10 85

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10 84.gif

10_84

10 86.gif

10_86

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10 85 Quick Notes


10 85 Further Notes and Views

Knot presentations

Planar diagram presentation X6271 X16,6,17,5 X18,11,19,12 X14,7,15,8 X8394 X4,9,5,10 X20,13,1,14 X10,17,11,18 X12,19,13,20 X2,16,3,15
Gauss code 1, -10, 5, -6, 2, -1, 4, -5, 6, -8, 3, -9, 7, -4, 10, -2, 8, -3, 9, -7
Dowker-Thistlethwaite code 6 8 16 14 4 18 20 2 10 12
Conway Notation [.4.20]

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-12][0]
Hyperbolic Volume 11.7978
A-Polynomial See Data:10 85/A-polynomial

[edit Notes for 10 85's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math]
Topological 4 genus [math]\displaystyle{ 2 }[/math]
Concordance genus [math]\displaystyle{ 4 }[/math]
Rasmussen s-Invariant 4

[edit Notes for 10 85's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-4 t^3+8 t^2-10 t+11-10 t^{-1} +8 t^{-2} -4 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+4 z^6+4 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 57, -4 }
Jones polynomial [math]\displaystyle{ -q+3-4 q^{-1} +7 q^{-2} -8 q^{-3} +9 q^{-4} -9 q^{-5} +7 q^{-6} -5 q^{-7} +3 q^{-8} - q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^4 z^8-a^6 z^6+6 a^4 z^6-a^2 z^6-4 a^6 z^4+12 a^4 z^4-4 a^2 z^4-4 a^6 z^2+9 a^4 z^2-3 a^2 z^2-a^6+a^4+a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}-z^2 a^{10}+5 z^5 a^9-4 z^3 a^9+z a^9+6 z^6 a^8-7 z^4 a^8+z^2 a^8+6 z^7 a^7-10 z^5 a^7+2 z^3 a^7+5 z^8 a^6-12 z^6 a^6+8 z^4 a^6-5 z^2 a^6+a^6+2 z^9 a^5-15 z^5 a^5+14 z^3 a^5-2 z a^5+8 z^8 a^4-32 z^6 a^4+37 z^4 a^4-14 z^2 a^4+a^4+2 z^9 a^3-5 z^7 a^3-4 z^5 a^3+11 z^3 a^3-2 z a^3+3 z^8 a^2-14 z^6 a^2+19 z^4 a^2-7 z^2 a^2-a^2+z^7 a-4 z^5 a+4 z^3 a-z a }[/math]
The A2 invariant [math]\displaystyle{ -q^{26}+q^{24}-q^{22}+q^{20}-q^{16}+q^{14}-3 q^{12}+2 q^{10}+2 q^6+2 q^4+1- q^{-2} }[/math]
The G2 invariant [math]\displaystyle{ q^{148}-2 q^{146}+3 q^{144}-4 q^{142}+2 q^{140}-q^{138}-2 q^{136}+8 q^{134}-11 q^{132}+14 q^{130}-12 q^{128}+6 q^{126}+2 q^{124}-11 q^{122}+20 q^{120}-24 q^{118}+21 q^{116}-14 q^{114}+q^{112}+9 q^{110}-16 q^{108}+22 q^{106}-24 q^{104}+21 q^{102}-17 q^{100}+3 q^{98}+12 q^{96}-26 q^{94}+33 q^{92}-29 q^{90}+14 q^{88}+9 q^{86}-28 q^{84}+33 q^{82}-18 q^{80}-7 q^{78}+34 q^{76}-45 q^{74}+30 q^{72}+5 q^{70}-40 q^{68}+64 q^{66}-65 q^{64}+40 q^{62}-2 q^{60}-38 q^{58}+62 q^{56}-69 q^{54}+51 q^{52}-22 q^{50}-13 q^{48}+37 q^{46}-47 q^{44}+43 q^{42}-20 q^{40}-10 q^{38}+33 q^{36}-39 q^{34}+27 q^{32}+7 q^{30}-38 q^{28}+60 q^{26}-49 q^{24}+21 q^{22}+22 q^{20}-54 q^{18}+68 q^{16}-53 q^{14}+23 q^{12}+9 q^{10}-34 q^8+42 q^6-33 q^4+18 q^2-2-8 q^{-2} +9 q^{-4} -9 q^{-6} +5 q^{-8} -2 q^{-10} + q^{-12} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_85.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{19}+2 q^{17}-2 q^{15}+2 q^{13}-2 q^{11}+q^7-q^5+3 q^3-q+2 q^{-1} - q^{-3} }[/math]
2 [math]\displaystyle{ q^{52}-2 q^{50}+4 q^{46}-5 q^{44}+q^{42}+4 q^{40}-8 q^{38}+5 q^{36}+6 q^{34}-10 q^{32}+q^{30}+9 q^{28}-5 q^{26}-6 q^{24}+6 q^{22}+3 q^{20}-8 q^{18}-q^{16}+9 q^{14}-5 q^{12}-6 q^{10}+11 q^8-10 q^4+9 q^2+6-9 q^{-2} + q^{-4} +6 q^{-6} -3 q^{-8} -2 q^{-10} + q^{-12} }[/math]
3 [math]\displaystyle{ -q^{99}+2 q^{97}-2 q^{93}-q^{91}+3 q^{89}+4 q^{87}-5 q^{85}-4 q^{83}-q^{81}+5 q^{79}+8 q^{77}-16 q^{73}-12 q^{71}+24 q^{69}+26 q^{67}-21 q^{65}-41 q^{63}+8 q^{61}+46 q^{59}+11 q^{57}-39 q^{55}-28 q^{53}+19 q^{51}+40 q^{49}+4 q^{47}-41 q^{45}-19 q^{43}+33 q^{41}+32 q^{39}-28 q^{37}-35 q^{35}+18 q^{33}+37 q^{31}-15 q^{29}-37 q^{27}+6 q^{25}+40 q^{23}+5 q^{21}-40 q^{19}-20 q^{17}+35 q^{15}+35 q^{13}-20 q^{11}-44 q^9-q^7+47 q^5+20 q^3-31 q-34 q^{-1} +16 q^{-3} +36 q^{-5} +3 q^{-7} -27 q^{-9} -13 q^{-11} +13 q^{-13} +15 q^{-15} -4 q^{-17} -9 q^{-19} -2 q^{-21} +3 q^{-23} +2 q^{-25} - q^{-27} }[/math]
4 [math]\displaystyle{ q^{160}-2 q^{158}+2 q^{154}-q^{152}+3 q^{150}-8 q^{148}+8 q^{144}+4 q^{142}+10 q^{140}-28 q^{138}-17 q^{136}+13 q^{134}+33 q^{132}+46 q^{130}-49 q^{128}-78 q^{126}-25 q^{124}+81 q^{122}+148 q^{120}-24 q^{118}-178 q^{116}-152 q^{114}+79 q^{112}+297 q^{110}+122 q^{108}-201 q^{106}-333 q^{104}-80 q^{102}+323 q^{100}+318 q^{98}-12 q^{96}-337 q^{94}-295 q^{92}+91 q^{90}+308 q^{88}+225 q^{86}-81 q^{84}-289 q^{82}-169 q^{80}+65 q^{78}+247 q^{76}+167 q^{74}-99 q^{72}-226 q^{70}-118 q^{68}+137 q^{66}+218 q^{64}+19 q^{62}-188 q^{60}-146 q^{58}+95 q^{56}+211 q^{54}+34 q^{52}-206 q^{50}-166 q^{48}+91 q^{46}+252 q^{44}+99 q^{42}-208 q^{40}-246 q^{38}-12 q^{36}+250 q^{34}+234 q^{32}-74 q^{30}-254 q^{28}-189 q^{26}+84 q^{24}+274 q^{22}+140 q^{20}-79 q^{18}-245 q^{16}-152 q^{14}+101 q^{12}+201 q^{10}+156 q^8-72 q^6-207 q^4-120 q^2+32+191 q^{-2} +124 q^{-4} -39 q^{-6} -132 q^{-8} -128 q^{-10} +33 q^{-12} +109 q^{-14} +83 q^{-16} + q^{-18} -90 q^{-20} -54 q^{-22} +44 q^{-26} +45 q^{-28} -7 q^{-30} -20 q^{-32} -20 q^{-34} -3 q^{-36} +12 q^{-38} +5 q^{-40} +2 q^{-42} -3 q^{-44} -2 q^{-46} + q^{-48} }[/math]
5 [math]\displaystyle{ -q^{235}+2 q^{233}-2 q^{229}+q^{227}-q^{225}+2 q^{223}+4 q^{221}-3 q^{219}-11 q^{217}-4 q^{215}+9 q^{213}+21 q^{211}+18 q^{209}-17 q^{207}-51 q^{205}-44 q^{203}+28 q^{201}+98 q^{199}+84 q^{197}-22 q^{195}-162 q^{193}-174 q^{191}+19 q^{189}+265 q^{187}+288 q^{185}+2 q^{183}-386 q^{181}-488 q^{179}-66 q^{177}+569 q^{175}+775 q^{173}+193 q^{171}-752 q^{169}-1154 q^{167}-475 q^{165}+857 q^{163}+1619 q^{161}+928 q^{159}-785 q^{157}-2007 q^{155}-1521 q^{153}+399 q^{151}+2165 q^{149}+2143 q^{147}+253 q^{145}-1932 q^{143}-2546 q^{141}-1047 q^{139}+1276 q^{137}+2531 q^{135}+1745 q^{133}-357 q^{131}-2058 q^{129}-2080 q^{127}-558 q^{125}+1227 q^{123}+1975 q^{121}+1235 q^{119}-325 q^{117}-1519 q^{115}-1513 q^{113}-399 q^{111}+914 q^{109}+1450 q^{107}+818 q^{105}-423 q^{103}-1220 q^{101}-910 q^{99}+164 q^{97}+985 q^{95}+831 q^{93}-128 q^{91}-927 q^{89}-727 q^{87}+260 q^{85}+1010 q^{83}+736 q^{81}-370 q^{79}-1229 q^{77}-907 q^{75}+388 q^{73}+1447 q^{71}+1210 q^{69}-207 q^{67}-1564 q^{65}-1578 q^{63}-155 q^{61}+1483 q^{59}+1896 q^{57}+664 q^{55}-1165 q^{53}-2042 q^{51}-1220 q^{49}+603 q^{47}+1928 q^{45}+1677 q^{43}+95 q^{41}-1500 q^{39}-1879 q^{37}-811 q^{35}+804 q^{33}+1735 q^{31}+1340 q^{29}-3 q^{27}-1218 q^{25}-1496 q^{23}-730 q^{21}+473 q^{19}+1260 q^{17}+1136 q^{15}+277 q^{13}-665 q^{11}-1117 q^9-817 q^7-14 q^5+728 q^3+944 q+553 q^{-1} -149 q^{-3} -708 q^{-5} -767 q^{-7} -339 q^{-9} +263 q^{-11} +643 q^{-13} +568 q^{-15} +157 q^{-17} -309 q^{-19} -522 q^{-21} -378 q^{-23} -11 q^{-25} +294 q^{-27} +362 q^{-29} +207 q^{-31} -55 q^{-33} -231 q^{-35} -221 q^{-37} -78 q^{-39} +68 q^{-41} +139 q^{-43} +113 q^{-45} +17 q^{-47} -55 q^{-49} -68 q^{-51} -38 q^{-53} +26 q^{-57} +28 q^{-59} +8 q^{-61} -5 q^{-63} -8 q^{-65} -5 q^{-67} -2 q^{-69} +3 q^{-71} +2 q^{-73} - q^{-75} }[/math]
6 [math]\displaystyle{ q^{324}-2 q^{322}+2 q^{318}-q^{316}+q^{314}-4 q^{312}+2 q^{310}-q^{308}+6 q^{306}+11 q^{304}-12 q^{302}-11 q^{300}-18 q^{298}+2 q^{296}+17 q^{294}+47 q^{292}+44 q^{290}-41 q^{288}-75 q^{286}-90 q^{284}-17 q^{282}+75 q^{280}+182 q^{278}+153 q^{276}-76 q^{274}-218 q^{272}-267 q^{270}-81 q^{268}+190 q^{266}+433 q^{264}+310 q^{262}-211 q^{260}-543 q^{258}-523 q^{256}+33 q^{254}+712 q^{252}+1028 q^{250}+384 q^{248}-974 q^{246}-1765 q^{244}-1298 q^{242}+534 q^{240}+2539 q^{238}+3158 q^{236}+1183 q^{234}-2494 q^{232}-5070 q^{230}-4381 q^{228}-69 q^{226}+5342 q^{224}+8081 q^{222}+5240 q^{220}-2104 q^{218}-9094 q^{216}-10797 q^{214}-5175 q^{212}+4807 q^{210}+12834 q^{208}+12918 q^{206}+4288 q^{204}-7748 q^{202}-15562 q^{200}-13829 q^{198}-3187 q^{196}+9931 q^{194}+17032 q^{192}+13631 q^{190}+2016 q^{188}-10703 q^{186}-16838 q^{184}-12833 q^{182}-1379 q^{180}+10291 q^{178}+15360 q^{176}+11501 q^{174}+1425 q^{172}-8783 q^{170}-13354 q^{168}-10152 q^{166}-1653 q^{164}+6958 q^{162}+11125 q^{160}+8833 q^{158}+1862 q^{156}-5568 q^{154}-9192 q^{152}-7229 q^{150}-1383 q^{148}+4763 q^{146}+7392 q^{144}+5244 q^{142}+18 q^{140}-4554 q^{138}-5416 q^{136}-2475 q^{134}+1912 q^{132}+4281 q^{130}+3028 q^{128}-816 q^{126}-3930 q^{124}-3547 q^{122}+90 q^{120}+4183 q^{118}+5212 q^{116}+2034 q^{114}-3488 q^{112}-7119 q^{110}-5611 q^{108}+407 q^{106}+6874 q^{104}+9026 q^{102}+4957 q^{100}-3165 q^{98}-9823 q^{96}-10083 q^{94}-3372 q^{92}+6017 q^{90}+11908 q^{88}+10233 q^{86}+1578 q^{84}-8540 q^{82}-13387 q^{80}-9661 q^{78}+278 q^{76}+10299 q^{74}+14089 q^{72}+9013 q^{70}-1786 q^{68}-11492 q^{66}-14080 q^{64}-8233 q^{62}+2568 q^{60}+11791 q^{58}+13804 q^{56}+7594 q^{54}-2862 q^{52}-11230 q^{50}-13011 q^{48}-7287 q^{46}+2356 q^{44}+10184 q^{42}+11892 q^{40}+6984 q^{38}-1264 q^{36}-8438 q^{34}-10547 q^{32}-6917 q^{30}+35 q^{28}+6326 q^{26}+8811 q^{24}+6784 q^{22}+1401 q^{20}-4079 q^{18}-7044 q^{16}-6261 q^{14}-2637 q^{12}+1832 q^{10}+5138 q^8+5569 q^6+3479 q^4-60 q^2-3124-4569 q^{-2} -3845 q^{-4} -1314 q^{-6} +1442 q^{-8} +3385 q^{-10} +3544 q^{-12} +2218 q^{-14} -97 q^{-16} -2170 q^{-18} -2964 q^{-20} -2421 q^{-22} -772 q^{-24} +969 q^{-26} +2230 q^{-28} +2212 q^{-30} +1189 q^{-32} -191 q^{-34} -1351 q^{-36} -1702 q^{-38} -1309 q^{-40} -243 q^{-42} +673 q^{-44} +1138 q^{-46} +1032 q^{-48} +474 q^{-50} -190 q^{-52} -698 q^{-54} -689 q^{-56} -435 q^{-58} -31 q^{-60} +290 q^{-62} +421 q^{-64} +329 q^{-66} +72 q^{-68} -89 q^{-70} -195 q^{-72} -177 q^{-74} -93 q^{-76} +20 q^{-78} +83 q^{-80} +69 q^{-82} +51 q^{-84} +7 q^{-86} -20 q^{-88} -34 q^{-90} -16 q^{-92} + q^{-96} +8 q^{-98} +5 q^{-100} +2 q^{-102} -3 q^{-104} -2 q^{-106} + q^{-108} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{26}+q^{24}-q^{22}+q^{20}-q^{16}+q^{14}-3 q^{12}+2 q^{10}+2 q^6+2 q^4+1- q^{-2} }[/math]
1,1 [math]\displaystyle{ q^{76}-4 q^{74}+8 q^{72}-12 q^{70}+20 q^{68}-32 q^{66}+42 q^{64}-50 q^{62}+60 q^{60}-72 q^{58}+74 q^{56}-68 q^{54}+70 q^{52}-72 q^{50}+72 q^{48}-76 q^{46}+82 q^{44}-86 q^{42}+70 q^{40}-40 q^{38}-10 q^{36}+84 q^{34}-158 q^{32}+230 q^{30}-287 q^{28}+326 q^{26}-338 q^{24}+310 q^{22}-271 q^{20}+202 q^{18}-120 q^{16}+24 q^{14}+69 q^{12}-136 q^{10}+200 q^8-216 q^6+213 q^4-178 q^2+140-96 q^{-2} +54 q^{-4} -30 q^{-6} +12 q^{-8} -4 q^{-10} + q^{-12} }[/math]
2,0 [math]\displaystyle{ q^{66}-q^{64}+q^{60}-q^{58}+q^{56}-2 q^{54}-q^{52}+2 q^{48}-3 q^{44}+3 q^{42}+4 q^{40}-2 q^{38}-6 q^{36}+5 q^{34}+2 q^{32}-3 q^{30}+q^{26}-2 q^{24}-5 q^{22}+q^{20}-q^{18}-3 q^{16}+3 q^{14}+6 q^{12}-q^{10}+q^8+7 q^6+4 q^4-3 q^2-1+2 q^{-2} -3 q^{-6} - q^{-8} + q^{-10} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{62}-2 q^{60}-q^{58}+5 q^{56}-3 q^{54}-4 q^{52}+8 q^{50}-q^{48}-7 q^{46}+7 q^{44}+q^{42}-9 q^{40}+4 q^{38}+2 q^{36}-6 q^{34}+q^{32}+3 q^{30}+2 q^{28}-4 q^{26}+5 q^{22}-8 q^{20}-2 q^{18}+7 q^{16}-3 q^{14}+2 q^{12}+9 q^{10}-2 q^8+5 q^4-5 q^2+1+ q^{-2} -2 q^{-4} + q^{-6} }[/math]
1,0,0 [math]\displaystyle{ -q^{33}+q^{31}-2 q^{29}+2 q^{27}-2 q^{25}+2 q^{23}-q^{21}-q^{17}-q^{15}+q^{13}+4 q^9+3 q^5-q^3+q- q^{-1} }[/math]
1,0,1 [math]\displaystyle{ q^{100}-4 q^{98}+6 q^{96}-2 q^{94}-7 q^{92}+17 q^{90}-20 q^{88}+7 q^{86}+16 q^{84}-31 q^{82}+29 q^{80}-13 q^{78}-12 q^{76}+27 q^{74}-24 q^{72}+16 q^{70}-3 q^{68}-4 q^{66}-2 q^{64}+21 q^{62}-41 q^{60}+42 q^{58}-19 q^{56}-32 q^{54}+83 q^{52}-117 q^{50}+117 q^{48}-89 q^{46}+44 q^{44}+9 q^{42}-53 q^{40}+96 q^{38}-114 q^{36}+120 q^{34}-109 q^{32}+76 q^{30}-56 q^{28}+12 q^{26}+7 q^{24}-57 q^{22}+93 q^{20}-110 q^{18}+126 q^{16}-84 q^{14}+59 q^{12}+10 q^{10}-41 q^8+75 q^6-75 q^4+49 q^2-24-6 q^{-2} +15 q^{-4} -16 q^{-6} +10 q^{-8} -4 q^{-10} + q^{-12} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{76}-q^{74}-2 q^{72}+3 q^{70}+q^{68}-6 q^{66}+q^{64}+8 q^{62}-2 q^{60}-4 q^{58}+7 q^{56}+3 q^{54}-7 q^{52}-4 q^{50}+4 q^{48}-7 q^{46}-8 q^{44}+6 q^{42}+2 q^{40}-7 q^{38}+6 q^{36}+9 q^{34}-8 q^{32}-3 q^{30}+4 q^{28}-q^{26}-6 q^{24}+2 q^{22}+6 q^{20}+3 q^{18}+2 q^{16}+6 q^{14}+2 q^{12}+2 q^8-q^6-2 q^4+q^2- q^{-2} + q^{-4} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{40}+q^{38}-2 q^{36}+q^{34}-q^{32}+q^{28}-q^{26}+q^{24}-2 q^{22}+q^{20}-2 q^{18}+2 q^{16}+3 q^{12}+2 q^{10}+q^8+2 q^6-q^4+q^2-1 }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{62}+2 q^{60}-3 q^{58}+5 q^{56}-7 q^{54}+8 q^{52}-10 q^{50}+11 q^{48}-11 q^{46}+11 q^{44}-9 q^{42}+5 q^{40}-6 q^{36}+12 q^{34}-17 q^{32}+21 q^{30}-24 q^{28}+24 q^{26}-24 q^{24}+19 q^{22}-14 q^{20}+8 q^{18}-q^{16}-3 q^{14}+10 q^{12}-11 q^{10}+14 q^8-12 q^6+13 q^4-9 q^2+7-5 q^{-2} +2 q^{-4} - q^{-6} }[/math]
1,0 [math]\displaystyle{ q^{100}-2 q^{96}-2 q^{94}+q^{92}+5 q^{90}+2 q^{88}-5 q^{86}-6 q^{84}+q^{82}+9 q^{80}+5 q^{78}-6 q^{76}-9 q^{74}+10 q^{70}+5 q^{68}-8 q^{66}-10 q^{64}+3 q^{62}+11 q^{60}-11 q^{56}-4 q^{54}+9 q^{52}+7 q^{50}-5 q^{48}-7 q^{46}+4 q^{44}+7 q^{42}-3 q^{40}-8 q^{38}+2 q^{36}+8 q^{34}-q^{32}-12 q^{30}-6 q^{28}+10 q^{26}+11 q^{24}-3 q^{22}-13 q^{20}+14 q^{16}+10 q^{14}-7 q^{12}-10 q^{10}+3 q^8+11 q^6+q^4-7 q^2-4+4 q^{-2} +3 q^{-4} -2 q^{-6} -2 q^{-8} + q^{-12} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{86}-2 q^{84}+q^{82}-2 q^{80}+5 q^{78}-5 q^{76}+4 q^{74}-6 q^{72}+9 q^{70}-7 q^{68}+7 q^{66}-9 q^{64}+9 q^{62}-7 q^{60}+6 q^{58}-6 q^{56}+2 q^{54}-5 q^{50}+7 q^{48}-12 q^{46}+13 q^{44}-16 q^{42}+18 q^{40}-19 q^{38}+20 q^{36}-18 q^{34}+16 q^{32}-15 q^{30}+11 q^{28}-7 q^{26}+4 q^{24}-q^{22}-q^{20}+9 q^{18}-5 q^{16}+11 q^{14}-9 q^{12}+13 q^{10}-10 q^8+9 q^6-9 q^4+7 q^2-5+3 q^{-2} -2 q^{-4} + q^{-6} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{148}-2 q^{146}+3 q^{144}-4 q^{142}+2 q^{140}-q^{138}-2 q^{136}+8 q^{134}-11 q^{132}+14 q^{130}-12 q^{128}+6 q^{126}+2 q^{124}-11 q^{122}+20 q^{120}-24 q^{118}+21 q^{116}-14 q^{114}+q^{112}+9 q^{110}-16 q^{108}+22 q^{106}-24 q^{104}+21 q^{102}-17 q^{100}+3 q^{98}+12 q^{96}-26 q^{94}+33 q^{92}-29 q^{90}+14 q^{88}+9 q^{86}-28 q^{84}+33 q^{82}-18 q^{80}-7 q^{78}+34 q^{76}-45 q^{74}+30 q^{72}+5 q^{70}-40 q^{68}+64 q^{66}-65 q^{64}+40 q^{62}-2 q^{60}-38 q^{58}+62 q^{56}-69 q^{54}+51 q^{52}-22 q^{50}-13 q^{48}+37 q^{46}-47 q^{44}+43 q^{42}-20 q^{40}-10 q^{38}+33 q^{36}-39 q^{34}+27 q^{32}+7 q^{30}-38 q^{28}+60 q^{26}-49 q^{24}+21 q^{22}+22 q^{20}-54 q^{18}+68 q^{16}-53 q^{14}+23 q^{12}+9 q^{10}-34 q^8+42 q^6-33 q^4+18 q^2-2-8 q^{-2} +9 q^{-4} -9 q^{-6} +5 q^{-8} -2 q^{-10} + q^{-12} }[/math]

.

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 85"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-4 t^3+8 t^2-10 t+11-10 t^{-1} +8 t^{-2} -4 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+4 z^6+4 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]= { 57, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q+3-4 q^{-1} +7 q^{-2} -8 q^{-3} +9 q^{-4} -9 q^{-5} +7 q^{-6} -5 q^{-7} +3 q^{-8} - q^{-9} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ a^4 z^8-a^6 z^6+6 a^4 z^6-a^2 z^6-4 a^6 z^4+12 a^4 z^4-4 a^2 z^4-4 a^6 z^2+9 a^4 z^2-3 a^2 z^2-a^6+a^4+a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}-z^2 a^{10}+5 z^5 a^9-4 z^3 a^9+z a^9+6 z^6 a^8-7 z^4 a^8+z^2 a^8+6 z^7 a^7-10 z^5 a^7+2 z^3 a^7+5 z^8 a^6-12 z^6 a^6+8 z^4 a^6-5 z^2 a^6+a^6+2 z^9 a^5-15 z^5 a^5+14 z^3 a^5-2 z a^5+8 z^8 a^4-32 z^6 a^4+37 z^4 a^4-14 z^2 a^4+a^4+2 z^9 a^3-5 z^7 a^3-4 z^5 a^3+11 z^3 a^3-2 z a^3+3 z^8 a^2-14 z^6 a^2+19 z^4 a^2-7 z^2 a^2-a^2+z^7 a-4 z^5 a+4 z^3 a-z a }[/math]

Vassiliev invariants

V2 and V3: (2, -3)
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{ -24 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{172}{3} }[/math] [math]\displaystyle{ -\frac{52}{3} }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -208 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{1376}{3} }[/math] [math]\displaystyle{ -\frac{416}{3} }[/math] [math]\displaystyle{ \frac{12631}{15} }[/math] [math]\displaystyle{ \frac{316}{15} }[/math] [math]\displaystyle{ -\frac{1436}{45} }[/math] [math]\displaystyle{ -\frac{55}{9} }[/math] [math]\displaystyle{ -\frac{569}{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]-4 is the signature of 10 85. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -7-6-5-4-3-2-10123χ
3          1-1
1         2 2
-1        21 -1
-3       52  3
-5      43   -1
-7     54    1
-9    44     0
-11   35      -2
-13  24       2
-15 13        -2
-17 2         2
-191          -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, 85]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 85]]
Out[3]=  
PD[X[6, 2, 7, 1], X[16, 6, 17, 5], X[18, 11, 19, 12], X[14, 7, 15, 8], 

 X[8, 3, 9, 4], X[4, 9, 5, 10], X[20, 13, 1, 14], X[10, 17, 11, 18], 



  X[12, 19, 13, 20], X[2, 16, 3, 15]]
In[4]:=
GaussCode[Knot[10, 85]]
Out[4]=  
GaussCode[1, -10, 5, -6, 2, -1, 4, -5, 6, -8, 3, -9, 7, -4, 10, -2, 8, 
  -3, 9, -7]
In[5]:=
BR[Knot[10, 85]]
Out[5]=  
BR[3, {-1, -1, -1, -1, 2, -1, -1, 2, -1, 2}]
In[6]:=
alex = Alexander[Knot[10, 85]][t]
Out[6]=  
      -4   4    8    10             2      3    4

11 + t   - -- + -- - -- - 10 t + 8 t  - 4 t  + t



           3    2   t


           t    t
In[7]:=
Conway[Knot[10, 85]][z]
Out[7]=  
       2      4      6    8
1 + 2 z  + 4 z  + 4 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 85]}
In[9]:=
{KnotDet[Knot[10, 85]], KnotSignature[Knot[10, 85]]}
Out[9]=  
{57, -4}
In[10]:=
J=Jones[Knot[10, 85]][q]
Out[10]=  
     -9   3    5    7    9    9    8    7    4

3 - q   + -- - -- + -- - -- + -- - -- + -- - - - q



          8    7    6    5    4    3    2   q


          q    q    q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 85]}
In[12]:=
A2Invariant[Knot[10, 85]][q]
Out[12]=  
     -26    -24    -22    -20    -16    -14    3     2    2    2     2

1 - q    + q    - q    + q    - q    + q    - --- + --- + -- + -- - q



                                              12    10    6    4


                                              q     q     q    q
In[13]:=
Kauffman[Knot[10, 85]][a, z]
Out[13]=  
  2    4    6            3        5      9        2  2       4  2

-a  + a  + a  - a z - 2 a  z - 2 a  z + a  z - 7 a  z  - 14 a  z  - 



    6  2    8  2    10  2        3       3  3       5  3      7  3
 5 a  z  + a  z  - a   z  + 4 a z  + 11 a  z  + 14 a  z  + 2 a  z  - 

    9  3    11  3       2  4       4  4      6  4      8  4
 4 a  z  + a   z  + 19 a  z  + 37 a  z  + 8 a  z  - 7 a  z  + 

    10  4        5      3  5       5  5       7  5      9  5
 3 a   z  - 4 a z  - 4 a  z  - 15 a  z  - 10 a  z  + 5 a  z  - 

     2  6       4  6       6  6      8  6      7      3  7      7  7
 14 a  z  - 32 a  z  - 12 a  z  + 6 a  z  + a z  - 5 a  z  + 6 a  z  + 

    2  8      4  8      6  8      3  9      5  9


  3 a  z  + 8 a  z  + 5 a  z  + 2 a  z  + 2 a  z
In[14]:=
{Vassiliev[2][Knot[10, 85]], Vassiliev[3][Knot[10, 85]]}
Out[14]=  
{0, -3}
In[15]:=
Kh[Knot[10, 85]][q, t]
Out[15]=  
3    5      1        2        1        3        2        4

-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 



5    3    19  7    17  6    15  6    15  5    13  5    13  4



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



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

  2
 t         2    3  3
 -- + 2 q t  + q  t


  q
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