8 10

8_9

8_11

Visit 8 10's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 10's page at Knotilus! Visit 8 10's page at the original Knot Atlas! 8 10 Quick Notes

8 10 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,1,12,16 X15,11,16,10 X7283
Gauss code	-1, 8, -2, 1, -4, 5, -8, 2, -3, 7, -6, 4, -5, 3, -7, 6
Dowker-Thistlethwaite code	4 8 12 2 14 16 6 10
Conway Notation	[3,21,2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-8] Hyperbolic Volume 8.65115 A-Polynomial See Data:8 10/A-polynomial

[edit Notes for 8 10'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{ 1 }[/math] Rasmussen s-Invariant 2

[edit Notes for 8 10's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-3 t^2+6 t-7+6 t^{-1} -3 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+3 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 27, 2 }
Jones polynomial	[math]\displaystyle{ -q^6+2 q^5-4 q^4+5 q^3-4 q^2+5 q-3+2 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +5 z^4 a^{-2} -z^4 a^{-4} -z^4+9 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+6 a^{-2} -3 a^{-4} -2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +5 z^6 a^{-2} +3 z^6 a^{-4} +2 z^6+a z^5+z^5 a^{-1} +3 z^5 a^{-3} +3 z^5 a^{-5} -13 z^4 a^{-2} -5 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4-3 a z^3-8 z^3 a^{-1} -9 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +5 z^2+2 a z+5 z a^{-1} +6 z a^{-3} +2 z a^{-5} -z a^{-7} -6 a^{-2} -3 a^{-4} -2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^6-q^2+2 q^{-2} + q^{-4} +4 q^{-6} + q^{-8} + q^{-10} - q^{-12} -2 q^{-14} - q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{32}-q^{30}+3 q^{28}-4 q^{26}+2 q^{24}-q^{22}-5 q^{20}+9 q^{18}-12 q^{16}+10 q^{14}-6 q^{12}-5 q^{10}+12 q^8-16 q^6+14 q^4-9 q^2-2+9 q^{-2} -13 q^{-4} +9 q^{-6} - q^{-8} -4 q^{-10} +12 q^{-12} -7 q^{-14} +3 q^{-16} +7 q^{-18} -10 q^{-20} +19 q^{-22} -14 q^{-24} +9 q^{-26} +7 q^{-28} -12 q^{-30} +23 q^{-32} -19 q^{-34} +14 q^{-36} - q^{-38} -7 q^{-40} +13 q^{-42} -17 q^{-44} +11 q^{-46} - q^{-48} -7 q^{-50} +8 q^{-52} -8 q^{-54} -2 q^{-56} +8 q^{-58} -14 q^{-60} +9 q^{-62} -6 q^{-64} -3 q^{-66} +9 q^{-68} -14 q^{-70} +14 q^{-72} -8 q^{-74} +2 q^{-76} +2 q^{-78} -7 q^{-80} +6 q^{-82} -6 q^{-84} +5 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_10.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^5+q^3-q+2 q^{-1} + q^{-3} + q^{-5} + q^{-7} -2 q^{-9} + q^{-11} - q^{-13} }[/math]
2	[math]\displaystyle{ q^{16}-q^{14}-2 q^{12}+3 q^{10}-5 q^6+3 q^4+3 q^2-5+2 q^{-2} +5 q^{-4} -2 q^{-6} +4 q^{-10} + q^{-12} -2 q^{-14} - q^{-16} +4 q^{-18} -4 q^{-20} -4 q^{-22} +6 q^{-24} -2 q^{-26} -4 q^{-28} +3 q^{-30} - q^{-34} + q^{-36} }[/math]
3	[math]\displaystyle{ -q^{33}+q^{31}+2 q^{29}-4 q^{25}-2 q^{23}+6 q^{21}+5 q^{19}-6 q^{17}-10 q^{15}+3 q^{13}+13 q^{11}+q^9-15 q^7-6 q^5+12 q^3+12 q-11 q^{-1} -11 q^{-3} +8 q^{-5} +16 q^{-7} -3 q^{-9} -13 q^{-11} +2 q^{-13} +14 q^{-15} + q^{-17} -10 q^{-19} -3 q^{-21} +6 q^{-23} +8 q^{-25} -4 q^{-27} -11 q^{-29} -4 q^{-31} +14 q^{-33} +6 q^{-35} -13 q^{-37} -13 q^{-39} +13 q^{-41} +13 q^{-43} -10 q^{-45} -13 q^{-47} +4 q^{-49} +11 q^{-51} - q^{-53} -6 q^{-55} + q^{-57} +3 q^{-59} - q^{-63} + q^{-67} - q^{-69} }[/math]
4	[math]\displaystyle{ q^{56}-q^{54}-2 q^{52}+q^{48}+6 q^{46}-6 q^{42}-6 q^{40}-4 q^{38}+15 q^{36}+12 q^{34}-2 q^{32}-16 q^{30}-25 q^{28}+9 q^{26}+26 q^{24}+24 q^{22}-4 q^{20}-45 q^{18}-22 q^{16}+11 q^{14}+45 q^{12}+32 q^{10}-33 q^8-46 q^6-26 q^4+37 q^2+60+ q^{-2} -40 q^{-4} -49 q^{-6} +13 q^{-8} +60 q^{-10} +25 q^{-12} -26 q^{-14} -52 q^{-16} - q^{-18} +46 q^{-20} +28 q^{-22} -16 q^{-24} -42 q^{-26} -9 q^{-28} +27 q^{-30} +29 q^{-32} -4 q^{-34} -30 q^{-36} -22 q^{-38} +5 q^{-40} +32 q^{-42} +21 q^{-44} -7 q^{-46} -40 q^{-48} -31 q^{-50} +28 q^{-52} +46 q^{-54} +27 q^{-56} -41 q^{-58} -66 q^{-60} +3 q^{-62} +48 q^{-64} +55 q^{-66} -17 q^{-68} -64 q^{-70} -19 q^{-72} +21 q^{-74} +50 q^{-76} +10 q^{-78} -32 q^{-80} -19 q^{-82} -4 q^{-84} +22 q^{-86} +10 q^{-88} -7 q^{-90} -3 q^{-92} -6 q^{-94} +4 q^{-96} +2 q^{-98} -2 q^{-100} + q^{-102} -2 q^{-104} + q^{-106} - q^{-110} + q^{-112} }[/math]
5	[math]\displaystyle{ -q^{85}+q^{83}+2 q^{81}-q^{77}-3 q^{75}-4 q^{73}+8 q^{69}+8 q^{67}+2 q^{65}-7 q^{63}-16 q^{61}-14 q^{59}+4 q^{57}+26 q^{55}+29 q^{53}+10 q^{51}-23 q^{49}-49 q^{47}-39 q^{45}+7 q^{43}+59 q^{41}+71 q^{39}+31 q^{37}-43 q^{35}-97 q^{33}-82 q^{31}+2 q^{29}+98 q^{27}+127 q^{25}+61 q^{23}-66 q^{21}-153 q^{19}-129 q^{17}+2 q^{15}+149 q^{13}+181 q^{11}+66 q^9-107 q^7-208 q^5-143 q^3+53 q+209 q^{-1} +185 q^{-3} +16 q^{-5} -176 q^{-7} -215 q^{-9} -60 q^{-11} +146 q^{-13} +214 q^{-15} +94 q^{-17} -100 q^{-19} -198 q^{-21} -105 q^{-23} +78 q^{-25} +170 q^{-27} +104 q^{-29} -55 q^{-31} -147 q^{-33} -93 q^{-35} +40 q^{-37} +124 q^{-39} +82 q^{-41} -32 q^{-43} -108 q^{-45} -81 q^{-47} +14 q^{-49} +92 q^{-51} +84 q^{-53} +12 q^{-55} -69 q^{-57} -102 q^{-59} -55 q^{-61} +46 q^{-63} +112 q^{-65} +105 q^{-67} +9 q^{-69} -122 q^{-71} -164 q^{-73} -60 q^{-75} +104 q^{-77} +204 q^{-79} +139 q^{-81} -71 q^{-83} -236 q^{-85} -193 q^{-87} +11 q^{-89} +218 q^{-91} +241 q^{-93} +50 q^{-95} -176 q^{-97} -246 q^{-99} -100 q^{-101} +113 q^{-103} +213 q^{-105} +131 q^{-107} -46 q^{-109} -160 q^{-111} -128 q^{-113} + q^{-115} +98 q^{-117} +99 q^{-119} +27 q^{-121} -47 q^{-123} -64 q^{-125} -31 q^{-127} +17 q^{-129} +32 q^{-131} +20 q^{-133} - q^{-135} -13 q^{-137} -12 q^{-139} -4 q^{-141} +6 q^{-143} +5 q^{-145} - q^{-151} -2 q^{-153} + q^{-155} +2 q^{-157} - q^{-159} + q^{-163} - q^{-165} }[/math]
6	[math]\displaystyle{ q^{120}-q^{118}-2 q^{116}+q^{112}+3 q^{110}+q^{108}+4 q^{106}-2 q^{104}-10 q^{102}-7 q^{100}-2 q^{98}+8 q^{96}+10 q^{94}+21 q^{92}+8 q^{90}-16 q^{88}-30 q^{86}-32 q^{84}-12 q^{82}+8 q^{80}+61 q^{78}+66 q^{76}+31 q^{74}-25 q^{72}-81 q^{70}-100 q^{68}-88 q^{66}+27 q^{64}+121 q^{62}+165 q^{60}+121 q^{58}+8 q^{56}-137 q^{54}-260 q^{52}-194 q^{50}-43 q^{48}+171 q^{46}+308 q^{44}+312 q^{42}+121 q^{40}-207 q^{38}-392 q^{36}-416 q^{34}-182 q^{32}+171 q^{30}+505 q^{28}+550 q^{26}+238 q^{24}-187 q^{22}-585 q^{20}-648 q^{18}-349 q^{16}+245 q^{14}+694 q^{12}+721 q^{10}+362 q^8-285 q^6-773 q^4-829 q^2-300+394 q^{-2} +838 q^{-4} +803 q^{-6} +225 q^{-8} -492 q^{-10} -923 q^{-12} -684 q^{-14} -47 q^{-16} +604 q^{-18} +871 q^{-20} +539 q^{-22} -128 q^{-24} -712 q^{-26} -727 q^{-28} -295 q^{-30} +311 q^{-32} +677 q^{-34} +545 q^{-36} +55 q^{-38} -455 q^{-40} -555 q^{-42} -298 q^{-44} +159 q^{-46} +453 q^{-48} +393 q^{-50} +60 q^{-52} -302 q^{-54} -376 q^{-56} -203 q^{-58} +123 q^{-60} +320 q^{-62} +280 q^{-64} +44 q^{-66} -222 q^{-68} -309 q^{-70} -203 q^{-72} +56 q^{-74} +255 q^{-76} +318 q^{-78} +173 q^{-80} -83 q^{-82} -316 q^{-84} -375 q^{-86} -189 q^{-88} +105 q^{-90} +417 q^{-92} +486 q^{-94} +266 q^{-96} -200 q^{-98} -580 q^{-100} -612 q^{-102} -288 q^{-104} +321 q^{-106} +768 q^{-108} +775 q^{-110} +203 q^{-112} -517 q^{-114} -928 q^{-116} -808 q^{-118} -103 q^{-120} +689 q^{-122} +1086 q^{-124} +708 q^{-126} -81 q^{-128} -797 q^{-130} -1043 q^{-132} -579 q^{-134} +221 q^{-136} +876 q^{-138} +865 q^{-140} +366 q^{-142} -297 q^{-144} -761 q^{-146} -678 q^{-148} -199 q^{-150} +362 q^{-152} +563 q^{-154} +434 q^{-156} +85 q^{-158} -283 q^{-160} -399 q^{-162} -258 q^{-164} +21 q^{-166} +177 q^{-168} +223 q^{-170} +139 q^{-172} -26 q^{-174} -120 q^{-176} -119 q^{-178} -38 q^{-180} +10 q^{-182} +53 q^{-184} +63 q^{-186} +17 q^{-188} -18 q^{-190} -28 q^{-192} -12 q^{-194} -10 q^{-196} +5 q^{-198} +17 q^{-200} +6 q^{-202} -2 q^{-204} -5 q^{-206} + q^{-208} -4 q^{-210} - q^{-212} +5 q^{-214} - q^{-218} -2 q^{-220} + q^{-222} - q^{-226} + q^{-228} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^6-q^2+2 q^{-2} + q^{-4} +4 q^{-6} + q^{-8} + q^{-10} - q^{-12} -2 q^{-14} - q^{-18} }[/math]
1,1	[math]\displaystyle{ q^{20}-2 q^{18}+6 q^{16}-12 q^{14}+19 q^{12}-28 q^{10}+36 q^8-44 q^6+39 q^4-40 q^2+26-10 q^{-2} -8 q^{-4} +34 q^{-6} -40 q^{-8} +70 q^{-10} -67 q^{-12} +80 q^{-14} -70 q^{-16} +60 q^{-18} -49 q^{-20} +24 q^{-22} -12 q^{-24} -12 q^{-26} +23 q^{-28} -34 q^{-30} +34 q^{-32} -34 q^{-34} +32 q^{-36} -26 q^{-38} +18 q^{-40} -14 q^{-42} +11 q^{-44} -6 q^{-46} +4 q^{-48} -2 q^{-50} + q^{-52} }[/math]
2,0	[math]\displaystyle{ q^{18}-q^{14}+q^{10}-q^8-3 q^6-q^4+q^2-2-2 q^{-2} +3 q^{-4} + q^{-6} + q^{-8} +3 q^{-10} +6 q^{-12} +4 q^{-14} +5 q^{-16} +6 q^{-18} +2 q^{-20} -4 q^{-22} -3 q^{-24} -3 q^{-26} -7 q^{-28} -4 q^{-30} - q^{-36} +2 q^{-40} + q^{-42} + q^{-46} }[/math]
3,0	[math]\displaystyle{ -q^{36}+q^{32}+2 q^{30}-3 q^{26}-q^{24}+3 q^{22}+6 q^{20}-7 q^{16}-6 q^{14}+3 q^{12}+8 q^{10}-12 q^6-9 q^4+3 q^2+10- q^{-2} -9 q^{-4} -3 q^{-6} +8 q^{-8} +10 q^{-10} - q^{-12} -2 q^{-14} +3 q^{-16} +10 q^{-18} +4 q^{-20} +4 q^{-24} +11 q^{-26} +5 q^{-28} + q^{-30} +9 q^{-34} +4 q^{-36} -11 q^{-38} -18 q^{-40} -9 q^{-42} +3 q^{-44} -4 q^{-46} -13 q^{-48} -14 q^{-50} +4 q^{-52} +10 q^{-54} +4 q^{-56} -5 q^{-58} -4 q^{-60} +7 q^{-62} +10 q^{-64} +4 q^{-66} - q^{-68} -2 q^{-70} +2 q^{-72} + q^{-74} - q^{-76} - q^{-78} - q^{-80} - q^{-84} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{14}-q^{12}+q^{10}+q^8-4 q^6-2 q^2-7+ q^{-2} +3 q^{-4} +9 q^{-8} +10 q^{-10} +6 q^{-12} +3 q^{-14} + q^{-16} -2 q^{-18} -6 q^{-20} -6 q^{-22} + q^{-24} -4 q^{-26} -3 q^{-28} +5 q^{-30} -2 q^{-32} -2 q^{-34} +3 q^{-36} - q^{-40} + q^{-42} }[/math]
1,0,0	[math]\displaystyle{ -q^7-2 q^3- q^{-1} +2 q^{-3} +2 q^{-5} +4 q^{-7} +4 q^{-9} +2 q^{-11} + q^{-13} -2 q^{-15} - q^{-17} -3 q^{-19} - q^{-23} }[/math]
1,0,1	[math]\displaystyle{ q^{24}-2 q^{22}+5 q^{20}-5 q^{18}+2 q^{16}+6 q^{14}-16 q^{12}+24 q^{10}-21 q^8+7 q^6+7 q^4-39 q^2+35-47 q^{-2} +14 q^{-4} +4 q^{-6} -21 q^{-8} +47 q^{-10} -7 q^{-12} +34 q^{-14} +21 q^{-16} +16 q^{-18} +19 q^{-22} -26 q^{-24} -13 q^{-26} +7 q^{-28} -54 q^{-30} +29 q^{-32} -29 q^{-34} -2 q^{-36} +27 q^{-38} -32 q^{-40} +31 q^{-42} -7 q^{-44} -10 q^{-46} +21 q^{-48} -19 q^{-50} +5 q^{-52} +5 q^{-54} -9 q^{-56} +9 q^{-58} -3 q^{-60} - q^{-62} +3 q^{-64} -2 q^{-66} + q^{-68} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{16}+q^{12}+2 q^{10}-2 q^6-q^4-6 q^2-10-7 q^{-2} -4 q^{-4} -3 q^{-6} + q^{-8} +14 q^{-10} +17 q^{-12} +15 q^{-14} +16 q^{-16} +17 q^{-18} +2 q^{-20} -3 q^{-22} -4 q^{-24} -11 q^{-26} -14 q^{-28} -8 q^{-30} -4 q^{-32} -6 q^{-34} -2 q^{-36} +3 q^{-38} +2 q^{-40} -2 q^{-42} +2 q^{-44} +3 q^{-46} + q^{-52} }[/math]
1,0,0,0	[math]\displaystyle{ -q^8-2 q^4-q^2-1- q^{-2} +2 q^{-4} +2 q^{-6} +5 q^{-8} +4 q^{-10} +5 q^{-12} +2 q^{-14} + q^{-16} -2 q^{-18} -2 q^{-20} -2 q^{-22} -3 q^{-24} - q^{-28} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{14}+q^{12}-3 q^{10}+3 q^8-4 q^6+4 q^4-4 q^2+3- q^{-2} + q^{-4} +4 q^{-6} -3 q^{-8} +8 q^{-10} -6 q^{-12} +9 q^{-14} -7 q^{-16} +6 q^{-18} -4 q^{-20} +2 q^{-22} - q^{-24} -2 q^{-26} +3 q^{-28} -5 q^{-30} +4 q^{-32} -4 q^{-34} +3 q^{-36} -2 q^{-38} + q^{-40} - q^{-42} }[/math]
1,0	[math]\displaystyle{ q^{24}-q^{20}-q^{18}+2 q^{16}+2 q^{14}-2 q^{12}-4 q^{10}-q^8+3 q^6+q^4-5 q^2-5+ q^{-2} +4 q^{-4} +2 q^{-6} -3 q^{-8} + q^{-10} +5 q^{-12} +6 q^{-14} + q^{-16} +2 q^{-18} +4 q^{-20} +5 q^{-22} -3 q^{-26} - q^{-28} +2 q^{-30} - q^{-32} -5 q^{-34} -4 q^{-36} + q^{-38} +2 q^{-40} -3 q^{-42} -5 q^{-44} +5 q^{-48} + q^{-50} -3 q^{-52} -3 q^{-54} + q^{-56} +3 q^{-58} + q^{-60} - q^{-62} - q^{-64} + q^{-68} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{18}-q^{16}+2 q^{14}-2 q^{12}+3 q^{10}-4 q^8+q^6-6 q^4-6-2 q^{-2} -2 q^{-4} + q^{-6} +5 q^{-8} +3 q^{-10} +13 q^{-12} +5 q^{-14} +14 q^{-16} +9 q^{-20} -5 q^{-22} +3 q^{-24} -9 q^{-26} -3 q^{-28} -7 q^{-30} -2 q^{-32} -2 q^{-34} -3 q^{-36} + q^{-38} -3 q^{-40} +5 q^{-42} -3 q^{-44} +2 q^{-46} -3 q^{-48} +3 q^{-50} - q^{-52} + q^{-54} - q^{-56} + q^{-58} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{32}-q^{30}+3 q^{28}-4 q^{26}+2 q^{24}-q^{22}-5 q^{20}+9 q^{18}-12 q^{16}+10 q^{14}-6 q^{12}-5 q^{10}+12 q^8-16 q^6+14 q^4-9 q^2-2+9 q^{-2} -13 q^{-4} +9 q^{-6} - q^{-8} -4 q^{-10} +12 q^{-12} -7 q^{-14} +3 q^{-16} +7 q^{-18} -10 q^{-20} +19 q^{-22} -14 q^{-24} +9 q^{-26} +7 q^{-28} -12 q^{-30} +23 q^{-32} -19 q^{-34} +14 q^{-36} - q^{-38} -7 q^{-40} +13 q^{-42} -17 q^{-44} +11 q^{-46} - q^{-48} -7 q^{-50} +8 q^{-52} -8 q^{-54} -2 q^{-56} +8 q^{-58} -14 q^{-60} +9 q^{-62} -6 q^{-64} -3 q^{-66} +9 q^{-68} -14 q^{-70} +14 q^{-72} -8 q^{-74} +2 q^{-76} +2 q^{-78} -7 q^{-80} +6 q^{-82} -6 q^{-84} +5 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }[/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["8 10"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-3 t^2+6 t-7+6 t^{-1} -3 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+3 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]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^6+2 q^5-4 q^4+5 q^3-4 q^2+5 q-3+2 q^{-1} - q^{-2} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^6 a^{-2} +5 z^4 a^{-2} -z^4 a^{-4} -z^4+9 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+6 a^{-2} -3 a^{-4} -2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +5 z^6 a^{-2} +3 z^6 a^{-4} +2 z^6+a z^5+z^5 a^{-1} +3 z^5 a^{-3} +3 z^5 a^{-5} -13 z^4 a^{-2} -5 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4-3 a z^3-8 z^3 a^{-1} -9 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +5 z^2+2 a z+5 z a^{-1} +6 z a^{-3} +2 z a^{-5} -z a^{-7} -6 a^{-2} -3 a^{-4} -2 }[/math]

Vassiliev invariants

V2 and V3:

(3, 3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 12 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 110 }[/math] [math]\displaystyle{ 10 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 432 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 1320 }[/math] [math]\displaystyle{ 120 }[/math] [math]\displaystyle{ \frac{19711}{10} }[/math] [math]\displaystyle{ \frac{618}{5} }[/math] [math]\displaystyle{ \frac{9062}{15} }[/math] [math]\displaystyle{ \frac{161}{6} }[/math] [math]\displaystyle{ \frac{511}{10} }[/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]2 is the signature of 8 10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 χ 13                 1 -1 11               1   1 9             3 1   -2 7           2 1     1 5         2 3       1 3       3 2         1 1     1 3           2 -1   1 2             -1 -3   1               1 -5 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=3 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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[8, 10]]
Out[2]=	8
In[3]:=	PD[Knot[8, 10]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[9, 15, 10, 14], X[5, 13, 6, 12], X[13, 7, 14, 6], X[11, 1, 12, 16], X[15, 11, 16, 10], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[8, 10]]
Out[4]=	GaussCode[-1, 8, -2, 1, -4, 5, -8, 2, -3, 7, -6, 4, -5, 3, -7, 6]
In[5]:=	BR[Knot[8, 10]]
Out[5]=	BR[3, {1, 1, 1, -2, 1, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[8, 10]][t]
Out[6]=	-3 3 6 2 3 -7 + t - -- + - + 6 t - 3 t + t 2 t t
In[7]:=	Conway[Knot[8, 10]][z]
Out[7]=	2 4 6 1 + 3 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 10], Knot[10, 143], Knot[11, NonAlternating, 106]}
In[9]:=	{KnotDet[Knot[8, 10]], KnotSignature[Knot[8, 10]]}
Out[9]=	{27, 2}
In[10]:=	J=Jones[Knot[8, 10]][q]
Out[10]=	-2 2 2 3 4 5 6 -3 - q + - + 5 q - 4 q + 5 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 10]}
In[12]:=	A2Invariant[Knot[8, 10]][q]
Out[12]=	-6 -2 2 4 6 8 10 12 14 18 -q - q + 2 q + q + 4 q + q + q - q - 2 q - q
In[13]:=	Kauffman[Knot[8, 10]][a, z]
Out[13]=	2 2 3 6 z 2 z 6 z 5 z 2 z 6 z -2 - -- - -- - -- + --- + --- + --- + 2 a z + 5 z - -- + ---- + 4 2 7 5 3 a 6 4 a a a a a a a 2 3 3 3 3 4 4 12 z z 3 z 9 z 8 z 3 4 2 z 5 z ----- + -- - ---- - ---- - ---- - 3 a z - 6 z + ---- - ---- - 2 7 5 3 a 6 4 a a a a a a 4 5 5 5 6 6 7 7 13 z 3 z 3 z z 5 6 3 z 5 z z z ----- + ---- + ---- + -- + a z + 2 z + ---- + ---- + -- + -- 2 5 3 a 4 2 3 a a a a a a a
In[14]:=	{Vassiliev[2][Knot[8, 10]], Vassiliev[3][Knot[8, 10]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[8, 10]][q, t]
Out[15]=	3 1 1 1 2 q 3 5 3 q + 3 q + ----- + ----- + ---- + --- + - + 2 q t + 2 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 13 5 3 q t + 2 q t + q t + 3 q t + q t + q t + q t