10 109

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10 108.gif

10_108

10 110.gif

10_110

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10 109 Quick Notes


10 109 Further Notes and Views

Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X18,11,19,12 X16,7,17,8 X8,17,9,18 X20,15,1,16 X12,19,13,20 X14,6,15,5 X2,10,3,9 X4,14,5,13
Gauss code 1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, -6
Dowker-Thistlethwaite code 6 10 14 16 2 18 4 20 8 12
Conway Notation [2.2.2.2]

Three dimensional invariants

Symmetry type Negative amphicheiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 14.9002
A-Polynomial See Data:10 109/A-polynomial

[edit Notes for 10 109'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 10 109's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-4 t^3+10 t^2-17 t+21-17 t^{-1} +10 t^{-2} -4 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+4 z^6+6 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 85, 0 }
Jones polynomial [math]\displaystyle{ -q^5+3 q^4-7 q^3+11 q^2-13 q+15-13 q^{-1} +11 q^{-2} -7 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} +14 z^4-6 a^2 z^2-6 z^2 a^{-2} +15 z^2-3 a^2-3 a^{-2} +7 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +5 a^2 z^8+5 z^8 a^{-2} +10 z^8+5 a^3 z^7+6 a z^7+6 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-7 a^2 z^6-7 z^6 a^{-2} +3 z^6 a^{-4} -20 z^6+a^5 z^5-8 a^3 z^5-16 a z^5-16 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+6 a^2 z^4+6 z^4 a^{-2} -5 z^4 a^{-4} +22 z^4-2 a^5 z^3+4 a^3 z^3+13 a z^3+13 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2-7 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} -18 z^2+a^5 z-a^3 z-5 a z-5 z a^{-1} -z a^{-3} +z a^{-5} +3 a^2+3 a^{-2} +7 }[/math]
The A2 invariant [math]\displaystyle{ -q^{14}+q^{12}-3 q^{10}+q^8-q^4+5 q^2-1+5 q^{-2} - q^{-4} + q^{-8} -3 q^{-10} + q^{-12} - q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-8 q^{70}+q^{68}+14 q^{66}-32 q^{64}+51 q^{62}-62 q^{60}+51 q^{58}-20 q^{56}-39 q^{54}+113 q^{52}-171 q^{50}+187 q^{48}-138 q^{46}+19 q^{44}+124 q^{42}-249 q^{40}+297 q^{38}-239 q^{36}+89 q^{34}+90 q^{32}-231 q^{30}+264 q^{28}-177 q^{26}+13 q^{24}+150 q^{22}-232 q^{20}+187 q^{18}-37 q^{16}-151 q^{14}+301 q^{12}-336 q^{10}+248 q^8-53 q^6-170 q^4+353 q^2-417+353 q^{-2} -170 q^{-4} -53 q^{-6} +248 q^{-8} -336 q^{-10} +301 q^{-12} -151 q^{-14} -37 q^{-16} +187 q^{-18} -232 q^{-20} +150 q^{-22} +13 q^{-24} -177 q^{-26} +264 q^{-28} -231 q^{-30} +90 q^{-32} +89 q^{-34} -239 q^{-36} +297 q^{-38} -249 q^{-40} +124 q^{-42} +19 q^{-44} -138 q^{-46} +187 q^{-48} -171 q^{-50} +113 q^{-52} -39 q^{-54} -20 q^{-56} +51 q^{-58} -62 q^{-60} +51 q^{-62} -32 q^{-64} +14 q^{-66} + q^{-68} -8 q^{-70} +9 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_109.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{11}+2 q^9-4 q^7+4 q^5-2 q^3+2 q+2 q^{-1} -2 q^{-3} +4 q^{-5} -4 q^{-7} +2 q^{-9} - q^{-11} }[/math]
2 [math]\displaystyle{ q^{32}-2 q^{30}+7 q^{26}-10 q^{24}-6 q^{22}+26 q^{20}-14 q^{18}-27 q^{16}+38 q^{14}-38 q^{10}+26 q^8+16 q^6-26 q^4+21-26 q^{-4} +16 q^{-6} +26 q^{-8} -38 q^{-10} +38 q^{-14} -27 q^{-16} -14 q^{-18} +26 q^{-20} -6 q^{-22} -10 q^{-24} +7 q^{-26} -2 q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ -q^{63}+2 q^{61}-3 q^{57}-q^{55}+9 q^{53}+4 q^{51}-23 q^{49}-13 q^{47}+44 q^{45}+41 q^{43}-62 q^{41}-100 q^{39}+65 q^{37}+172 q^{35}-28 q^{33}-238 q^{31}-52 q^{29}+280 q^{27}+139 q^{25}-267 q^{23}-226 q^{21}+215 q^{19}+277 q^{17}-135 q^{15}-291 q^{13}+55 q^{11}+264 q^9+30 q^7-219 q^5-98 q^3+162 q+162 q^{-1} -98 q^{-3} -219 q^{-5} +30 q^{-7} +264 q^{-9} +55 q^{-11} -291 q^{-13} -135 q^{-15} +277 q^{-17} +215 q^{-19} -226 q^{-21} -267 q^{-23} +139 q^{-25} +280 q^{-27} -52 q^{-29} -238 q^{-31} -28 q^{-33} +172 q^{-35} +65 q^{-37} -100 q^{-39} -62 q^{-41} +41 q^{-43} +44 q^{-45} -13 q^{-47} -23 q^{-49} +4 q^{-51} +9 q^{-53} - q^{-55} -3 q^{-57} +2 q^{-61} - q^{-63} }[/math]
4 [math]\displaystyle{ q^{104}-2 q^{102}+3 q^{98}-3 q^{96}+2 q^{94}-7 q^{92}+4 q^{90}+20 q^{88}-11 q^{86}-14 q^{84}-47 q^{82}+15 q^{80}+121 q^{78}+49 q^{76}-62 q^{74}-276 q^{72}-140 q^{70}+316 q^{68}+462 q^{66}+219 q^{64}-653 q^{62}-905 q^{60}-12 q^{58}+1048 q^{56}+1376 q^{54}-258 q^{52}-1835 q^{50}-1447 q^{48}+613 q^{46}+2634 q^{44}+1386 q^{42}-1520 q^{40}-2875 q^{38}-1093 q^{36}+2454 q^{34}+2872 q^{32}+93 q^{30}-2801 q^{28}-2520 q^{26}+986 q^{24}+2865 q^{22}+1476 q^{20}-1528 q^{18}-2602 q^{16}-408 q^{14}+1822 q^{12}+1882 q^{10}-243 q^8-1912 q^6-1218 q^4+745 q^2+1861+745 q^{-2} -1218 q^{-4} -1912 q^{-6} -243 q^{-8} +1882 q^{-10} +1822 q^{-12} -408 q^{-14} -2602 q^{-16} -1528 q^{-18} +1476 q^{-20} +2865 q^{-22} +986 q^{-24} -2520 q^{-26} -2801 q^{-28} +93 q^{-30} +2872 q^{-32} +2454 q^{-34} -1093 q^{-36} -2875 q^{-38} -1520 q^{-40} +1386 q^{-42} +2634 q^{-44} +613 q^{-46} -1447 q^{-48} -1835 q^{-50} -258 q^{-52} +1376 q^{-54} +1048 q^{-56} -12 q^{-58} -905 q^{-60} -653 q^{-62} +219 q^{-64} +462 q^{-66} +316 q^{-68} -140 q^{-70} -276 q^{-72} -62 q^{-74} +49 q^{-76} +121 q^{-78} +15 q^{-80} -47 q^{-82} -14 q^{-84} -11 q^{-86} +20 q^{-88} +4 q^{-90} -7 q^{-92} +2 q^{-94} -3 q^{-96} +3 q^{-98} -2 q^{-102} + q^{-104} }[/math]
5 [math]\displaystyle{ -q^{155}+2 q^{153}-3 q^{149}+3 q^{147}+2 q^{145}-4 q^{143}-q^{141}-q^{139}-7 q^{137}+11 q^{135}+28 q^{133}+4 q^{131}-31 q^{129}-63 q^{127}-57 q^{125}+36 q^{123}+184 q^{121}+222 q^{119}+11 q^{117}-352 q^{115}-592 q^{113}-343 q^{111}+438 q^{109}+1253 q^{107}+1227 q^{105}-88 q^{103}-1972 q^{101}-2832 q^{99}-1334 q^{97}+2094 q^{95}+4991 q^{93}+4286 q^{91}-701 q^{89}-6719 q^{87}-8568 q^{85}-3193 q^{83}+6606 q^{81}+13159 q^{79}+9577 q^{77}-3359 q^{75}-16099 q^{73}-17271 q^{71}-3553 q^{69}+15601 q^{67}+24222 q^{65}+12985 q^{63}-10867 q^{61}-27958 q^{59}-22765 q^{57}+2383 q^{55}+27293 q^{53}+30347 q^{51}+7663 q^{49}-22195 q^{47}-33817 q^{45}-16957 q^{43}+14195 q^{41}+32822 q^{39}+23342 q^{37}-5386 q^{35}-28197 q^{33}-26026 q^{31}-2235 q^{29}+21581 q^{27}+25347 q^{25}+7640 q^{23}-14764 q^{21}-22543 q^{19}-10608 q^{17}+8924 q^{15}+18941 q^{13}+11978 q^{11}-4543 q^9-15841 q^7-12669 q^5+1379 q^3+13741 q+13741 q^{-1} +1379 q^{-3} -12669 q^{-5} -15841 q^{-7} -4543 q^{-9} +11978 q^{-11} +18941 q^{-13} +8924 q^{-15} -10608 q^{-17} -22543 q^{-19} -14764 q^{-21} +7640 q^{-23} +25347 q^{-25} +21581 q^{-27} -2235 q^{-29} -26026 q^{-31} -28197 q^{-33} -5386 q^{-35} +23342 q^{-37} +32822 q^{-39} +14195 q^{-41} -16957 q^{-43} -33817 q^{-45} -22195 q^{-47} +7663 q^{-49} +30347 q^{-51} +27293 q^{-53} +2383 q^{-55} -22765 q^{-57} -27958 q^{-59} -10867 q^{-61} +12985 q^{-63} +24222 q^{-65} +15601 q^{-67} -3553 q^{-69} -17271 q^{-71} -16099 q^{-73} -3359 q^{-75} +9577 q^{-77} +13159 q^{-79} +6606 q^{-81} -3193 q^{-83} -8568 q^{-85} -6719 q^{-87} -701 q^{-89} +4286 q^{-91} +4991 q^{-93} +2094 q^{-95} -1334 q^{-97} -2832 q^{-99} -1972 q^{-101} -88 q^{-103} +1227 q^{-105} +1253 q^{-107} +438 q^{-109} -343 q^{-111} -592 q^{-113} -352 q^{-115} +11 q^{-117} +222 q^{-119} +184 q^{-121} +36 q^{-123} -57 q^{-125} -63 q^{-127} -31 q^{-129} +4 q^{-131} +28 q^{-133} +11 q^{-135} -7 q^{-137} - q^{-139} - q^{-141} -4 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}+3 q^{210}-3 q^{208}-2 q^{206}+12 q^{202}-2 q^{200}-12 q^{198}+7 q^{196}-14 q^{194}-15 q^{192}+4 q^{190}+62 q^{188}+45 q^{186}-22 q^{184}-28 q^{182}-130 q^{180}-166 q^{178}-61 q^{176}+268 q^{174}+447 q^{172}+364 q^{170}+140 q^{168}-596 q^{166}-1263 q^{164}-1334 q^{162}-123 q^{160}+1590 q^{158}+2958 q^{156}+3267 q^{154}+805 q^{152}-3423 q^{150}-7276 q^{148}-7082 q^{146}-2166 q^{144}+6351 q^{142}+14700 q^{140}+15389 q^{138}+5690 q^{136}-11964 q^{134}-26799 q^{132}-29393 q^{130}-12890 q^{128}+18819 q^{126}+46840 q^{124}+52118 q^{122}+23431 q^{120}-27179 q^{118}-74733 q^{116}-85092 q^{114}-40712 q^{112}+39679 q^{110}+112342 q^{108}+126233 q^{106}+63203 q^{104}-55078 q^{102}-158976 q^{100}-176923 q^{98}-84719 q^{96}+76363 q^{94}+210285 q^{92}+230455 q^{90}+102841 q^{88}-104731 q^{86}-266784 q^{84}-275385 q^{82}-109284 q^{80}+138428 q^{78}+319539 q^{76}+306278 q^{74}+99016 q^{72}-180371 q^{70}-356738 q^{68}-313153 q^{66}-71885 q^{64}+221829 q^{62}+374479 q^{60}+292206 q^{58}+26369 q^{56}-251667 q^{54}-365377 q^{52}-246762 q^{50}+25519 q^{48}+266682 q^{46}+329262 q^{44}+180904 q^{42}-70774 q^{40}-260866 q^{38}-272938 q^{36}-110226 q^{34}+105095 q^{32}+235464 q^{30}+203724 q^{28}+47380 q^{26}-123571 q^{24}-197604 q^{22}-135835 q^{20}+5058 q^{18}+128690 q^{16}+154331 q^{14}+76682 q^{12}-46056 q^{10}-127127 q^8-115073 q^6-24477 q^4+80664 q^2+125109+80664 q^{-2} -24477 q^{-4} -115073 q^{-6} -127127 q^{-8} -46056 q^{-10} +76682 q^{-12} +154331 q^{-14} +128690 q^{-16} +5058 q^{-18} -135835 q^{-20} -197604 q^{-22} -123571 q^{-24} +47380 q^{-26} +203724 q^{-28} +235464 q^{-30} +105095 q^{-32} -110226 q^{-34} -272938 q^{-36} -260866 q^{-38} -70774 q^{-40} +180904 q^{-42} +329262 q^{-44} +266682 q^{-46} +25519 q^{-48} -246762 q^{-50} -365377 q^{-52} -251667 q^{-54} +26369 q^{-56} +292206 q^{-58} +374479 q^{-60} +221829 q^{-62} -71885 q^{-64} -313153 q^{-66} -356738 q^{-68} -180371 q^{-70} +99016 q^{-72} +306278 q^{-74} +319539 q^{-76} +138428 q^{-78} -109284 q^{-80} -275385 q^{-82} -266784 q^{-84} -104731 q^{-86} +102841 q^{-88} +230455 q^{-90} +210285 q^{-92} +76363 q^{-94} -84719 q^{-96} -176923 q^{-98} -158976 q^{-100} -55078 q^{-102} +63203 q^{-104} +126233 q^{-106} +112342 q^{-108} +39679 q^{-110} -40712 q^{-112} -85092 q^{-114} -74733 q^{-116} -27179 q^{-118} +23431 q^{-120} +52118 q^{-122} +46840 q^{-124} +18819 q^{-126} -12890 q^{-128} -29393 q^{-130} -26799 q^{-132} -11964 q^{-134} +5690 q^{-136} +15389 q^{-138} +14700 q^{-140} +6351 q^{-142} -2166 q^{-144} -7082 q^{-146} -7276 q^{-148} -3423 q^{-150} +805 q^{-152} +3267 q^{-154} +2958 q^{-156} +1590 q^{-158} -123 q^{-160} -1334 q^{-162} -1263 q^{-164} -596 q^{-166} +140 q^{-168} +364 q^{-170} +447 q^{-172} +268 q^{-174} -61 q^{-176} -166 q^{-178} -130 q^{-180} -28 q^{-182} -22 q^{-184} +45 q^{-186} +62 q^{-188} +4 q^{-190} -15 q^{-192} -14 q^{-194} +7 q^{-196} -12 q^{-198} -2 q^{-200} +12 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}-3 q^{10}+q^8-q^4+5 q^2-1+5 q^{-2} - q^{-4} + q^{-8} -3 q^{-10} + q^{-12} - q^{-14} }[/math]
1,1 [math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+60 q^{36}-116 q^{34}+204 q^{32}-334 q^{30}+514 q^{28}-738 q^{26}+976 q^{24}-1190 q^{22}+1334 q^{20}-1352 q^{18}+1186 q^{16}-840 q^{14}+312 q^{12}+332 q^{10}-1030 q^8+1690 q^6-2220 q^4+2582 q^2-2694+2582 q^{-2} -2220 q^{-4} +1690 q^{-6} -1030 q^{-8} +332 q^{-10} +312 q^{-12} -840 q^{-14} +1186 q^{-16} -1352 q^{-18} +1334 q^{-20} -1190 q^{-22} +976 q^{-24} -738 q^{-26} +514 q^{-28} -334 q^{-30} +204 q^{-32} -116 q^{-34} +60 q^{-36} -28 q^{-38} +12 q^{-40} -4 q^{-42} + q^{-44} }[/math]
2,0 [math]\displaystyle{ q^{38}-q^{36}+4 q^{32}-3 q^{30}-5 q^{28}+7 q^{26}+3 q^{24}-9 q^{22}-7 q^{20}+9 q^{18}+5 q^{16}-19 q^{14}+2 q^{12}+14 q^{10}-8 q^8-6 q^6+12 q^4+7 q^2-6+7 q^{-2} +12 q^{-4} -6 q^{-6} -8 q^{-8} +14 q^{-10} +2 q^{-12} -19 q^{-14} +5 q^{-16} +9 q^{-18} -7 q^{-20} -9 q^{-22} +3 q^{-24} +7 q^{-26} -5 q^{-28} -3 q^{-30} +4 q^{-32} - q^{-36} + q^{-38} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{34}-2 q^{32}+q^{30}+5 q^{28}-9 q^{26}+2 q^{24}+15 q^{22}-22 q^{20}+2 q^{18}+23 q^{16}-32 q^{14}-q^{12}+22 q^{10}-22 q^8-3 q^6+19 q^4+4 q^2+4 q^{-2} +19 q^{-4} -3 q^{-6} -22 q^{-8} +22 q^{-10} - q^{-12} -32 q^{-14} +23 q^{-16} +2 q^{-18} -22 q^{-20} +15 q^{-22} +2 q^{-24} -9 q^{-26} +5 q^{-28} + q^{-30} -2 q^{-32} + q^{-34} }[/math]
1,0,0 [math]\displaystyle{ -q^{17}+q^{15}-4 q^{13}+2 q^{11}-4 q^9+2 q^7-q^5+4 q^3+3 q+3 q^{-1} +4 q^{-3} - q^{-5} +2 q^{-7} -4 q^{-9} +2 q^{-11} -4 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}+23 q^{46}-72 q^{44}+110 q^{42}-85 q^{40}-44 q^{38}+252 q^{36}-425 q^{34}+405 q^{32}-93 q^{30}-430 q^{28}+923 q^{26}-1076 q^{24}+723 q^{22}+48 q^{20}-926 q^{18}+1449 q^{16}-1414 q^{14}+762 q^{12}+89 q^{10}-756 q^8+914 q^6-568 q^4+142 q^2+113+142 q^{-2} -568 q^{-4} +914 q^{-6} -756 q^{-8} +89 q^{-10} +762 q^{-12} -1414 q^{-14} +1449 q^{-16} -926 q^{-18} +48 q^{-20} +723 q^{-22} -1076 q^{-24} +923 q^{-26} -430 q^{-28} -93 q^{-30} +405 q^{-32} -425 q^{-34} +252 q^{-36} -44 q^{-38} -85 q^{-40} +110 q^{-42} -72 q^{-44} +23 q^{-46} +7 q^{-48} -14 q^{-50} +10 q^{-52} -4 q^{-54} + q^{-56} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{40}-q^{38}+5 q^{34}-3 q^{32}-4 q^{30}+13 q^{28}-3 q^{26}-15 q^{24}+10 q^{22}+9 q^{20}-25 q^{18}-14 q^{16}+13 q^{14}-7 q^{12}-28 q^{10}+9 q^8+30 q^6-14 q^4+6 q^2+46+6 q^{-2} -14 q^{-4} +30 q^{-6} +9 q^{-8} -28 q^{-10} -7 q^{-12} +13 q^{-14} -14 q^{-16} -25 q^{-18} +9 q^{-20} +10 q^{-22} -15 q^{-24} -3 q^{-26} +13 q^{-28} -4 q^{-30} -3 q^{-32} +5 q^{-34} - q^{-38} + q^{-40} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{20}+q^{18}-4 q^{16}+q^{14}-3 q^{12}-2 q^{10}+q^8-q^6+5 q^4+2 q^2+7+2 q^{-2} +5 q^{-4} - q^{-6} + q^{-8} -2 q^{-10} -3 q^{-12} + q^{-14} -4 q^{-16} + q^{-18} - q^{-20} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+9 q^{28}-15 q^{26}+22 q^{24}-29 q^{22}+34 q^{20}-36 q^{18}+31 q^{16}-24 q^{14}+11 q^{12}+4 q^{10}-22 q^8+41 q^6-53 q^4+66 q^2-66+66 q^{-2} -53 q^{-4} +41 q^{-6} -22 q^{-8} +4 q^{-10} +11 q^{-12} -24 q^{-14} +31 q^{-16} -36 q^{-18} +34 q^{-20} -29 q^{-22} +22 q^{-24} -15 q^{-26} +9 q^{-28} -5 q^{-30} +2 q^{-32} - q^{-34} }[/math]
1,0 [math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+7 q^{46}-12 q^{42}-9 q^{40}+12 q^{38}+22 q^{36}-3 q^{34}-31 q^{32}-15 q^{30}+28 q^{28}+29 q^{26}-16 q^{24}-39 q^{22}-4 q^{20}+34 q^{18}+15 q^{16}-26 q^{14}-22 q^{12}+17 q^{10}+24 q^8-6 q^6-22 q^4+6 q^2+27+6 q^{-2} -22 q^{-4} -6 q^{-6} +24 q^{-8} +17 q^{-10} -22 q^{-12} -26 q^{-14} +15 q^{-16} +34 q^{-18} -4 q^{-20} -39 q^{-22} -16 q^{-24} +29 q^{-26} +28 q^{-28} -15 q^{-30} -31 q^{-32} -3 q^{-34} +22 q^{-36} +12 q^{-38} -9 q^{-40} -12 q^{-42} +7 q^{-46} +3 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}+8 q^{38}-12 q^{36}+14 q^{34}-17 q^{32}+24 q^{30}-28 q^{28}+26 q^{26}-27 q^{24}+26 q^{22}-24 q^{20}+8 q^{18}-11 q^{16}-3 q^{14}+11 q^{12}-28 q^{10}+31 q^8-34 q^6+56 q^4-43 q^2+58-43 q^{-2} +56 q^{-4} -34 q^{-6} +31 q^{-8} -28 q^{-10} +11 q^{-12} -3 q^{-14} -11 q^{-16} +8 q^{-18} -24 q^{-20} +26 q^{-22} -27 q^{-24} +26 q^{-26} -28 q^{-28} +24 q^{-30} -17 q^{-32} +14 q^{-34} -12 q^{-36} +8 q^{-38} -4 q^{-40} +3 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}+9 q^{72}-8 q^{70}+q^{68}+14 q^{66}-32 q^{64}+51 q^{62}-62 q^{60}+51 q^{58}-20 q^{56}-39 q^{54}+113 q^{52}-171 q^{50}+187 q^{48}-138 q^{46}+19 q^{44}+124 q^{42}-249 q^{40}+297 q^{38}-239 q^{36}+89 q^{34}+90 q^{32}-231 q^{30}+264 q^{28}-177 q^{26}+13 q^{24}+150 q^{22}-232 q^{20}+187 q^{18}-37 q^{16}-151 q^{14}+301 q^{12}-336 q^{10}+248 q^8-53 q^6-170 q^4+353 q^2-417+353 q^{-2} -170 q^{-4} -53 q^{-6} +248 q^{-8} -336 q^{-10} +301 q^{-12} -151 q^{-14} -37 q^{-16} +187 q^{-18} -232 q^{-20} +150 q^{-22} +13 q^{-24} -177 q^{-26} +264 q^{-28} -231 q^{-30} +90 q^{-32} +89 q^{-34} -239 q^{-36} +297 q^{-38} -249 q^{-40} +124 q^{-42} +19 q^{-44} -138 q^{-46} +187 q^{-48} -171 q^{-50} +113 q^{-52} -39 q^{-54} -20 q^{-56} +51 q^{-58} -62 q^{-60} +51 q^{-62} -32 q^{-64} +14 q^{-66} + q^{-68} -8 q^{-70} +9 q^{-72} -8 q^{-74} +5 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["10 109"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-4 t^3+10 t^2-17 t+21-17 t^{-1} +10 t^{-2} -4 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+4 z^6+6 z^4+3 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]= { 85, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+3 q^4-7 q^3+11 q^2-13 q+15-13 q^{-1} +11 q^{-2} -7 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} +14 z^4-6 a^2 z^2-6 z^2 a^{-2} +15 z^2-3 a^2-3 a^{-2} +7 }[/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} +5 a^2 z^8+5 z^8 a^{-2} +10 z^8+5 a^3 z^7+6 a z^7+6 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-7 a^2 z^6-7 z^6 a^{-2} +3 z^6 a^{-4} -20 z^6+a^5 z^5-8 a^3 z^5-16 a z^5-16 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+6 a^2 z^4+6 z^4 a^{-2} -5 z^4 a^{-4} +22 z^4-2 a^5 z^3+4 a^3 z^3+13 a z^3+13 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2-7 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} -18 z^2+a^5 z-a^3 z-5 a z-5 z a^{-1} -z a^{-3} +z a^{-5} +3 a^2+3 a^{-2} +7 }[/math]

Vassiliev invariants

V2 and V3: (3, 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{ 12 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 62 }[/math] [math]\displaystyle{ -14 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 744 }[/math] [math]\displaystyle{ -168 }[/math] [math]\displaystyle{ \frac{6831}{10} }[/math] [math]\displaystyle{ \frac{1298}{5} }[/math] [math]\displaystyle{ -\frac{3538}{15} }[/math] [math]\displaystyle{ \frac{209}{6} }[/math] [math]\displaystyle{ -\frac{1169}{10} }[/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 10 109. 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        51 -4
5       62  4
3      75   -2
1     86    2
-1    68     2
-3   57      -2
-5  26       4
-7 15        -4
-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, 109]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 109]]
Out[3]=  
PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[18, 11, 19, 12], X[16, 7, 17, 8], 

 X[8, 17, 9, 18], X[20, 15, 1, 16], X[12, 19, 13, 20], 



  X[14, 6, 15, 5], X[2, 10, 3, 9], X[4, 14, 5, 13]]
In[4]:=
GaussCode[Knot[10, 109]]
Out[4]=  
GaussCode[1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, 
  -3, 7, -6]
In[5]:=
BR[Knot[10, 109]]
Out[5]=  
BR[3, {-1, -1, 2, -1, 2, 2, -1, -1, 2, 2}]
In[6]:=
alex = Alexander[Knot[10, 109]][t]
Out[6]=  
      -4   4    10   17              2      3    4

21 + t   - -- + -- - -- - 17 t + 10 t  - 4 t  + t



           3    2   t


           t    t
In[7]:=
Conway[Knot[10, 109]][z]
Out[7]=  
       2      4      6    8
1 + 3 z  + 6 z  + 4 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 109]}
In[9]:=
{KnotDet[Knot[10, 109]], KnotSignature[Knot[10, 109]]}
Out[9]=  
{85, 0}
In[10]:=
J=Jones[Knot[10, 109]][q]
Out[10]=  
      -5   3    7    11   13              2      3      4    5

15 - q   + -- - -- + -- - -- - 13 q + 11 q  - 7 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, 81], Knot[10, 109]}
In[12]:=
A2Invariant[Knot[10, 109]][q]
Out[12]=  
      -14    -12    3     -8    -4   5       2    4    8      10

-1 - q    + q    - --- + q   - q   + -- + 5 q  - q  + q  - 3 q   + 



                   10                2
                  q                 q

  12    14


  q   - q
In[13]:=
Kauffman[Knot[10, 109]][a, z]
Out[13]=  
                                                                 2

   3       2   z    z    5 z            3      5         2   2 z



7 + -- + 3 a  + -- - -- - --- - 5 a z - a  z + a  z - 18 z  + ---- - 



    2           5    3    a                                    4
   a           a    a                                         a

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

                      4      4                        5      5
    5  3       4   5 z    6 z       2  4      4  4   z    8 z
 2 a  z  + 22 z  - ---- + ---- + 6 a  z  - 5 a  z  + -- - ---- - 
                     4      2                         5     3
                    a      a                         a     a

     5                                          6      6
 16 z          5      3  5    5  5       6   3 z    7 z       2  6
 ----- - 16 a z  - 8 a  z  + a  z  - 20 z  + ---- - ---- - 7 a  z  + 
   a                                           4      2
                                              a      a

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

    9
 2 z         9
 ---- + 2 a z


   a
In[14]:=
{Vassiliev[2][Knot[10, 109]], Vassiliev[3][Knot[10, 109]]}
Out[14]=  
{0, 0}
In[15]:=
Kh[Knot[10, 109]][q, t]
Out[15]=  
8           1        2       1       5       2       6       5

- + 8 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

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

  7  4      9  4    11  5


  q  t  + 2 q  t  + q   t
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