10 79

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10 78.gif

10_78

10 80.gif

10_80

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10 79 Quick Notes


10 79 Further Notes and Views

Knot presentations

Planar diagram presentation X6271 X8493 X12,6,13,5 X18,13,19,14 X16,9,17,10 X10,17,11,18 X20,15,1,16 X14,19,15,20 X2837 X4,12,5,11
Gauss code 1, -9, 2, -10, 3, -1, 9, -2, 5, -6, 10, -3, 4, -8, 7, -5, 6, -4, 8, -7
Dowker-Thistlethwaite code 6 8 12 2 16 4 18 20 10 14
Conway Notation [(3,2)(3,2)]

Three dimensional invariants

Symmetry type Negative amphicheiral
Unknotting number [math]\displaystyle{ \{2,3\} }[/math]
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 12.5403
A-Polynomial See Data:10 79/A-polynomial

[edit Notes for 10 79'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 79's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-3 t^3+7 t^2-12 t+15-12 t^{-1} +7 t^{-2} -3 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+5 z^6+9 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 61, 0 }
Jones polynomial [math]\displaystyle{ -q^5+2 q^4-5 q^3+8 q^2-9 q+11-9 q^{-1} +8 q^{-2} -5 q^{-3} +2 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +7 z^6-5 a^2 z^4-5 z^4 a^{-2} +19 z^4-9 a^2 z^2-9 z^2 a^{-2} +23 z^2-5 a^2-5 a^{-2} +11 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^9+z^9 a^{-1} +3 a^2 z^8+3 z^8 a^{-2} +6 z^8+3 a^3 z^7+4 a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +2 a^4 z^6-7 a^2 z^6-7 z^6 a^{-2} +2 z^6 a^{-4} -18 z^6+a^5 z^5-6 a^3 z^5-15 a z^5-15 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} -4 a^4 z^4+12 a^2 z^4+12 z^4 a^{-2} -4 z^4 a^{-4} +32 z^4-3 a^5 z^3+4 a^3 z^3+22 a z^3+22 z^3 a^{-1} +4 z^3 a^{-3} -3 z^3 a^{-5} +a^4 z^2-13 a^2 z^2-13 z^2 a^{-2} +z^2 a^{-4} -28 z^2+2 a^5 z-2 a^3 z-11 a z-11 z a^{-1} -2 z a^{-3} +2 z a^{-5} +5 a^2+5 a^{-2} +11 }[/math]
The A2 invariant [math]\displaystyle{ -q^{14}-3 q^{10}+5 q^2+1+5 q^{-2} -3 q^{-10} - q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}-q^{78}+3 q^{76}-4 q^{74}+4 q^{72}-3 q^{70}-q^{68}+8 q^{66}-15 q^{64}+21 q^{62}-23 q^{60}+16 q^{58}-4 q^{56}-18 q^{54}+44 q^{52}-63 q^{50}+64 q^{48}-47 q^{46}+q^{44}+45 q^{42}-88 q^{40}+101 q^{38}-82 q^{36}+29 q^{34}+29 q^{32}-79 q^{30}+87 q^{28}-58 q^{26}+4 q^{24}+50 q^{22}-75 q^{20}+60 q^{18}-7 q^{16}-53 q^{14}+105 q^{12}-113 q^{10}+85 q^8-15 q^6-59 q^4+126 q^2-143+126 q^{-2} -59 q^{-4} -15 q^{-6} +85 q^{-8} -113 q^{-10} +105 q^{-12} -53 q^{-14} -7 q^{-16} +60 q^{-18} -75 q^{-20} +50 q^{-22} +4 q^{-24} -58 q^{-26} +87 q^{-28} -79 q^{-30} +29 q^{-32} +29 q^{-34} -82 q^{-36} +101 q^{-38} -88 q^{-40} +45 q^{-42} + q^{-44} -47 q^{-46} +64 q^{-48} -63 q^{-50} +44 q^{-52} -18 q^{-54} -4 q^{-56} +16 q^{-58} -23 q^{-60} +21 q^{-62} -15 q^{-64} +8 q^{-66} - q^{-68} -3 q^{-70} +4 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_79.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{11}+q^9-3 q^7+3 q^5-q^3+2 q+2 q^{-1} - q^{-3} +3 q^{-5} -3 q^{-7} + q^{-9} - q^{-11} }[/math]
2 [math]\displaystyle{ q^{32}-q^{30}+4 q^{26}-5 q^{24}-4 q^{22}+12 q^{20}-7 q^{18}-14 q^{16}+18 q^{14}-19 q^{10}+13 q^8+8 q^6-12 q^4+2 q^2+11+2 q^{-2} -12 q^{-4} +8 q^{-6} +13 q^{-8} -19 q^{-10} +18 q^{-14} -14 q^{-16} -7 q^{-18} +12 q^{-20} -4 q^{-22} -5 q^{-24} +4 q^{-26} - q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ -q^{63}+q^{61}-q^{57}-q^{55}+4 q^{53}+2 q^{51}-7 q^{49}-5 q^{47}+15 q^{45}+14 q^{43}-19 q^{41}-33 q^{39}+20 q^{37}+55 q^{35}-9 q^{33}-76 q^{31}-21 q^{29}+91 q^{27}+46 q^{25}-86 q^{23}-78 q^{21}+70 q^{19}+91 q^{17}-45 q^{15}-98 q^{13}+23 q^{11}+87 q^9+8 q^7-70 q^5-27 q^3+52 q+52 q^{-1} -27 q^{-3} -70 q^{-5} +8 q^{-7} +87 q^{-9} +23 q^{-11} -98 q^{-13} -45 q^{-15} +91 q^{-17} +70 q^{-19} -78 q^{-21} -86 q^{-23} +46 q^{-25} +91 q^{-27} -21 q^{-29} -76 q^{-31} -9 q^{-33} +55 q^{-35} +20 q^{-37} -33 q^{-39} -19 q^{-41} +14 q^{-43} +15 q^{-45} -5 q^{-47} -7 q^{-49} +2 q^{-51} +4 q^{-53} - q^{-55} - q^{-57} + q^{-61} - q^{-63} }[/math]
4 [math]\displaystyle{ q^{104}-q^{102}+q^{98}-2 q^{96}+2 q^{94}-3 q^{92}+q^{90}+6 q^{88}-7 q^{86}-q^{84}-12 q^{82}+6 q^{80}+33 q^{78}+2 q^{76}-14 q^{74}-66 q^{72}-23 q^{70}+85 q^{68}+91 q^{66}+46 q^{64}-158 q^{62}-192 q^{60}+20 q^{58}+220 q^{56}+307 q^{54}-74 q^{52}-402 q^{50}-304 q^{48}+115 q^{46}+599 q^{44}+302 q^{42}-334 q^{40}-635 q^{38}-276 q^{36}+565 q^{34}+644 q^{32}+28 q^{30}-629 q^{28}-593 q^{26}+245 q^{24}+644 q^{22}+325 q^{20}-350 q^{18}-590 q^{16}-59 q^{14}+404 q^{12}+399 q^{10}-64 q^8-411 q^6-239 q^4+158 q^2+385+158 q^{-2} -239 q^{-4} -411 q^{-6} -64 q^{-8} +399 q^{-10} +404 q^{-12} -59 q^{-14} -590 q^{-16} -350 q^{-18} +325 q^{-20} +644 q^{-22} +245 q^{-24} -593 q^{-26} -629 q^{-28} +28 q^{-30} +644 q^{-32} +565 q^{-34} -276 q^{-36} -635 q^{-38} -334 q^{-40} +302 q^{-42} +599 q^{-44} +115 q^{-46} -304 q^{-48} -402 q^{-50} -74 q^{-52} +307 q^{-54} +220 q^{-56} +20 q^{-58} -192 q^{-60} -158 q^{-62} +46 q^{-64} +91 q^{-66} +85 q^{-68} -23 q^{-70} -66 q^{-72} -14 q^{-74} +2 q^{-76} +33 q^{-78} +6 q^{-80} -12 q^{-82} - q^{-84} -7 q^{-86} +6 q^{-88} + q^{-90} -3 q^{-92} +2 q^{-94} -2 q^{-96} + q^{-98} - q^{-102} + q^{-104} }[/math]
5 [math]\displaystyle{ -q^{155}+q^{153}-q^{149}+2 q^{147}+q^{145}-3 q^{143}+q^{139}-2 q^{137}+5 q^{135}+9 q^{133}-5 q^{131}-12 q^{129}-13 q^{127}-9 q^{125}+17 q^{123}+47 q^{121}+36 q^{119}-20 q^{117}-84 q^{115}-110 q^{113}-31 q^{111}+120 q^{109}+232 q^{107}+174 q^{105}-82 q^{103}-376 q^{101}-436 q^{99}-124 q^{97}+417 q^{95}+797 q^{93}+587 q^{91}-230 q^{89}-1087 q^{87}-1252 q^{85}-365 q^{83}+1081 q^{81}+1983 q^{79}+1368 q^{77}-589 q^{75}-2447 q^{73}-2568 q^{71}-506 q^{69}+2363 q^{67}+3692 q^{65}+1995 q^{63}-1644 q^{61}-4278 q^{59}-3550 q^{57}+293 q^{55}+4214 q^{53}+4767 q^{51}+1252 q^{49}-3436 q^{47}-5320 q^{45}-2715 q^{43}+2227 q^{41}+5174 q^{39}+3668 q^{37}-874 q^{35}-4452 q^{33}-4047 q^{31}-264 q^{29}+3401 q^{27}+3869 q^{25}+1072 q^{23}-2321 q^{21}-3380 q^{19}-1467 q^{17}+1398 q^{15}+2745 q^{13}+1651 q^{11}-705 q^9-2228 q^7-1719 q^5+218 q^3+1879 q+1879 q^{-1} +218 q^{-3} -1719 q^{-5} -2228 q^{-7} -705 q^{-9} +1651 q^{-11} +2745 q^{-13} +1398 q^{-15} -1467 q^{-17} -3380 q^{-19} -2321 q^{-21} +1072 q^{-23} +3869 q^{-25} +3401 q^{-27} -264 q^{-29} -4047 q^{-31} -4452 q^{-33} -874 q^{-35} +3668 q^{-37} +5174 q^{-39} +2227 q^{-41} -2715 q^{-43} -5320 q^{-45} -3436 q^{-47} +1252 q^{-49} +4767 q^{-51} +4214 q^{-53} +293 q^{-55} -3550 q^{-57} -4278 q^{-59} -1644 q^{-61} +1995 q^{-63} +3692 q^{-65} +2363 q^{-67} -506 q^{-69} -2568 q^{-71} -2447 q^{-73} -589 q^{-75} +1368 q^{-77} +1983 q^{-79} +1081 q^{-81} -365 q^{-83} -1252 q^{-85} -1087 q^{-87} -230 q^{-89} +587 q^{-91} +797 q^{-93} +417 q^{-95} -124 q^{-97} -436 q^{-99} -376 q^{-101} -82 q^{-103} +174 q^{-105} +232 q^{-107} +120 q^{-109} -31 q^{-111} -110 q^{-113} -84 q^{-115} -20 q^{-117} +36 q^{-119} +47 q^{-121} +17 q^{-123} -9 q^{-125} -13 q^{-127} -12 q^{-129} -5 q^{-131} +9 q^{-133} +5 q^{-135} -2 q^{-137} + q^{-139} -3 q^{-143} + q^{-145} +2 q^{-147} - q^{-149} + q^{-153} - q^{-155} }[/math]
6 [math]\displaystyle{ q^{216}-q^{214}+q^{210}-2 q^{208}-q^{206}+6 q^{202}-2 q^{200}-5 q^{198}+3 q^{196}-5 q^{194}-4 q^{192}+2 q^{190}+23 q^{188}+9 q^{186}-14 q^{184}-8 q^{182}-31 q^{180}-33 q^{178}-5 q^{176}+79 q^{174}+90 q^{172}+41 q^{170}-133 q^{166}-221 q^{164}-192 q^{162}+77 q^{160}+318 q^{158}+424 q^{156}+397 q^{154}-38 q^{152}-612 q^{150}-1014 q^{148}-724 q^{146}+18 q^{144}+1001 q^{142}+1846 q^{140}+1590 q^{138}+215 q^{136}-1824 q^{134}-3053 q^{132}-2931 q^{130}-871 q^{128}+2618 q^{126}+5189 q^{124}+5175 q^{122}+1689 q^{120}-3379 q^{118}-7953 q^{116}-8542 q^{114}-3498 q^{112}+4686 q^{110}+11716 q^{108}+12553 q^{106}+6128 q^{104}-6006 q^{102}-16477 q^{100}-17997 q^{98}-8512 q^{96}+7925 q^{94}+21510 q^{92}+24060 q^{90}+10906 q^{88}-10682 q^{86}-27701 q^{84}-29020 q^{82}-12009 q^{80}+13861 q^{78}+33801 q^{76}+32854 q^{74}+11153 q^{72}-18682 q^{70}-37998 q^{68}-33953 q^{66}-8484 q^{64}+23696 q^{62}+40371 q^{60}+31753 q^{58}+3049 q^{56}-27098 q^{54}-39538 q^{52}-26785 q^{50}+3146 q^{48}+28958 q^{46}+35370 q^{44}+18968 q^{42}-8079 q^{40}-28122 q^{38}-28875 q^{36}-10823 q^{34}+11680 q^{32}+24759 q^{30}+20638 q^{28}+4229 q^{26}-13088 q^{24}-20018 q^{22}-13027 q^{20}+972 q^{18}+12802 q^{16}+14704 q^{14}+7084 q^{12}-4556 q^{10}-11920 q^8-10508 q^6-2176 q^4+7427 q^2+11325+7427 q^{-2} -2176 q^{-4} -10508 q^{-6} -11920 q^{-8} -4556 q^{-10} +7084 q^{-12} +14704 q^{-14} +12802 q^{-16} +972 q^{-18} -13027 q^{-20} -20018 q^{-22} -13088 q^{-24} +4229 q^{-26} +20638 q^{-28} +24759 q^{-30} +11680 q^{-32} -10823 q^{-34} -28875 q^{-36} -28122 q^{-38} -8079 q^{-40} +18968 q^{-42} +35370 q^{-44} +28958 q^{-46} +3146 q^{-48} -26785 q^{-50} -39538 q^{-52} -27098 q^{-54} +3049 q^{-56} +31753 q^{-58} +40371 q^{-60} +23696 q^{-62} -8484 q^{-64} -33953 q^{-66} -37998 q^{-68} -18682 q^{-70} +11153 q^{-72} +32854 q^{-74} +33801 q^{-76} +13861 q^{-78} -12009 q^{-80} -29020 q^{-82} -27701 q^{-84} -10682 q^{-86} +10906 q^{-88} +24060 q^{-90} +21510 q^{-92} +7925 q^{-94} -8512 q^{-96} -17997 q^{-98} -16477 q^{-100} -6006 q^{-102} +6128 q^{-104} +12553 q^{-106} +11716 q^{-108} +4686 q^{-110} -3498 q^{-112} -8542 q^{-114} -7953 q^{-116} -3379 q^{-118} +1689 q^{-120} +5175 q^{-122} +5189 q^{-124} +2618 q^{-126} -871 q^{-128} -2931 q^{-130} -3053 q^{-132} -1824 q^{-134} +215 q^{-136} +1590 q^{-138} +1846 q^{-140} +1001 q^{-142} +18 q^{-144} -724 q^{-146} -1014 q^{-148} -612 q^{-150} -38 q^{-152} +397 q^{-154} +424 q^{-156} +318 q^{-158} +77 q^{-160} -192 q^{-162} -221 q^{-164} -133 q^{-166} +41 q^{-170} +90 q^{-172} +79 q^{-174} -5 q^{-176} -33 q^{-178} -31 q^{-180} -8 q^{-182} -14 q^{-184} +9 q^{-186} +23 q^{-188} +2 q^{-190} -4 q^{-192} -5 q^{-194} +3 q^{-196} -5 q^{-198} -2 q^{-200} +6 q^{-202} - q^{-206} -2 q^{-208} + q^{-210} - q^{-214} + q^{-216} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{14}-3 q^{10}+5 q^2+1+5 q^{-2} -3 q^{-10} - q^{-14} }[/math]
1,1 [math]\displaystyle{ q^{44}-2 q^{42}+6 q^{40}-12 q^{38}+25 q^{36}-42 q^{34}+68 q^{32}-106 q^{30}+155 q^{28}-214 q^{26}+276 q^{24}-332 q^{22}+363 q^{20}-368 q^{18}+312 q^{16}-224 q^{14}+67 q^{12}+102 q^{10}-294 q^8+476 q^6-610 q^4+724 q^2-734+724 q^{-2} -610 q^{-4} +476 q^{-6} -294 q^{-8} +102 q^{-10} +67 q^{-12} -224 q^{-14} +312 q^{-16} -368 q^{-18} +363 q^{-20} -332 q^{-22} +276 q^{-24} -214 q^{-26} +155 q^{-28} -106 q^{-30} +68 q^{-32} -42 q^{-34} +25 q^{-36} -12 q^{-38} +6 q^{-40} -2 q^{-42} + q^{-44} }[/math]
2,0 [math]\displaystyle{ q^{38}+q^{34}+3 q^{32}-2 q^{28}+3 q^{26}-7 q^{22}-7 q^{20}+q^{18}-q^{16}-13 q^{14}-2 q^{12}+5 q^{10}-4 q^8-q^6+12 q^4+11 q^2+6+11 q^{-2} +12 q^{-4} - q^{-6} -4 q^{-8} +5 q^{-10} -2 q^{-12} -13 q^{-14} - q^{-16} + q^{-18} -7 q^{-20} -7 q^{-22} +3 q^{-26} -2 q^{-28} +3 q^{-32} + q^{-34} + q^{-38} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{34}-q^{32}+q^{30}+3 q^{28}-4 q^{26}+q^{24}+7 q^{22}-12 q^{20}+9 q^{16}-20 q^{14}-5 q^{12}+7 q^{10}-13 q^8-q^6+15 q^4+10 q^2+10+10 q^{-2} +15 q^{-4} - q^{-6} -13 q^{-8} +7 q^{-10} -5 q^{-12} -20 q^{-14} +9 q^{-16} -12 q^{-20} +7 q^{-22} + q^{-24} -4 q^{-26} +3 q^{-28} + q^{-30} - q^{-32} + q^{-34} }[/math]
1,0,0 [math]\displaystyle{ -q^{17}-4 q^{13}-4 q^9+q^7+5 q^3+5 q+5 q^{-1} +5 q^{-3} + q^{-7} -4 q^{-9} -4 q^{-13} - q^{-17} }[/math]
1,0,1 [math]\displaystyle{ q^{56}-2 q^{54}+5 q^{52}-5 q^{50}+2 q^{48}+10 q^{46}-24 q^{44}+34 q^{42}-24 q^{40}-14 q^{38}+67 q^{36}-113 q^{34}+108 q^{32}-22 q^{30}-107 q^{28}+245 q^{26}-276 q^{24}+187 q^{22}+8 q^{20}-265 q^{18}+360 q^{16}-401 q^{14}+173 q^{12}+9 q^{10}-205 q^8+256 q^6-120 q^4+85 q^2+71+85 q^{-2} -120 q^{-4} +256 q^{-6} -205 q^{-8} +9 q^{-10} +173 q^{-12} -401 q^{-14} +360 q^{-16} -265 q^{-18} +8 q^{-20} +187 q^{-22} -276 q^{-24} +245 q^{-26} -107 q^{-28} -22 q^{-30} +108 q^{-32} -113 q^{-34} +67 q^{-36} -14 q^{-38} -24 q^{-40} +34 q^{-42} -24 q^{-44} +10 q^{-46} +2 q^{-48} -5 q^{-50} +5 q^{-52} -2 q^{-54} + q^{-56} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{40}+q^{36}+4 q^{34}+q^{32}+8 q^{28}-q^{26}-8 q^{24}+q^{22}-2 q^{20}-22 q^{18}-19 q^{16}-7 q^{14}-15 q^{12}-21 q^{10}+3 q^8+23 q^6+9 q^4+26 q^2+46+26 q^{-2} +9 q^{-4} +23 q^{-6} +3 q^{-8} -21 q^{-10} -15 q^{-12} -7 q^{-14} -19 q^{-16} -22 q^{-18} -2 q^{-20} + q^{-22} -8 q^{-24} - q^{-26} +8 q^{-28} + q^{-32} +4 q^{-34} + q^{-36} + q^{-40} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{20}-4 q^{16}-q^{14}-4 q^{12}-3 q^{10}+6 q^4+5 q^2+9+5 q^{-2} +6 q^{-4} -3 q^{-10} -4 q^{-12} - q^{-14} -4 q^{-16} - q^{-20} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{34}+q^{32}-3 q^{30}+5 q^{28}-8 q^{26}+11 q^{24}-15 q^{22}+16 q^{20}-18 q^{18}+15 q^{16}-12 q^{14}+5 q^{12}+3 q^{10}-11 q^8+21 q^6-25 q^4+34 q^2-32+34 q^{-2} -25 q^{-4} +21 q^{-6} -11 q^{-8} +3 q^{-10} +5 q^{-12} -12 q^{-14} +15 q^{-16} -18 q^{-18} +16 q^{-20} -15 q^{-22} +11 q^{-24} -8 q^{-26} +5 q^{-28} -3 q^{-30} + q^{-32} - q^{-34} }[/math]
1,0 [math]\displaystyle{ q^{56}-q^{52}-q^{50}+2 q^{48}+4 q^{46}-6 q^{42}-4 q^{40}+6 q^{38}+11 q^{36}-3 q^{34}-16 q^{32}-8 q^{30}+13 q^{28}+13 q^{26}-10 q^{24}-21 q^{22}-4 q^{20}+15 q^{18}+6 q^{16}-13 q^{14}-11 q^{12}+9 q^{10}+13 q^8-8 q^4+7 q^2+17+7 q^{-2} -8 q^{-4} +13 q^{-8} +9 q^{-10} -11 q^{-12} -13 q^{-14} +6 q^{-16} +15 q^{-18} -4 q^{-20} -21 q^{-22} -10 q^{-24} +13 q^{-26} +13 q^{-28} -8 q^{-30} -16 q^{-32} -3 q^{-34} +11 q^{-36} +6 q^{-38} -4 q^{-40} -6 q^{-42} +4 q^{-46} +2 q^{-48} - q^{-50} - q^{-52} + q^{-56} }[/math]

D4 Invariants.

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

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{80}-q^{78}+3 q^{76}-4 q^{74}+4 q^{72}-3 q^{70}-q^{68}+8 q^{66}-15 q^{64}+21 q^{62}-23 q^{60}+16 q^{58}-4 q^{56}-18 q^{54}+44 q^{52}-63 q^{50}+64 q^{48}-47 q^{46}+q^{44}+45 q^{42}-88 q^{40}+101 q^{38}-82 q^{36}+29 q^{34}+29 q^{32}-79 q^{30}+87 q^{28}-58 q^{26}+4 q^{24}+50 q^{22}-75 q^{20}+60 q^{18}-7 q^{16}-53 q^{14}+105 q^{12}-113 q^{10}+85 q^8-15 q^6-59 q^4+126 q^2-143+126 q^{-2} -59 q^{-4} -15 q^{-6} +85 q^{-8} -113 q^{-10} +105 q^{-12} -53 q^{-14} -7 q^{-16} +60 q^{-18} -75 q^{-20} +50 q^{-22} +4 q^{-24} -58 q^{-26} +87 q^{-28} -79 q^{-30} +29 q^{-32} +29 q^{-34} -82 q^{-36} +101 q^{-38} -88 q^{-40} +45 q^{-42} + q^{-44} -47 q^{-46} +64 q^{-48} -63 q^{-50} +44 q^{-52} -18 q^{-54} -4 q^{-56} +16 q^{-58} -23 q^{-60} +21 q^{-62} -15 q^{-64} +8 q^{-66} - q^{-68} -3 q^{-70} +4 q^{-72} -4 q^{-74} +3 q^{-76} - 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 79"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-3 t^3+7 t^2-12 t+15-12 t^{-1} +7 t^{-2} -3 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+5 z^6+9 z^4+5 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]= { 61, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+2 q^4-5 q^3+8 q^2-9 q+11-9 q^{-1} +8 q^{-2} -5 q^{-3} +2 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} +7 z^6-5 a^2 z^4-5 z^4 a^{-2} +19 z^4-9 a^2 z^2-9 z^2 a^{-2} +23 z^2-5 a^2-5 a^{-2} +11 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^9+z^9 a^{-1} +3 a^2 z^8+3 z^8 a^{-2} +6 z^8+3 a^3 z^7+4 a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +2 a^4 z^6-7 a^2 z^6-7 z^6 a^{-2} +2 z^6 a^{-4} -18 z^6+a^5 z^5-6 a^3 z^5-15 a z^5-15 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} -4 a^4 z^4+12 a^2 z^4+12 z^4 a^{-2} -4 z^4 a^{-4} +32 z^4-3 a^5 z^3+4 a^3 z^3+22 a z^3+22 z^3 a^{-1} +4 z^3 a^{-3} -3 z^3 a^{-5} +a^4 z^2-13 a^2 z^2-13 z^2 a^{-2} +z^2 a^{-4} -28 z^2+2 a^5 z-2 a^3 z-11 a z-11 z a^{-1} -2 z a^{-3} +2 z a^{-5} +5 a^2+5 a^{-2} +11 }[/math]

Vassiliev invariants

V2 and V3: (5, 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{ 20 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{694}{3} }[/math] [math]\displaystyle{ \frac{74}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{13880}{3} }[/math] [math]\displaystyle{ \frac{1480}{3} }[/math] [math]\displaystyle{ \frac{27247}{6} }[/math] [math]\displaystyle{ \frac{1154}{3} }[/math] [math]\displaystyle{ \frac{11678}{9} }[/math] [math]\displaystyle{ \frac{1045}{18} }[/math] [math]\displaystyle{ \frac{559}{6} }[/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 79. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-1012345χ
11          1-1
9         1 1
7        41 -3
5       41  3
3      54   -1
1     64    2
-1    46     2
-3   45      -1
-5  14       3
-7 14        -3
-9 1         1
-111          -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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, 79]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 79]]
Out[3]=  
PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[18, 13, 19, 14], 

 X[16, 9, 17, 10], X[10, 17, 11, 18], X[20, 15, 1, 16], 



  X[14, 19, 15, 20], X[2, 8, 3, 7], X[4, 12, 5, 11]]
In[4]:=
GaussCode[Knot[10, 79]]
Out[4]=  
GaussCode[1, -9, 2, -10, 3, -1, 9, -2, 5, -6, 10, -3, 4, -8, 7, -5, 6, 
  -4, 8, -7]
In[5]:=
BR[Knot[10, 79]]
Out[5]=  
BR[3, {-1, -1, -1, 2, 2, -1, -1, 2, 2, 2}]
In[6]:=
alex = Alexander[Knot[10, 79]][t]
Out[6]=  
      -4   3    7    12             2      3    4

15 + t   - -- + -- - -- - 12 t + 7 t  - 3 t  + t



           3    2   t


           t    t
In[7]:=
Conway[Knot[10, 79]][z]
Out[7]=  
       2      4      6    8
1 + 5 z  + 9 z  + 5 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 79]}
In[9]:=
{KnotDet[Knot[10, 79]], KnotSignature[Knot[10, 79]]}
Out[9]=  
{61, 0}
In[10]:=
J=Jones[Knot[10, 79]][q]
Out[10]=  
      -5   2    5    8    9            2      3      4    5

11 - q   + -- - -- + -- - - - 9 q + 8 q  - 5 q  + 2 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, 79]}
In[12]:=
A2Invariant[Knot[10, 79]][q]
Out[12]=  
     -14    3    5       2      10    14

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



           10    2


           q     q
In[13]:=
Kauffman[Knot[10, 79]][a, z]
Out[13]=  
     5       2   2 z   2 z   11 z               3        5         2

11 + -- + 5 a  + --- - --- - ---- - 11 a z - 2 a  z + 2 a  z - 28 z  + 



     2           5     3     a
    a           a     a

  2       2                         3      3       3
 z    13 z        2  2    4  2   3 z    4 z    22 z          3
 -- - ----- - 13 a  z  + a  z  - ---- + ---- + ----- + 22 a z  + 
  4     2                          5      3      a
 a     a                          a      a

                                4       4                         5
    3  3      5  3       4   4 z    12 z        2  4      4  4   z
 4 a  z  - 3 a  z  + 32 z  - ---- + ----- + 12 a  z  - 4 a  z  + -- - 
                               4      2                           5
                              a      a                           a

    5       5                                          6      6
 6 z    15 z          5      3  5    5  5       6   2 z    7 z
 ---- - ----- - 15 a z  - 6 a  z  + a  z  - 18 z  + ---- - ---- - 
   3      a                                           4      2
  a                                                  a      a

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

            9
    2  8   z       9
 3 a  z  + -- + a z


            a
In[14]:=
{Vassiliev[2][Knot[10, 79]], Vassiliev[3][Knot[10, 79]]}
Out[14]=  
{0, 0}
In[15]:=
Kh[Knot[10, 79]][q, t]
Out[15]=  
6           1        1       1       4       1       4       4

- + 6 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

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

  7  4    9  4    11  5


  q  t  + q  t  + q   t
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