10 99

10_98

10_100

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10 99 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X16,11,17,12 X14,7,15,8 X8,15,9,16 X20,13,1,14 X12,19,13,20 X18,6,19,5 X2,10,3,9 X4,18,5,17
Gauss code	1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 6, -4, 5, -3, 10, -8, 7, -6
Dowker-Thistlethwaite code	6 10 18 14 2 16 20 8 4 12
Conway Notation	[.2.2.20.20]

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 14.3343 A-Polynomial See Data:10 99/A-polynomial

[edit Notes for 10 99's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 99's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-4 t^3+10 t^2-16 t+19-16 t^{-1} +10 t^{-2} -4 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+4 z^6+6 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{t^4-2 t^3+3 t^2-2 t+1\right\} }[/math]
Determinant and Signature	{ 81, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-7 q^3+10 q^2-12 q+15-12 q^{-1} +10 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} +16 z^2-4 a^2-4 a^{-2} +9 }[/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+5 a z^7+5 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-9 a^2 z^6-9 z^6 a^{-2} +3 z^6 a^{-4} -24 z^6+a^5 z^5-9 a^3 z^5-18 a z^5-18 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+9 a^2 z^4+9 z^4 a^{-2} -5 z^4 a^{-4} +28 z^4-2 a^5 z^3+5 a^3 z^3+21 a z^3+21 z^3 a^{-1} +5 z^3 a^{-3} -2 z^3 a^{-5} +a^4 z^2-8 a^2 z^2-8 z^2 a^{-2} +z^2 a^{-4} -18 z^2+a^5 z-3 a^3 z-10 a z-10 z a^{-1} -3 z a^{-3} +z a^{-5} +4 a^2+4 a^{-2} +9 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+q^{12}-3 q^{10}-q^6-q^4+6 q^2+1+6 q^{-2} - q^{-4} - q^{-6} -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}-7 q^{70}+14 q^{66}-30 q^{64}+46 q^{62}-56 q^{60}+42 q^{58}-13 q^{56}-38 q^{54}+99 q^{52}-139 q^{50}+152 q^{48}-109 q^{46}+11 q^{44}+103 q^{42}-204 q^{40}+232 q^{38}-184 q^{36}+63 q^{34}+71 q^{32}-177 q^{30}+209 q^{28}-138 q^{26}+4 q^{24}+117 q^{22}-185 q^{20}+147 q^{18}-29 q^{16}-127 q^{14}+240 q^{12}-255 q^{10}+205 q^8-47 q^6-139 q^4+285 q^2-329+285 q^{-2} -139 q^{-4} -47 q^{-6} +205 q^{-8} -255 q^{-10} +240 q^{-12} -127 q^{-14} -29 q^{-16} +147 q^{-18} -185 q^{-20} +117 q^{-22} +4 q^{-24} -138 q^{-26} +209 q^{-28} -177 q^{-30} +71 q^{-32} +63 q^{-34} -184 q^{-36} +232 q^{-38} -204 q^{-40} +103 q^{-42} +11 q^{-44} -109 q^{-46} +152 q^{-48} -139 q^{-50} +99 q^{-52} -38 q^{-54} -13 q^{-56} +42 q^{-58} -56 q^{-60} +46 q^{-62} -30 q^{-64} +14 q^{-66} -7 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_99.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-4 q^7+3 q^5-2 q^3+3 q+3 q^{-1} -2 q^{-3} +3 q^{-5} -4 q^{-7} +2 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}+7 q^{26}-9 q^{24}-6 q^{22}+23 q^{20}-11 q^{18}-23 q^{16}+31 q^{14}-q^{12}-34 q^{10}+21 q^8+13 q^6-22 q^4+3 q^2+21+3 q^{-2} -22 q^{-4} +13 q^{-6} +21 q^{-8} -34 q^{-10} - q^{-12} +31 q^{-14} -23 q^{-16} -11 q^{-18} +23 q^{-20} -6 q^{-22} -9 q^{-24} +7 q^{-26} -2 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}-3 q^{57}-q^{55}+8 q^{53}+4 q^{51}-20 q^{49}-12 q^{47}+37 q^{45}+34 q^{43}-48 q^{41}-80 q^{39}+49 q^{37}+133 q^{35}-19 q^{33}-181 q^{31}-39 q^{29}+211 q^{27}+108 q^{25}-202 q^{23}-177 q^{21}+165 q^{19}+209 q^{17}-105 q^{15}-229 q^{13}+43 q^{11}+202 q^9+24 q^7-172 q^5-68 q^3+130 q+130 q^{-1} -68 q^{-3} -172 q^{-5} +24 q^{-7} +202 q^{-9} +43 q^{-11} -229 q^{-13} -105 q^{-15} +209 q^{-17} +165 q^{-19} -177 q^{-21} -202 q^{-23} +108 q^{-25} +211 q^{-27} -39 q^{-29} -181 q^{-31} -19 q^{-33} +133 q^{-35} +49 q^{-37} -80 q^{-39} -48 q^{-41} +34 q^{-43} +37 q^{-45} -12 q^{-47} -20 q^{-49} +4 q^{-51} +8 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}-6 q^{92}+4 q^{90}+17 q^{88}-12 q^{86}-11 q^{84}-38 q^{82}+16 q^{80}+100 q^{78}+30 q^{76}-53 q^{74}-215 q^{72}-94 q^{70}+252 q^{68}+329 q^{66}+150 q^{64}-488 q^{62}-640 q^{60}+20 q^{58}+739 q^{56}+964 q^{54}-208 q^{52}-1286 q^{50}-985 q^{48}+428 q^{46}+1843 q^{44}+953 q^{42}-1060 q^{40}-1997 q^{38}-773 q^{36}+1726 q^{34}+2004 q^{32}+67 q^{30}-1963 q^{28}-1781 q^{26}+704 q^{24}+2004 q^{22}+1033 q^{20}-1077 q^{18}-1842 q^{16}-271 q^{14}+1275 q^{12}+1312 q^{10}-174 q^8-1338 q^6-831 q^4+526 q^2+1297+526 q^{-2} -831 q^{-4} -1338 q^{-6} -174 q^{-8} +1312 q^{-10} +1275 q^{-12} -271 q^{-14} -1842 q^{-16} -1077 q^{-18} +1033 q^{-20} +2004 q^{-22} +704 q^{-24} -1781 q^{-26} -1963 q^{-28} +67 q^{-30} +2004 q^{-32} +1726 q^{-34} -773 q^{-36} -1997 q^{-38} -1060 q^{-40} +953 q^{-42} +1843 q^{-44} +428 q^{-46} -985 q^{-48} -1286 q^{-50} -208 q^{-52} +964 q^{-54} +739 q^{-56} +20 q^{-58} -640 q^{-60} -488 q^{-62} +150 q^{-64} +329 q^{-66} +252 q^{-68} -94 q^{-70} -215 q^{-72} -53 q^{-74} +30 q^{-76} +100 q^{-78} +16 q^{-80} -38 q^{-82} -11 q^{-84} -12 q^{-86} +17 q^{-88} +4 q^{-90} -6 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}-2 q^{141}-q^{139}-4 q^{137}+12 q^{135}+25 q^{133}-q^{131}-34 q^{129}-56 q^{127}-38 q^{125}+47 q^{123}+161 q^{121}+168 q^{119}-28 q^{117}-311 q^{115}-454 q^{113}-203 q^{111}+409 q^{109}+956 q^{107}+833 q^{105}-200 q^{103}-1506 q^{101}-1950 q^{99}-750 q^{97}+1630 q^{95}+3441 q^{93}+2727 q^{91}-748 q^{89}-4624 q^{87}-5553 q^{85}-1809 q^{83}+4581 q^{81}+8584 q^{79}+5970 q^{77}-2476 q^{75}-10507 q^{73}-10990 q^{71}-2036 q^{69}+10210 q^{67}+15522 q^{65}+8175 q^{63}-7130 q^{61}-17994 q^{59}-14553 q^{57}+1623 q^{55}+17617 q^{53}+19518 q^{51}+4895 q^{49}-14319 q^{47}-21827 q^{45}-10943 q^{43}+9175 q^{41}+21198 q^{39}+15066 q^{37}-3453 q^{35}-18189 q^{33}-16820 q^{31}-1429 q^{29}+13860 q^{27}+16264 q^{25}+4923 q^{23}-9432 q^{21}-14396 q^{19}-6737 q^{17}+5644 q^{15}+11929 q^{13}+7548 q^{11}-2824 q^9-9911 q^7-7870 q^5+865 q^3+8536 q+8536 q^{-1} +865 q^{-3} -7870 q^{-5} -9911 q^{-7} -2824 q^{-9} +7548 q^{-11} +11929 q^{-13} +5644 q^{-15} -6737 q^{-17} -14396 q^{-19} -9432 q^{-21} +4923 q^{-23} +16264 q^{-25} +13860 q^{-27} -1429 q^{-29} -16820 q^{-31} -18189 q^{-33} -3453 q^{-35} +15066 q^{-37} +21198 q^{-39} +9175 q^{-41} -10943 q^{-43} -21827 q^{-45} -14319 q^{-47} +4895 q^{-49} +19518 q^{-51} +17617 q^{-53} +1623 q^{-55} -14553 q^{-57} -17994 q^{-59} -7130 q^{-61} +8175 q^{-63} +15522 q^{-65} +10210 q^{-67} -2036 q^{-69} -10990 q^{-71} -10507 q^{-73} -2476 q^{-75} +5970 q^{-77} +8584 q^{-79} +4581 q^{-81} -1809 q^{-83} -5553 q^{-85} -4624 q^{-87} -748 q^{-89} +2727 q^{-91} +3441 q^{-93} +1630 q^{-95} -750 q^{-97} -1950 q^{-99} -1506 q^{-101} -200 q^{-103} +833 q^{-105} +956 q^{-107} +409 q^{-109} -203 q^{-111} -454 q^{-113} -311 q^{-115} -28 q^{-117} +168 q^{-119} +161 q^{-121} +47 q^{-123} -38 q^{-125} -56 q^{-127} -34 q^{-129} - q^{-131} +25 q^{-133} +12 q^{-135} -4 q^{-137} - q^{-139} -2 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}-q^{200}-12 q^{198}+4 q^{196}-15 q^{194}-12 q^{192}+9 q^{190}+61 q^{188}+40 q^{186}-27 q^{184}-40 q^{182}-126 q^{180}-135 q^{178}-20 q^{176}+270 q^{174}+390 q^{172}+265 q^{170}+29 q^{168}-581 q^{166}-1037 q^{164}-953 q^{162}+124 q^{160}+1430 q^{158}+2280 q^{156}+2226 q^{154}+150 q^{152}-2922 q^{150}-5333 q^{148}-4570 q^{146}-694 q^{144}+5142 q^{142}+10252 q^{140}+9707 q^{138}+2449 q^{136}-9219 q^{134}-17832 q^{132}-18043 q^{130}-6371 q^{128}+13863 q^{126}+30171 q^{124}+31412 q^{122}+11974 q^{120}-19134 q^{118}-46737 q^{116}-50701 q^{114}-21835 q^{112}+26831 q^{110}+68777 q^{108}+74379 q^{106}+35022 q^{104}-35698 q^{102}-95905 q^{100}-104029 q^{98}-47632 q^{96}+47803 q^{94}+125414 q^{92}+135620 q^{90}+58788 q^{88}-63906 q^{86}-158481 q^{84}-162175 q^{82}-63243 q^{80}+82938 q^{78}+189641 q^{76}+180943 q^{74}+57919 q^{72}-107400 q^{70}-211584 q^{68}-185594 q^{66}-42662 q^{64}+131818 q^{62}+222293 q^{60}+173641 q^{58}+15966 q^{56}-149114 q^{54}-216952 q^{52}-147027 q^{50}+14658 q^{48}+157680 q^{46}+195215 q^{44}+107764 q^{42}-40913 q^{40}-153594 q^{38}-161370 q^{36}-65709 q^{34}+60640 q^{32}+137507 q^{30}+119529 q^{28}+28919 q^{26}-70629 q^{24}-114160 q^{22}-78954 q^{20}+1552 q^{18}+72515 q^{16}+87614 q^{14}+44310 q^{12}-24876 q^{10}-70728 q^8-64269 q^6-14025 q^4+44494 q^2+69071+44494 q^{-2} -14025 q^{-4} -64269 q^{-6} -70728 q^{-8} -24876 q^{-10} +44310 q^{-12} +87614 q^{-14} +72515 q^{-16} +1552 q^{-18} -78954 q^{-20} -114160 q^{-22} -70629 q^{-24} +28919 q^{-26} +119529 q^{-28} +137507 q^{-30} +60640 q^{-32} -65709 q^{-34} -161370 q^{-36} -153594 q^{-38} -40913 q^{-40} +107764 q^{-42} +195215 q^{-44} +157680 q^{-46} +14658 q^{-48} -147027 q^{-50} -216952 q^{-52} -149114 q^{-54} +15966 q^{-56} +173641 q^{-58} +222293 q^{-60} +131818 q^{-62} -42662 q^{-64} -185594 q^{-66} -211584 q^{-68} -107400 q^{-70} +57919 q^{-72} +180943 q^{-74} +189641 q^{-76} +82938 q^{-78} -63243 q^{-80} -162175 q^{-82} -158481 q^{-84} -63906 q^{-86} +58788 q^{-88} +135620 q^{-90} +125414 q^{-92} +47803 q^{-94} -47632 q^{-96} -104029 q^{-98} -95905 q^{-100} -35698 q^{-102} +35022 q^{-104} +74379 q^{-106} +68777 q^{-108} +26831 q^{-110} -21835 q^{-112} -50701 q^{-114} -46737 q^{-116} -19134 q^{-118} +11974 q^{-120} +31412 q^{-122} +30171 q^{-124} +13863 q^{-126} -6371 q^{-128} -18043 q^{-130} -17832 q^{-132} -9219 q^{-134} +2449 q^{-136} +9707 q^{-138} +10252 q^{-140} +5142 q^{-142} -694 q^{-144} -4570 q^{-146} -5333 q^{-148} -2922 q^{-150} +150 q^{-152} +2226 q^{-154} +2280 q^{-156} +1430 q^{-158} +124 q^{-160} -953 q^{-162} -1037 q^{-164} -581 q^{-166} +29 q^{-168} +265 q^{-170} +390 q^{-172} +270 q^{-174} -20 q^{-176} -135 q^{-178} -126 q^{-180} -40 q^{-182} -27 q^{-184} +40 q^{-186} +61 q^{-188} +9 q^{-190} -12 q^{-192} -15 q^{-194} +4 q^{-196} -12 q^{-198} - 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^6-q^4+6 q^2+1+6 q^{-2} - q^{-4} - q^{-6} -3 q^{-10} + q^{-12} - q^{-14} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+58 q^{36}-104 q^{34}+178 q^{32}-286 q^{30}+420 q^{28}-576 q^{26}+752 q^{24}-906 q^{22}+987 q^{20}-976 q^{18}+850 q^{16}-598 q^{14}+186 q^{12}+280 q^{10}-796 q^8+1266 q^6-1651 q^4+1934 q^2-1990+1934 q^{-2} -1651 q^{-4} +1266 q^{-6} -796 q^{-8} +280 q^{-10} +186 q^{-12} -598 q^{-14} +850 q^{-16} -976 q^{-18} +987 q^{-20} -906 q^{-22} +752 q^{-24} -576 q^{-26} +420 q^{-28} -286 q^{-30} +178 q^{-32} -104 q^{-34} +58 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}-2 q^{30}-4 q^{28}+6 q^{26}+4 q^{24}-6 q^{22}-6 q^{20}+7 q^{18}+q^{16}-20 q^{14}-5 q^{12}+5 q^{10}-12 q^8-6 q^6+17 q^4+16 q^2+8+16 q^{-2} +17 q^{-4} -6 q^{-6} -12 q^{-8} +5 q^{-10} -5 q^{-12} -20 q^{-14} + q^{-16} +7 q^{-18} -6 q^{-20} -6 q^{-22} +4 q^{-24} +6 q^{-26} -4 q^{-28} -2 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}-8 q^{26}+2 q^{24}+13 q^{22}-19 q^{20}+3 q^{18}+19 q^{16}-29 q^{14}-5 q^{12}+15 q^{10}-22 q^8-4 q^6+20 q^4+9 q^2+8+9 q^{-2} +20 q^{-4} -4 q^{-6} -22 q^{-8} +15 q^{-10} -5 q^{-12} -29 q^{-14} +19 q^{-16} +3 q^{-18} -19 q^{-20} +13 q^{-22} +2 q^{-24} -8 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}+q^{11}-5 q^9+q^7-q^5+5 q^3+5 q+5 q^{-1} +5 q^{-3} - q^{-5} + q^{-7} -5 q^{-9} + 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}+21 q^{46}-62 q^{44}+95 q^{42}-72 q^{40}-35 q^{38}+192 q^{36}-316 q^{34}+299 q^{32}-66 q^{30}-307 q^{28}+653 q^{26}-759 q^{24}+519 q^{22}+37 q^{20}-667 q^{18}+1029 q^{16}-1024 q^{14}+503 q^{12}+58 q^{10}-562 q^8+643 q^6-371 q^4+141 q^2+117+141 q^{-2} -371 q^{-4} +643 q^{-6} -562 q^{-8} +58 q^{-10} +503 q^{-12} -1024 q^{-14} +1029 q^{-16} -667 q^{-18} +37 q^{-20} +519 q^{-22} -759 q^{-24} +653 q^{-26} -307 q^{-28} -66 q^{-30} +299 q^{-32} -316 q^{-34} +192 q^{-36} -35 q^{-38} -72 q^{-40} +95 q^{-42} -62 q^{-44} +21 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}-2 q^{32}-3 q^{30}+12 q^{28}-q^{26}-11 q^{24}+9 q^{22}+8 q^{20}-24 q^{18}-18 q^{16}+q^{14}-17 q^{12}-34 q^{10}+2 q^8+28 q^6-2 q^4+23 q^2+58+23 q^{-2} -2 q^{-4} +28 q^{-6} +2 q^{-8} -34 q^{-10} -17 q^{-12} + q^{-14} -18 q^{-16} -24 q^{-18} +8 q^{-20} +9 q^{-22} -11 q^{-24} - q^{-26} +12 q^{-28} -3 q^{-30} -2 q^{-32} +5 q^{-34} - q^{-38} + q^{-40} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{20}+q^{18}-4 q^{16}-4 q^{12}-3 q^{10}-q^6+6 q^4+4 q^2+9+4 q^{-2} +6 q^{-4} - q^{-6} -3 q^{-10} -4 q^{-12} -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}-14 q^{26}+20 q^{24}-27 q^{22}+29 q^{20}-31 q^{18}+27 q^{16}-21 q^{14}+9 q^{12}+5 q^{10}-20 q^8+36 q^6-46 q^4+59 q^2-58+59 q^{-2} -46 q^{-4} +36 q^{-6} -20 q^{-8} +5 q^{-10} +9 q^{-12} -21 q^{-14} +27 q^{-16} -31 q^{-18} +29 q^{-20} -27 q^{-22} +20 q^{-24} -14 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}-11 q^{42}-8 q^{40}+11 q^{38}+20 q^{36}-4 q^{34}-27 q^{32}-12 q^{30}+25 q^{28}+25 q^{26}-15 q^{24}-35 q^{22}-5 q^{20}+27 q^{18}+11 q^{16}-24 q^{14}-21 q^{12}+14 q^{10}+22 q^8-3 q^6-17 q^4+9 q^2+27+9 q^{-2} -17 q^{-4} -3 q^{-6} +22 q^{-8} +14 q^{-10} -21 q^{-12} -24 q^{-14} +11 q^{-16} +27 q^{-18} -5 q^{-20} -35 q^{-22} -15 q^{-24} +25 q^{-26} +25 q^{-28} -12 q^{-30} -27 q^{-32} -4 q^{-34} +20 q^{-36} +11 q^{-38} -8 q^{-40} -11 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}-11 q^{36}+13 q^{34}-16 q^{32}+22 q^{30}-24 q^{28}+23 q^{26}-23 q^{24}+22 q^{22}-22 q^{20}+4 q^{18}-12 q^{16}-7 q^{14}+6 q^{12}-27 q^{10}+27 q^8-28 q^6+54 q^4-32 q^2+58-32 q^{-2} +54 q^{-4} -28 q^{-6} +27 q^{-8} -27 q^{-10} +6 q^{-12} -7 q^{-14} -12 q^{-16} +4 q^{-18} -22 q^{-20} +22 q^{-22} -23 q^{-24} +23 q^{-26} -24 q^{-28} +22 q^{-30} -16 q^{-32} +13 q^{-34} -11 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}-7 q^{70}+14 q^{66}-30 q^{64}+46 q^{62}-56 q^{60}+42 q^{58}-13 q^{56}-38 q^{54}+99 q^{52}-139 q^{50}+152 q^{48}-109 q^{46}+11 q^{44}+103 q^{42}-204 q^{40}+232 q^{38}-184 q^{36}+63 q^{34}+71 q^{32}-177 q^{30}+209 q^{28}-138 q^{26}+4 q^{24}+117 q^{22}-185 q^{20}+147 q^{18}-29 q^{16}-127 q^{14}+240 q^{12}-255 q^{10}+205 q^8-47 q^6-139 q^4+285 q^2-329+285 q^{-2} -139 q^{-4} -47 q^{-6} +205 q^{-8} -255 q^{-10} +240 q^{-12} -127 q^{-14} -29 q^{-16} +147 q^{-18} -185 q^{-20} +117 q^{-22} +4 q^{-24} -138 q^{-26} +209 q^{-28} -177 q^{-30} +71 q^{-32} +63 q^{-34} -184 q^{-36} +232 q^{-38} -204 q^{-40} +103 q^{-42} +11 q^{-44} -109 q^{-46} +152 q^{-48} -139 q^{-50} +99 q^{-52} -38 q^{-54} -13 q^{-56} +42 q^{-58} -56 q^{-60} +46 q^{-62} -30 q^{-64} +14 q^{-66} -7 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 99"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^4-4 t^3+10 t^2-16 t+19-16 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+4 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{ \left\{t^4-2 t^3+3 t^2-2 t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, 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+10 q^2-12 q+15-12 q^{-1} +10 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} +16 z^2-4 a^2-4 a^{-2} +9 }[/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+5 a z^7+5 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-9 a^2 z^6-9 z^6 a^{-2} +3 z^6 a^{-4} -24 z^6+a^5 z^5-9 a^3 z^5-18 a z^5-18 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+9 a^2 z^4+9 z^4 a^{-2} -5 z^4 a^{-4} +28 z^4-2 a^5 z^3+5 a^3 z^3+21 a z^3+21 z^3 a^{-1} +5 z^3 a^{-3} -2 z^3 a^{-5} +a^4 z^2-8 a^2 z^2-8 z^2 a^{-2} +z^2 a^{-4} -18 z^2+a^5 z-3 a^3 z-10 a z-10 z a^{-1} -3 z a^{-3} +z a^{-5} +4 a^2+4 a^{-2} +9 }[/math]

Vassiliev invariants

V2 and V3:

(4, 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{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{440}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{7040}{3} }[/math] [math]\displaystyle{ \frac{640}{3} }[/math] [math]\displaystyle{ \frac{34622}{15} }[/math] [math]\displaystyle{ \frac{984}{5} }[/math] [math]\displaystyle{ \frac{29528}{45} }[/math] [math]\displaystyle{ \frac{34}{9} }[/math] [math]\displaystyle{ \frac{542}{15} }[/math]

V_2,1 through V_6,9 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 99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   2   2 7                 5 1   -4 5               5 2     3 3             7 5       -2 1           8 5         3 -1         5 8           3 -3       5 7             -2 -5     2 5               3 -7   1 5                 -4 -9   2                   2 -11 1                     -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 99]]
Out[2]=	10
In[3]:=	PD[Knot[10, 99]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[16, 11, 17, 12], X[14, 7, 15, 8], X[8, 15, 9, 16], X[20, 13, 1, 14], X[12, 19, 13, 20], X[18, 6, 19, 5], X[2, 10, 3, 9], X[4, 18, 5, 17]]
In[4]:=	GaussCode[Knot[10, 99]]
Out[4]=	GaussCode[1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 6, -4, 5, -3, 10, -8, 7, -6]
In[5]:=	BR[Knot[10, 99]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, 2, -1, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 99]][t]
Out[6]=	-4 4 10 16 2 3 4 19 + t - -- + -- - -- - 16 t + 10 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 99]][z]
Out[7]=	2 4 6 8 1 + 4 z + 6 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 99]}
In[9]:=	{KnotDet[Knot[10, 99]], KnotSignature[Knot[10, 99]]}
Out[9]=	{81, 0}
In[10]:=	J=Jones[Knot[10, 99]][q]
Out[10]=	-5 3 7 10 12 2 3 4 5 15 - q + -- - -- + -- - -- - 12 q + 10 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, 99]}
In[12]:=	A2Invariant[Knot[10, 99]][q]
Out[12]=	-14 -12 3 -6 -4 6 2 4 6 10 12 1 - q + q - --- - q - q + -- + 6 q - q - q - 3 q + q - 10 2 q q 14 q
In[13]:=	Kauffman[Knot[10, 99]][a, z]
Out[13]=	2 4 2 z 3 z 10 z 3 5 2 z 9 + -- + 4 a + -- - --- - ---- - 10 a z - 3 a z + a z - 18 z + -- - 2 5 3 a 4 a a a a 2 3 3 3 8 z 2 2 4 2 2 z 5 z 21 z 3 3 3 ---- - 8 a z + a z - ---- + ---- + ----- + 21 a z + 5 a z - 2 5 3 a a a a 4 4 5 5 5 3 4 5 z 9 z 2 4 4 4 z 9 z 2 a z + 28 z - ---- + ---- + 9 a z - 5 a z + -- - ---- - 4 2 5 3 a a a a 5 6 6 18 z 5 3 5 5 5 6 3 z 9 z 2 6 ----- - 18 a z - 9 a z + a z - 24 z + ---- - ---- - 9 a z + a 4 2 a a 7 7 8 4 6 5 z 5 z 7 3 7 8 5 z 2 8 3 a z + ---- + ---- + 5 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, 99]], Vassiliev[3][Knot[10, 99]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 99]][q, t]
Out[15]=	8 1 2 1 5 2 5 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 5 3 3 2 5 2 5 3 7 3 ---- + --- + 5 q t + 7 q t + 5 q t + 5 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