^{-30} +2 q^{-32} - q^{-34} }[/math]
1,0 [math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+6 q^{46}-q^{44}-9 q^{42}-6 q^{40}+9 q^{38}+15 q^{36}-4 q^{34}-21 q^{32}-9 q^{30}+19 q^{28}+20 q^{26}-11 q^{24}-26 q^{22}-3 q^{20}+23 q^{18}+11 q^{16}-17 q^{14}-15 q^{12}+11 q^{10}+17 q^8-4 q^6-15 q^4+4 q^2+18+4 q^{-2} -15 q^{-4} -4 q^{-6} +16 q^{-8} +11 q^{-10} -15 q^{-12} -17 q^{-14} +10 q^{-16} +23 q^{-18} -2 q^{-20} -26 q^{-22} -12 q^{-24} +20 q^{-26} +20 q^{-28} -9 q^{-30} -22 q^{-32} -4 q^{-34} +16 q^{-36} +9 q^{-38} -6 q^{-40} -9 q^{-42} +6 q^{-46} +2 q^{-48} -2 q^{-50} -2 q^{-52} + q^{-56} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{46}-2 q^{44}+3 q^{42}-4 q^{40}+7 q^{38}-9 q^{36}+10 q^{34}-13 q^{32}+17 q^{30}-19 q^{28}+17 q^{26}-18 q^{24}+17 q^{22}-16 q^{20}+5 q^{18}-6 q^{16}-2 q^{14}+8 q^{12}-19 q^{10}+22 q^8-23 q^6+39 q^4-30 q^2+40-30 q^{-2} +39 q^{-4} -24 q^{-6} +22 q^{-8} -20 q^{-10} +8 q^{-12} -3 q^{-14} -6 q^{-16} +4 q^{-18} -16 q^{-20} +17 q^{-22} -18 q^{-24} +18 q^{-26} -19 q^{-28} +18 q^{-30} -13 q^{-32} +11 q^{-34} -9 q^{-36} +7 q^{-38} -4 q^{-40} +2 q^{-42} -2 q^{-44} + q^{-46} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-5 q^{70}-3 q^{68}+16 q^{66}-27 q^{64}+36 q^{62}-38 q^{60}+23 q^{58}-q^{56}-34 q^{54}+70 q^{52}-92 q^{50}+95 q^{48}-66 q^{46}+4 q^{44}+68 q^{42}-130 q^{40}+151 q^{38}-122 q^{36}+46 q^{34}+44 q^{32}-116 q^{30}+138 q^{28}-93 q^{26}+9 q^{24}+77 q^{22}-122 q^{20}+97 q^{18}-19 q^{16}-82 q^{14}+161 q^{12}-175 q^{10}+132 q^8-27 q^6-94 q^4+190 q^2-223+188 q^{-2} -92 q^{-4} -27 q^{-6} +132 q^{-8} -176 q^{-10} +165 q^{-12} -89 q^{-14} -10 q^{-16} +93 q^{-18} -126 q^{-20} +87 q^{-22} -3 q^{-24} -84 q^{-26} +137 q^{-28} -124 q^{-30} +54 q^{-32} +40 q^{-34} -124 q^{-36} +160 q^{-38} -142 q^{-40} +74 q^{-42} +7 q^{-44} -77 q^{-46} +108 q^{-48} -102 q^{-50} +72 q^{-52} -28 q^{-54} -10 q^{-56} +32 q^{-58} -42 q^{-60} +36 q^{-62} -23 q^{-64} +12 q^{-66} -6 q^{-70} +7 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80} }[/math]

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Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 91"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-4 t^3+9 t^2-14 t+17-14 t^{-1} +9 t^{-2} -4 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+4 z^6+5 z^4+2 z^2+1 }[/math]
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 73, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+3 q^4-6 q^3+9 q^2-11 q+13-11 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +6 z^6-4 a^2 z^4-4 z^4 a^{-2} +13 z^4-5 a^2 z^2-5 z^2 a^{-2} +12 z^2-2 a^2-2 a^{-2} +5 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+5 z^8 a^{-2} +9 z^8+4 a^3 z^7+a z^7+2 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-7 a^2 z^6-13 z^6 a^{-2} +3 z^6 a^{-4} -26 z^6+a^5 z^5-6 a^3 z^5-7 a z^5-13 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+7 a^2 z^4+16 z^4 a^{-2} -6 z^4 a^{-4} +35 z^4-2 a^5 z^3+9 a z^3+18 z^3 a^{-1} +9 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2-7 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -19 z^2+a^5 z-4 a z-6 z a^{-1} -3 z a^{-3} +2 a^2+2 a^{-2} +5 }[/math]

Vassiliev invariants

V2 and V3: (2, 0)
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
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{28}{3} }[/math] [math]\displaystyle{ -\frac{76}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ -\frac{608}{3} }[/math] [math]\displaystyle{ \frac{511}{15} }[/math] [math]\displaystyle{ \frac{812}{5} }[/math] [math]\displaystyle{ -\frac{13556}{45} }[/math] [math]\displaystyle{ -\frac{175}{9} }[/math] [math]\displaystyle{ -\frac{1649}{15} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 10 91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -5-4-3-2-1012345χ
11          1-1
9         2 2
7        41 -3
5       52  3
3      64   -2
1     75    2
-1    57     2
-3   46      -2
-5  25       3
-7 14        -3
-9 2         2
-111          -1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[10, 91]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 91]]
Out[3]=  
PD[X[6, 2, 7, 1], X[20, 6, 1, 5], X[16, 9, 17, 10], X[10, 3, 11, 4], 

 X[2, 18, 3, 17], X[14, 7, 15, 8], X[8, 15, 9, 16], X[12, 20, 13, 19], 



  X[18, 12, 19, 11], X[4, 13, 5, 14]]
In[4]:=
GaussCode[Knot[10, 91]]
Out[4]=  
GaussCode[1, -5, 4, -10, 2, -1, 6, -7, 3, -4, 9, -8, 10, -6, 7, -3, 5, 
  -9, 8, -2]
In[5]:=
BR[Knot[10, 91]]
Out[5]=  
BR[3, {-1, -1, -1, 2, -1, 2, 2, -1, 2, 2}]
In[6]:=
alex = Alexander[Knot[10, 91]][t]
Out[6]=  
      -4   4    9    14             2      3    4

17 + t   - -- + -- - -- - 14 t + 9 t  - 4 t  + t



           3    2   t


           t    t
In[7]:=
Conway[Knot[10, 91]][z]
Out[7]=  
       2      4      6    8
1 + 2 z  + 5 z  + 4 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 91]}
In[9]:=
{KnotDet[Knot[10, 91]], KnotSignature[Knot[10, 91]]}
Out[9]=  
{73, 0}
In[10]:=
J=Jones[Knot[10, 91]][q]
Out[10]=  
      -5   3    6    9    11             2      3      4    5

13 - q   + -- - -- + -- - -- - 11 q + 9 q  - 6 q  + 3 q  - q



           4    3    2   q


           q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 43], Knot[10, 91]}
In[12]:=
A2Invariant[Knot[10, 91]][q]
Out[12]=  
      -14    -12    2     -8    -4   4       2    4    8      10

-1 - q    + q    - --- + q   - q   + -- + 4 q  - q  + q  - 2 q   + 



                   10                2
                  q                 q

  12    14


  q   - q
In[13]:=
Kauffman[Knot[10, 91]][a, z]
Out[13]=  
                                                    2      2

   2       2   3 z   6 z            5         2   z    9 z



5 + -- + 2 a  - --- - --- - 4 a z + a  z - 19 z  + -- - ---- - 



    2           3     a                            4     2
   a           a                                  a     a

                        3      3       3
    2  2      4  2   2 z    9 z    18 z         3      5  3       4
 7 a  z  + 2 a  z  - ---- + ---- + ----- + 9 a z  - 2 a  z  + 35 z  - 
                       5      3      a
                      a      a

    4       4                        5       5       5
 6 z    16 z       2  4      4  4   z    12 z    13 z         5
 ---- + ----- + 7 a  z  - 6 a  z  + -- - ----- - ----- - 7 a z  - 
   4      2                          5     3       a
  a      a                          a     a

                              6       6                          7
    3  5    5  5       6   3 z    13 z       2  6      4  6   5 z
 6 a  z  + a  z  - 26 z  + ---- - ----- - 7 a  z  + 3 a  z  + ---- + 
                             4      2                           3
                            a      a                           a

    7                              8                9
 2 z       7      3  7      8   5 z       2  8   2 z         9
 ---- + a z  + 4 a  z  + 9 z  + ---- + 4 a  z  + ---- + 2 a z
  a                               2               a


                                  a
In[14]:=
{Vassiliev[2][Knot[10, 91]], Vassiliev[3][Knot[10, 91]]}
Out[14]=  
{0, 0}
In[15]:=
Kh[Knot[10, 91]][q, t]
Out[15]=  
7           1        2       1       4       2       5       4

- + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
q          11  5    9  4    7  4    7  3    5  3    5  2    3  2



         q   t    q  t    q  t    q  t    q  t    q  t    q  t

  6      5               3        3  2      5  2      5  3      7  3
 ---- + --- + 5 q t + 6 q  t + 4 q  t  + 5 q  t  + 2 q  t  + 4 q  t  + 
  3     q t
 q  t

  7  4      9  4    11  5


  q  t  + 2 q  t  + q   t
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