10 17

10_16

10_18

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10 17 Quick Notes

10 17 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X12,4,13,3 X20,15,1,16 X16,7,17,8 X18,9,19,10 X8,17,9,18 X10,19,11,20 X14,6,15,5 X2,12,3,11 X4,14,5,13
Gauss code	1, -9, 2, -10, 8, -1, 4, -6, 5, -7, 9, -2, 10, -8, 3, -4, 6, -5, 7, -3
Dowker-Thistlethwaite code	6 12 14 16 18 2 4 20 8 10
Conway Notation	[4114]

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 1 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 8.53676 A-Polynomial See Data:10 17/A-polynomial

[edit Notes for 10 17'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 17's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-3 t^3+5 t^2-7 t+9-7 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+5 z^6+7 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 41, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+2 q^4-3 q^3+5 q^2-6 q+7-6 q^{-1} +5 q^{-2} -3 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} +17 z^4-7 a^2 z^2-7 z^2 a^{-2} +16 z^2-2 a^2-2 a^{-2} +5 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7-2 a z^7-2 z^7 a^{-1} +2 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^5-5 a^3 z^5-5 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+11 a^2 z^4+11 z^4 a^{-2} -6 z^4 a^{-4} +34 z^4-3 a^5 z^3+2 a^3 z^3+6 a z^3+6 z^3 a^{-1} +2 z^3 a^{-3} -3 z^3 a^{-5} +3 a^4 z^2-8 a^2 z^2-8 z^2 a^{-2} +3 z^2 a^{-4} -22 z^2+a^5 z-3 a z-3 z a^{-1} +z a^{-5} +2 a^2+2 a^{-2} +5 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}-q^{10}+q^8+q^6+2 q^2-1+2 q^{-2} + q^{-6} + q^{-8} - q^{-10} - q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-q^{78}+2 q^{76}-3 q^{74}+2 q^{72}-q^{70}-2 q^{68}+6 q^{66}-7 q^{64}+8 q^{62}-7 q^{60}+2 q^{58}+2 q^{56}-8 q^{54}+11 q^{52}-15 q^{50}+12 q^{48}-7 q^{46}-2 q^{44}+9 q^{42}-16 q^{40}+19 q^{38}-14 q^{36}+4 q^{34}+5 q^{32}-14 q^{30}+15 q^{28}-7 q^{26}-q^{24}+11 q^{22}-12 q^{20}+9 q^{18}+2 q^{16}-12 q^{14}+21 q^{12}-20 q^{10}+14 q^8-q^6-12 q^4+23 q^2-25+23 q^{-2} -12 q^{-4} - q^{-6} +14 q^{-8} -20 q^{-10} +21 q^{-12} -12 q^{-14} +2 q^{-16} +9 q^{-18} -12 q^{-20} +11 q^{-22} - q^{-24} -7 q^{-26} +15 q^{-28} -14 q^{-30} +5 q^{-32} +4 q^{-34} -14 q^{-36} +19 q^{-38} -16 q^{-40} +9 q^{-42} -2 q^{-44} -7 q^{-46} +12 q^{-48} -15 q^{-50} +11 q^{-52} -8 q^{-54} +2 q^{-56} +2 q^{-58} -7 q^{-60} +8 q^{-62} -7 q^{-64} +6 q^{-66} -2 q^{-68} - q^{-70} +2 q^{-72} -3 q^{-74} +2 q^{-76} - q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_17.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+q^9-q^7+2 q^5-q^3+q+ q^{-1} - q^{-3} +2 q^{-5} - q^{-7} + q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-q^{30}-q^{28}+2 q^{26}-2 q^{24}-2 q^{22}+4 q^{20}-q^{18}-4 q^{16}+6 q^{14}+q^{12}-6 q^{10}+4 q^8+2 q^6-4 q^4+q^2+3+ q^{-2} -4 q^{-4} +2 q^{-6} +4 q^{-8} -6 q^{-10} + q^{-12} +6 q^{-14} -4 q^{-16} - q^{-18} +4 q^{-20} -2 q^{-22} -2 q^{-24} +2 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+q^{61}+q^{59}-q^{55}+q^{51}-q^{49}-q^{47}+2 q^{45}+q^{43}-4 q^{41}-3 q^{39}+3 q^{37}+7 q^{35}-3 q^{33}-9 q^{31}-2 q^{29}+13 q^{27}+7 q^{25}-11 q^{23}-12 q^{21}+9 q^{19}+15 q^{17}-6 q^{15}-14 q^{13}+2 q^{11}+12 q^9+q^7-9 q^5-2 q^3+6 q+6 q^{-1} -2 q^{-3} -9 q^{-5} + q^{-7} +12 q^{-9} +2 q^{-11} -14 q^{-13} -6 q^{-15} +15 q^{-17} +9 q^{-19} -12 q^{-21} -11 q^{-23} +7 q^{-25} +13 q^{-27} -2 q^{-29} -9 q^{-31} -3 q^{-33} +7 q^{-35} +3 q^{-37} -3 q^{-39} -4 q^{-41} + q^{-43} +2 q^{-45} - q^{-47} - q^{-49} + q^{-51} - q^{-55} + q^{-59} + q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-q^{102}-q^{100}-q^{96}+3 q^{94}+q^{90}-6 q^{86}+2 q^{84}+5 q^{80}+7 q^{78}-8 q^{76}-5 q^{74}-9 q^{72}+6 q^{70}+19 q^{68}+q^{66}-5 q^{64}-25 q^{62}-10 q^{60}+19 q^{58}+19 q^{56}+19 q^{54}-22 q^{52}-32 q^{50}-10 q^{48}+13 q^{46}+50 q^{44}+12 q^{42}-29 q^{40}-43 q^{38}-24 q^{36}+49 q^{34}+46 q^{32}+4 q^{30}-44 q^{28}-52 q^{26}+22 q^{24}+45 q^{22}+25 q^{20}-21 q^{18}-45 q^{16}-2 q^{14}+22 q^{12}+23 q^{10}-2 q^8-22 q^6-8 q^4+8 q^2+17+8 q^{-2} -8 q^{-4} -22 q^{-6} -2 q^{-8} +23 q^{-10} +22 q^{-12} -2 q^{-14} -45 q^{-16} -21 q^{-18} +25 q^{-20} +45 q^{-22} +22 q^{-24} -52 q^{-26} -44 q^{-28} +4 q^{-30} +46 q^{-32} +49 q^{-34} -24 q^{-36} -43 q^{-38} -29 q^{-40} +12 q^{-42} +50 q^{-44} +13 q^{-46} -10 q^{-48} -32 q^{-50} -22 q^{-52} +19 q^{-54} +19 q^{-56} +19 q^{-58} -10 q^{-60} -25 q^{-62} -5 q^{-64} + q^{-66} +19 q^{-68} +6 q^{-70} -9 q^{-72} -5 q^{-74} -8 q^{-76} +7 q^{-78} +5 q^{-80} +2 q^{-84} -6 q^{-86} + q^{-90} +3 q^{-94} - q^{-96} - q^{-100} - q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+q^{153}+q^{151}+q^{147}-q^{145}-3 q^{143}-2 q^{141}+q^{139}+2 q^{137}+5 q^{135}+4 q^{133}-4 q^{131}-9 q^{129}-6 q^{127}+q^{125}+9 q^{123}+14 q^{121}+6 q^{119}-12 q^{117}-19 q^{115}-13 q^{113}+6 q^{111}+25 q^{109}+28 q^{107}+4 q^{105}-29 q^{103}-40 q^{101}-21 q^{99}+18 q^{97}+50 q^{95}+47 q^{93}+3 q^{91}-47 q^{89}-69 q^{87}-43 q^{85}+22 q^{83}+77 q^{81}+82 q^{79}+31 q^{77}-60 q^{75}-116 q^{73}-89 q^{71}+9 q^{69}+116 q^{67}+154 q^{65}+66 q^{63}-89 q^{61}-184 q^{59}-142 q^{57}+20 q^{55}+185 q^{53}+206 q^{51}+52 q^{49}-150 q^{47}-232 q^{45}-119 q^{43}+92 q^{41}+223 q^{39}+162 q^{37}-36 q^{35}-187 q^{33}-165 q^{31}-12 q^{29}+131 q^{27}+148 q^{25}+40 q^{23}-82 q^{21}-112 q^{19}-45 q^{17}+42 q^{15}+73 q^{13}+42 q^{11}-15 q^9-49 q^7-31 q^5+5 q^3+33 q+33 q^{-1} +5 q^{-3} -31 q^{-5} -49 q^{-7} -15 q^{-9} +42 q^{-11} +73 q^{-13} +42 q^{-15} -45 q^{-17} -112 q^{-19} -82 q^{-21} +40 q^{-23} +148 q^{-25} +131 q^{-27} -12 q^{-29} -165 q^{-31} -187 q^{-33} -36 q^{-35} +162 q^{-37} +223 q^{-39} +92 q^{-41} -119 q^{-43} -232 q^{-45} -150 q^{-47} +52 q^{-49} +206 q^{-51} +185 q^{-53} +20 q^{-55} -142 q^{-57} -184 q^{-59} -89 q^{-61} +66 q^{-63} +154 q^{-65} +116 q^{-67} +9 q^{-69} -89 q^{-71} -116 q^{-73} -60 q^{-75} +31 q^{-77} +82 q^{-79} +77 q^{-81} +22 q^{-83} -43 q^{-85} -69 q^{-87} -47 q^{-89} +3 q^{-91} +47 q^{-93} +50 q^{-95} +18 q^{-97} -21 q^{-99} -40 q^{-101} -29 q^{-103} +4 q^{-105} +28 q^{-107} +25 q^{-109} +6 q^{-111} -13 q^{-113} -19 q^{-115} -12 q^{-117} +6 q^{-119} +14 q^{-121} +9 q^{-123} + q^{-125} -6 q^{-127} -9 q^{-129} -4 q^{-131} +4 q^{-133} +5 q^{-135} +2 q^{-137} + q^{-139} -2 q^{-141} -3 q^{-143} - q^{-145} + q^{-147} + q^{-151} + q^{-153} - q^{-155} }[/math]
6	[math]\displaystyle{ q^{216}-q^{214}-q^{212}-q^{208}+q^{206}+q^{204}+5 q^{202}-3 q^{198}-2 q^{196}-6 q^{194}-3 q^{192}+q^{190}+13 q^{188}+6 q^{186}-2 q^{182}-13 q^{180}-12 q^{178}-6 q^{176}+18 q^{174}+15 q^{172}+7 q^{170}+3 q^{168}-17 q^{166}-24 q^{164}-18 q^{162}+21 q^{160}+28 q^{158}+22 q^{156}+13 q^{154}-22 q^{152}-47 q^{150}-49 q^{148}+10 q^{146}+44 q^{144}+58 q^{142}+57 q^{140}+4 q^{138}-61 q^{136}-105 q^{134}-58 q^{132}-7 q^{130}+55 q^{128}+121 q^{126}+113 q^{124}+39 q^{122}-74 q^{120}-122 q^{118}-160 q^{116}-118 q^{114}+25 q^{112}+168 q^{110}+241 q^{108}+184 q^{106}+62 q^{104}-182 q^{102}-359 q^{100}-331 q^{98}-111 q^{96}+211 q^{94}+447 q^{92}+512 q^{90}+203 q^{88}-264 q^{86}-609 q^{84}-625 q^{82}-264 q^{80}+284 q^{78}+778 q^{76}+756 q^{74}+273 q^{72}-403 q^{70}-858 q^{68}-811 q^{66}-265 q^{64}+536 q^{62}+938 q^{60}+766 q^{58}+111 q^{56}-589 q^{54}-919 q^{52}-667 q^{50}+61 q^{48}+643 q^{46}+793 q^{44}+429 q^{42}-151 q^{40}-597 q^{38}-624 q^{36}-207 q^{34}+234 q^{32}+473 q^{30}+373 q^{28}+83 q^{26}-224 q^{24}-337 q^{22}-188 q^{20}+14 q^{18}+170 q^{16}+175 q^{14}+92 q^{12}-42 q^{10}-121 q^8-92 q^6-18 q^4+62 q^2+81+62 q^{-2} -18 q^{-4} -92 q^{-6} -121 q^{-8} -42 q^{-10} +92 q^{-12} +175 q^{-14} +170 q^{-16} +14 q^{-18} -188 q^{-20} -337 q^{-22} -224 q^{-24} +83 q^{-26} +373 q^{-28} +473 q^{-30} +234 q^{-32} -207 q^{-34} -624 q^{-36} -597 q^{-38} -151 q^{-40} +429 q^{-42} +793 q^{-44} +643 q^{-46} +61 q^{-48} -667 q^{-50} -919 q^{-52} -589 q^{-54} +111 q^{-56} +766 q^{-58} +938 q^{-60} +536 q^{-62} -265 q^{-64} -811 q^{-66} -858 q^{-68} -403 q^{-70} +273 q^{-72} +756 q^{-74} +778 q^{-76} +284 q^{-78} -264 q^{-80} -625 q^{-82} -609 q^{-84} -264 q^{-86} +203 q^{-88} +512 q^{-90} +447 q^{-92} +211 q^{-94} -111 q^{-96} -331 q^{-98} -359 q^{-100} -182 q^{-102} +62 q^{-104} +184 q^{-106} +241 q^{-108} +168 q^{-110} +25 q^{-112} -118 q^{-114} -160 q^{-116} -122 q^{-118} -74 q^{-120} +39 q^{-122} +113 q^{-124} +121 q^{-126} +55 q^{-128} -7 q^{-130} -58 q^{-132} -105 q^{-134} -61 q^{-136} +4 q^{-138} +57 q^{-140} +58 q^{-142} +44 q^{-144} +10 q^{-146} -49 q^{-148} -47 q^{-150} -22 q^{-152} +13 q^{-154} +22 q^{-156} +28 q^{-158} +21 q^{-160} -18 q^{-162} -24 q^{-164} -17 q^{-166} +3 q^{-168} +7 q^{-170} +15 q^{-172} +18 q^{-174} -6 q^{-176} -12 q^{-178} -13 q^{-180} -2 q^{-182} +6 q^{-186} +13 q^{-188} + q^{-190} -3 q^{-192} -6 q^{-194} -2 q^{-196} -3 q^{-198} +5 q^{-202} + q^{-204} + q^{-206} - q^{-208} - q^{-212} - q^{-214} + q^{-216} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{14}-q^{10}+q^8+q^6+2 q^2-1+2 q^{-2} + q^{-6} + q^{-8} - q^{-10} - q^{-14} }[/math]
1,1	[math]\displaystyle{ q^{44}-2 q^{42}+4 q^{40}-8 q^{38}+13 q^{36}-18 q^{34}+22 q^{32}-28 q^{30}+33 q^{28}-38 q^{26}+40 q^{24}-44 q^{22}+44 q^{20}-40 q^{18}+32 q^{16}-14 q^{14}-5 q^{12}+26 q^{10}-50 q^8+70 q^6-85 q^4+98 q^2-94+98 q^{-2} -85 q^{-4} +70 q^{-6} -50 q^{-8} +26 q^{-10} -5 q^{-12} -14 q^{-14} +32 q^{-16} -40 q^{-18} +44 q^{-20} -44 q^{-22} +40 q^{-24} -38 q^{-26} +33 q^{-28} -28 q^{-30} +22 q^{-32} -18 q^{-34} +13 q^{-36} -8 q^{-38} +4 q^{-40} -2 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{38}-q^{30}-2 q^{28}-q^{26}-2 q^{22}+3 q^{18}+3 q^{16}-2 q^{14}+q^{12}+3 q^{10}-q^8-q^6+2 q^4+q^2-2+ q^{-2} +2 q^{-4} - q^{-6} - q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} +3 q^{-18} -2 q^{-22} - q^{-26} -2 q^{-28} - q^{-30} + q^{-38} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-q^{32}+q^{28}-2 q^{26}+q^{24}+2 q^{22}-4 q^{20}+3 q^{16}-7 q^{14}-q^{12}+4 q^{10}-3 q^8+5 q^4+3 q^2+2+3 q^{-2} +5 q^{-4} -3 q^{-8} +4 q^{-10} - q^{-12} -7 q^{-14} +3 q^{-16} -4 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-26} + q^{-28} - q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{17}-2 q^{13}+q^{11}-q^9+2 q^7+2 q^3+q+ q^{-1} +2 q^{-3} +2 q^{-7} - q^{-9} + q^{-11} -2 q^{-13} - q^{-17} }[/math]
1,0,1	[math]\displaystyle{ q^{56}-2 q^{54}+3 q^{52}-3 q^{50}+q^{48}+4 q^{46}-9 q^{44}+9 q^{42}-7 q^{40}+q^{38}+6 q^{36}-10 q^{34}+10 q^{32}-4 q^{30}-7 q^{28}+17 q^{26}-22 q^{24}+9 q^{22}+9 q^{20}-30 q^{18}+39 q^{16}-36 q^{14}+13 q^{12}+3 q^{10}-24 q^8+28 q^6-13 q^4+18 q^2+7+18 q^{-2} -13 q^{-4} +28 q^{-6} -24 q^{-8} +3 q^{-10} +13 q^{-12} -36 q^{-14} +39 q^{-16} -30 q^{-18} +9 q^{-20} +9 q^{-22} -22 q^{-24} +17 q^{-26} -7 q^{-28} -4 q^{-30} +10 q^{-32} -10 q^{-34} +6 q^{-36} + q^{-38} -7 q^{-40} +9 q^{-42} -9 q^{-44} +4 q^{-46} + q^{-48} -3 q^{-50} +3 q^{-52} -2 q^{-54} + q^{-56} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{40}+q^{34}-q^{30}+q^{28}-q^{26}-3 q^{24}-5 q^{18}-4 q^{16}-2 q^{12}-5 q^{10}+q^8+7 q^6+2 q^4+7 q^2+12+7 q^{-2} +2 q^{-4} +7 q^{-6} + q^{-8} -5 q^{-10} -2 q^{-12} -4 q^{-16} -5 q^{-18} -3 q^{-24} - q^{-26} + q^{-28} - q^{-30} + q^{-34} + q^{-40} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{20}-2 q^{16}-q^{12}+q^8+3 q^4+q^2+3+ q^{-2} +3 q^{-4} + q^{-8} - q^{-12} -2 q^{-16} - q^{-20} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+q^{32}-2 q^{30}+3 q^{28}-4 q^{26}+5 q^{24}-6 q^{22}+6 q^{20}-6 q^{18}+5 q^{16}-3 q^{14}+q^{12}+2 q^{10}-5 q^8+8 q^6-9 q^4+13 q^2-12+13 q^{-2} -9 q^{-4} +8 q^{-6} -5 q^{-8} +2 q^{-10} + q^{-12} -3 q^{-14} +5 q^{-16} -6 q^{-18} +6 q^{-20} -6 q^{-22} +5 q^{-24} -4 q^{-26} +3 q^{-28} -2 q^{-30} + q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-q^{52}-q^{50}+q^{48}+2 q^{46}-q^{44}-3 q^{42}-q^{40}+3 q^{38}+4 q^{36}-2 q^{34}-6 q^{32}-2 q^{30}+5 q^{28}+5 q^{26}-4 q^{24}-7 q^{22}-q^{20}+6 q^{18}+3 q^{16}-4 q^{14}-3 q^{12}+3 q^{10}+4 q^8-3 q^4+2 q^2+5+2 q^{-2} -3 q^{-4} +4 q^{-8} +3 q^{-10} -3 q^{-12} -4 q^{-14} +3 q^{-16} +6 q^{-18} - q^{-20} -7 q^{-22} -4 q^{-24} +5 q^{-26} +5 q^{-28} -2 q^{-30} -6 q^{-32} -2 q^{-34} +4 q^{-36} +3 q^{-38} - q^{-40} -3 q^{-42} - q^{-44} +2 q^{-46} + q^{-48} - q^{-50} - q^{-52} + q^{-56} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{46}-q^{44}+q^{42}-2 q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-4 q^{32}+5 q^{30}-5 q^{28}+4 q^{26}-5 q^{24}+3 q^{22}-5 q^{20}-q^{18}-2 q^{16}-2 q^{14}+2 q^{12}-6 q^{10}+7 q^8-4 q^6+13 q^4-5 q^2+14-5 q^{-2} +13 q^{-4} -4 q^{-6} +7 q^{-8} -6 q^{-10} +2 q^{-12} -2 q^{-14} -2 q^{-16} - q^{-18} -5 q^{-20} +3 q^{-22} -5 q^{-24} +4 q^{-26} -5 q^{-28} +5 q^{-30} -4 q^{-32} +3 q^{-34} -3 q^{-36} +3 q^{-38} -2 q^{-40} + q^{-42} - q^{-44} + q^{-46} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-q^{78}+2 q^{76}-3 q^{74}+2 q^{72}-q^{70}-2 q^{68}+6 q^{66}-7 q^{64}+8 q^{62}-7 q^{60}+2 q^{58}+2 q^{56}-8 q^{54}+11 q^{52}-15 q^{50}+12 q^{48}-7 q^{46}-2 q^{44}+9 q^{42}-16 q^{40}+19 q^{38}-14 q^{36}+4 q^{34}+5 q^{32}-14 q^{30}+15 q^{28}-7 q^{26}-q^{24}+11 q^{22}-12 q^{20}+9 q^{18}+2 q^{16}-12 q^{14}+21 q^{12}-20 q^{10}+14 q^8-q^6-12 q^4+23 q^2-25+23 q^{-2} -12 q^{-4} - q^{-6} +14 q^{-8} -20 q^{-10} +21 q^{-12} -12 q^{-14} +2 q^{-16} +9 q^{-18} -12 q^{-20} +11 q^{-22} - q^{-24} -7 q^{-26} +15 q^{-28} -14 q^{-30} +5 q^{-32} +4 q^{-34} -14 q^{-36} +19 q^{-38} -16 q^{-40} +9 q^{-42} -2 q^{-44} -7 q^{-46} +12 q^{-48} -15 q^{-50} +11 q^{-52} -8 q^{-54} +2 q^{-56} +2 q^{-58} -7 q^{-60} +8 q^{-62} -7 q^{-64} +6 q^{-66} -2 q^{-68} - q^{-70} +2 q^{-72} -3 q^{-74} +2 q^{-76} - q^{-78} + q^{-80} }[/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 17"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-3 t^3+5 t^2-7 t+9-7 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+5 z^6+7 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]=

{ 41, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^5+2 q^4-3 q^3+5 q^2-6 q+7-6 q^{-1} +5 q^{-2} -3 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} +17 z^4-7 a^2 z^2-7 z^2 a^{-2} +16 z^2-2 a^2-2 a^{-2} +5 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7-2 a z^7-2 z^7 a^{-1} +2 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^5-5 a^3 z^5-5 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+11 a^2 z^4+11 z^4 a^{-2} -6 z^4 a^{-4} +34 z^4-3 a^5 z^3+2 a^3 z^3+6 a z^3+6 z^3 a^{-1} +2 z^3 a^{-3} -3 z^3 a^{-5} +3 a^4 z^2-8 a^2 z^2-8 z^2 a^{-2} +3 z^2 a^{-4} -22 z^2+a^5 z-3 a z-3 z a^{-1} +z a^{-5} +2 a^2+2 a^{-2} +5 }[/math]

Vassiliev invariants

V2 and V3:

(2, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{20}{3} }[/math] [math]\displaystyle{ -\frac{124}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ -\frac{992}{3} }[/math] [math]\displaystyle{ -\frac{2009}{15} }[/math] [math]\displaystyle{ \frac{3476}{15} }[/math] [math]\displaystyle{ -\frac{25556}{45} }[/math] [math]\displaystyle{ \frac{89}{9} }[/math] [math]\displaystyle{ -\frac{2489}{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 17. 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										1		1
7									2	1		-1
5								3	1			2
3							3	2				-1
1						4	3					1
-1					3	4						1
-3				2	3							-1
-5			1	3								2
-7		1	2									-1
-9		1										1
-11	1											-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, 17]]
Out[2]=	10
In[3]:=	PD[Knot[10, 17]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[20, 15, 1, 16], X[16, 7, 17, 8], X[18, 9, 19, 10], X[8, 17, 9, 18], X[10, 19, 11, 20], X[14, 6, 15, 5], X[2, 12, 3, 11], X[4, 14, 5, 13]]
In[4]:=	GaussCode[Knot[10, 17]]
Out[4]=	GaussCode[1, -9, 2, -10, 8, -1, 4, -6, 5, -7, 9, -2, 10, -8, 3, -4, 6, -5, 7, -3]
In[5]:=	BR[Knot[10, 17]]
Out[5]=	BR[3, {-1, -1, -1, -1, 2, -1, 2, 2, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 17]][t]
Out[6]=	-4 3 5 7 2 3 4 9 + t - -- + -- - - - 7 t + 5 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 17]][z]
Out[7]=	2 4 6 8 1 + 2 z + 7 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 17]}
In[9]:=	{KnotDet[Knot[10, 17]], KnotSignature[Knot[10, 17]]}
Out[9]=	{41, 0}
In[10]:=	J=Jones[Knot[10, 17]][q]
Out[10]=	-5 2 3 5 6 2 3 4 5 7 - q + -- - -- + -- - - - 6 q + 5 q - 3 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, 17]}
In[12]:=	A2Invariant[Knot[10, 17]][q]
Out[12]=	-14 -10 -8 -6 2 2 6 8 10 14 -1 - q - q + q + q + -- + 2 q + q + q - q - q 2 q
In[13]:=	Kauffman[Knot[10, 17]][a, z]
Out[13]=	2 2 2 2 z 3 z 5 2 3 z 8 z 5 + -- + 2 a + -- - --- - 3 a z + a z - 22 z + ---- - ---- - 2 5 a 4 2 a a a a 3 3 3 2 2 4 2 3 z 2 z 6 z 3 3 3 5 3 8 a z + 3 a z - ---- + ---- + ---- + 6 a z + 2 a z - 3 a z + 5 3 a a a 4 4 5 5 4 6 z 11 z 2 4 4 4 z 5 z 3 5 34 z - ---- + ----- + 11 a z - 6 a z + -- - ---- - 5 a z + 4 2 5 3 a a a a 6 6 7 7 5 5 6 2 z 7 z 2 6 4 6 2 z 2 z a z - 18 z + ---- - ---- - 7 a z + 2 a z + ---- - ---- - 4 2 3 a a a a 8 9 7 3 7 8 2 z 2 8 z 9 2 a z + 2 a z + 4 z + ---- + 2 a z + -- + a z 2 a a
In[14]:=	{Vassiliev[2][Knot[10, 17]], Vassiliev[3][Knot[10, 17]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 17]][q, t]
Out[15]=	4 1 1 1 2 1 3 2 - + 4 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 3 3 3 3 2 5 2 5 3 7 3 ---- + --- + 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + 3 q t q t 7 4 9 4 11 5 q t + q t + q t