10 32

10_31

10_33

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10 32 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,14,6,15 X15,20,16,1 X7,17,8,16 X19,7,20,6 X9,19,10,18 X17,9,18,8 X13,10,14,11 X11,2,12,3
Gauss code	-1, 10, -2, 1, -3, 6, -5, 8, -7, 9, -10, 2, -9, 3, -4, 5, -8, 7, -6, 4
Dowker-Thistlethwaite code	4 12 14 16 18 2 10 20 8 6
Conway Notation	[311122]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-5] Hyperbolic Volume 12.0909 A-Polynomial See Data:10 32/A-polynomial

[edit Notes for 10 32'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{ 3 }[/math] Rasmussen s-Invariant 0

[edit Notes for 10 32's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+8 t^2-15 t+19-15 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^6-4 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 69, 0 }
Jones polynomial	[math]\displaystyle{ q^4-3 q^3+6 q^2-9 q+11-11 q^{-1} +11 q^{-2} -8 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^6-z^6+a^4 z^4-3 a^2 z^4+z^4 a^{-2} -3 z^4+2 a^4 z^2-2 a^2 z^2+2 z^2 a^{-2} -3 z^2+a^2+ a^{-2} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+6 a^2 z^8+3 z^8+3 a^5 z^7+5 a^3 z^7+7 a z^7+5 z^7 a^{-1} +a^6 z^6-7 a^4 z^6-10 a^2 z^6+5 z^6 a^{-2} +3 z^6-10 a^5 z^5-18 a^3 z^5-15 a z^5-4 z^5 a^{-1} +3 z^5 a^{-3} -3 a^6 z^4+3 a^4 z^4+2 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} -11 z^4+9 a^5 z^3+13 a^3 z^3+7 a z^3-3 z^3 a^{-3} +2 a^6 z^2+4 z^2 a^{-2} -z^2 a^{-4} +7 z^2-a^5 z-2 a^3 z-a z+z a^{-1} +z a^{-3} -a^2- a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{18}-q^{16}-2 q^{10}+3 q^8+q^4+q^2-2+2 q^{-2} -2 q^{-4} + q^{-6} + q^{-8} - q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-2 q^{92}+5 q^{90}-9 q^{88}+9 q^{86}-8 q^{84}-q^{82}+19 q^{80}-35 q^{78}+49 q^{76}-47 q^{74}+23 q^{72}+14 q^{70}-62 q^{68}+99 q^{66}-108 q^{64}+80 q^{62}-18 q^{60}-52 q^{58}+110 q^{56}-129 q^{54}+105 q^{52}-46 q^{50}-25 q^{48}+76 q^{46}-97 q^{44}+68 q^{42}-6 q^{40}-48 q^{38}+83 q^{36}-75 q^{34}+24 q^{32}+46 q^{30}-114 q^{28}+146 q^{26}-129 q^{24}+62 q^{22}+40 q^{20}-129 q^{18}+182 q^{16}-171 q^{14}+109 q^{12}-16 q^{10}-75 q^8+126 q^6-128 q^4+84 q^2-11-48 q^{-2} +72 q^{-4} -55 q^{-6} +4 q^{-8} +48 q^{-10} -84 q^{-12} +83 q^{-14} -50 q^{-16} -6 q^{-18} +65 q^{-20} -104 q^{-22} +113 q^{-24} -83 q^{-26} +37 q^{-28} +14 q^{-30} -58 q^{-32} +78 q^{-34} -76 q^{-36} +58 q^{-38} -25 q^{-40} -3 q^{-42} +23 q^{-44} -32 q^{-46} +30 q^{-48} -21 q^{-50} +12 q^{-52} -2 q^{-54} -4 q^{-56} +5 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_32.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-2 q^{11}+2 q^9-3 q^7+3 q^5+2 q^{-1} -3 q^{-3} +3 q^{-5} -2 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{38}-2 q^{36}-2 q^{34}+7 q^{32}-2 q^{30}-10 q^{28}+13 q^{26}+4 q^{24}-20 q^{22}+12 q^{20}+11 q^{18}-23 q^{16}+5 q^{14}+16 q^{12}-13 q^{10}-5 q^8+12 q^6+4 q^4-11 q^2-1+19 q^{-2} -12 q^{-4} -13 q^{-6} +23 q^{-8} -7 q^{-10} -14 q^{-12} +16 q^{-14} -2 q^{-16} -8 q^{-18} +6 q^{-20} -2 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{75}-2 q^{73}-2 q^{71}+3 q^{69}+7 q^{67}-2 q^{65}-16 q^{63}-2 q^{61}+25 q^{59}+14 q^{57}-33 q^{55}-34 q^{53}+33 q^{51}+59 q^{49}-24 q^{47}-79 q^{45}+3 q^{43}+94 q^{41}+23 q^{39}-97 q^{37}-48 q^{35}+89 q^{33}+73 q^{31}-78 q^{29}-86 q^{27}+53 q^{25}+97 q^{23}-34 q^{21}-98 q^{19}+8 q^{17}+91 q^{15}+19 q^{13}-72 q^{11}-44 q^9+49 q^7+71 q^5-18 q^3-85 q-20 q^{-1} +95 q^{-3} +49 q^{-5} -89 q^{-7} -73 q^{-9} +77 q^{-11} +83 q^{-13} -58 q^{-15} -79 q^{-17} +38 q^{-19} +68 q^{-21} -25 q^{-23} -52 q^{-25} +16 q^{-27} +36 q^{-29} -8 q^{-31} -24 q^{-33} +4 q^{-35} +15 q^{-37} -3 q^{-39} -7 q^{-41} + q^{-43} +3 q^{-45} -2 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{124}-2 q^{122}-2 q^{120}+3 q^{118}+3 q^{116}+7 q^{114}-9 q^{112}-16 q^{110}-q^{108}+10 q^{106}+43 q^{104}+q^{102}-50 q^{100}-48 q^{98}-15 q^{96}+112 q^{94}+85 q^{92}-35 q^{90}-140 q^{88}-161 q^{86}+113 q^{84}+233 q^{82}+137 q^{80}-138 q^{78}-384 q^{76}-81 q^{74}+256 q^{72}+405 q^{70}+97 q^{68}-466 q^{66}-383 q^{64}+23 q^{62}+524 q^{60}+442 q^{58}-280 q^{56}-544 q^{54}-319 q^{52}+396 q^{50}+649 q^{48}+23 q^{46}-487 q^{44}-542 q^{42}+158 q^{40}+644 q^{38}+253 q^{36}-334 q^{34}-592 q^{32}-37 q^{30}+516 q^{28}+380 q^{26}-154 q^{24}-528 q^{22}-213 q^{20}+304 q^{18}+462 q^{16}+84 q^{14}-367 q^{12}-402 q^{10}-25 q^8+449 q^6+378 q^4-53 q^2-511-426 q^{-2} +262 q^{-4} +564 q^{-6} +333 q^{-8} -397 q^{-10} -672 q^{-12} -38 q^{-14} +479 q^{-16} +553 q^{-18} -131 q^{-20} -603 q^{-22} -219 q^{-24} +220 q^{-26} +481 q^{-28} +57 q^{-30} -352 q^{-32} -188 q^{-34} +30 q^{-36} +271 q^{-38} +74 q^{-40} -152 q^{-42} -77 q^{-44} -22 q^{-46} +115 q^{-48} +33 q^{-50} -62 q^{-52} -17 q^{-54} -16 q^{-56} +45 q^{-58} +9 q^{-60} -25 q^{-62} + q^{-64} -7 q^{-66} +14 q^{-68} +2 q^{-70} -8 q^{-72} +2 q^{-74} -2 q^{-76} +3 q^{-78} -2 q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ q^{185}-2 q^{183}-2 q^{181}+3 q^{179}+3 q^{177}+3 q^{175}-9 q^{171}-16 q^{169}-q^{167}+20 q^{165}+28 q^{163}+21 q^{161}-17 q^{159}-64 q^{157}-68 q^{155}+4 q^{153}+99 q^{151}+139 q^{149}+72 q^{147}-103 q^{145}-255 q^{143}-220 q^{141}+40 q^{139}+346 q^{137}+439 q^{135}+175 q^{133}-347 q^{131}-706 q^{129}-529 q^{127}+164 q^{125}+883 q^{123}+1001 q^{121}+275 q^{119}-851 q^{117}-1465 q^{115}-940 q^{113}+490 q^{111}+1745 q^{109}+1709 q^{107}+227 q^{105}-1661 q^{103}-2409 q^{101}-1210 q^{99}+1161 q^{97}+2809 q^{95}+2254 q^{93}-246 q^{91}-2778 q^{89}-3170 q^{87}-895 q^{85}+2309 q^{83}+3731 q^{81}+2040 q^{79}-1464 q^{77}-3870 q^{75}-3031 q^{73}+487 q^{71}+3634 q^{69}+3639 q^{67}+495 q^{65}-3093 q^{63}-3938 q^{61}-1268 q^{59}+2472 q^{57}+3880 q^{55}+1789 q^{53}-1823 q^{51}-3658 q^{49}-2083 q^{47}+1305 q^{45}+3317 q^{43}+2211 q^{41}-865 q^{39}-2986 q^{37}-2283 q^{35}+463 q^{33}+2662 q^{31}+2385 q^{29}-14 q^{27}-2314 q^{25}-2529 q^{23}-577 q^{21}+1853 q^{19}+2709 q^{17}+1326 q^{15}-1178 q^{13}-2819 q^{11}-2194 q^9+284 q^7+2693 q^5+3057 q^3+861 q-2288 q^{-1} -3736 q^{-3} -2048 q^{-5} +1518 q^{-7} +4021 q^{-9} +3175 q^{-11} -506 q^{-13} -3876 q^{-15} -3933 q^{-17} -585 q^{-19} +3258 q^{-21} +4234 q^{-23} +1543 q^{-25} -2360 q^{-27} -4033 q^{-29} -2157 q^{-31} +1381 q^{-33} +3407 q^{-35} +2369 q^{-37} -513 q^{-39} -2586 q^{-41} -2207 q^{-43} -70 q^{-45} +1739 q^{-47} +1787 q^{-49} +378 q^{-51} -1030 q^{-53} -1290 q^{-55} -451 q^{-57} +545 q^{-59} +827 q^{-61} +370 q^{-63} -243 q^{-65} -476 q^{-67} -256 q^{-69} +97 q^{-71} +258 q^{-73} +144 q^{-75} -47 q^{-77} -122 q^{-79} -67 q^{-81} +21 q^{-83} +59 q^{-85} +33 q^{-87} -21 q^{-89} -31 q^{-91} -6 q^{-93} +13 q^{-95} +12 q^{-97} + q^{-99} -4 q^{-101} -10 q^{-103} - q^{-105} +9 q^{-107} + q^{-109} -3 q^{-111} + q^{-113} - q^{-115} -2 q^{-117} +3 q^{-119} -2 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{18}-q^{16}-2 q^{10}+3 q^8+q^4+q^2-2+2 q^{-2} -2 q^{-4} + q^{-6} + q^{-8} - q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{52}-4 q^{50}+12 q^{48}-30 q^{46}+60 q^{44}-110 q^{42}+180 q^{40}-260 q^{38}+354 q^{36}-444 q^{34}+508 q^{32}-532 q^{30}+503 q^{28}-422 q^{26}+276 q^{24}-70 q^{22}-163 q^{20}+404 q^{18}-638 q^{16}+834 q^{14}-968 q^{12}+1022 q^{10}-990 q^8+882 q^6-704 q^4+490 q^2-252+24 q^{-2} +176 q^{-4} -320 q^{-6} +416 q^{-8} -468 q^{-10} +469 q^{-12} -430 q^{-14} +370 q^{-16} -304 q^{-18} +233 q^{-20} -164 q^{-22} +110 q^{-24} -70 q^{-26} +40 q^{-28} -20 q^{-30} +10 q^{-32} -4 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{48}-q^{46}-2 q^{44}+q^{42}+3 q^{40}-q^{38}-5 q^{36}+3 q^{34}+8 q^{32}-2 q^{30}-7 q^{28}+4 q^{26}+4 q^{24}-8 q^{22}-7 q^{20}+7 q^{18}+4 q^{16}-8 q^{14}+3 q^{12}+7 q^{10}-4 q^8-2 q^6+9 q^4-7+5 q^{-2} +9 q^{-4} -7 q^{-6} -8 q^{-8} +9 q^{-10} +3 q^{-12} -9 q^{-14} - q^{-16} +6 q^{-18} -4 q^{-22} + q^{-24} +3 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{40}-2 q^{38}+q^{36}+2 q^{34}-7 q^{32}+6 q^{30}+4 q^{28}-12 q^{26}+10 q^{24}+5 q^{22}-18 q^{20}+10 q^{18}+9 q^{16}-16 q^{14}+7 q^{12}+10 q^{10}-6 q^8-4 q^6+3 q^4+6 q^2-10-4 q^{-2} +17 q^{-4} -10 q^{-6} -9 q^{-8} +20 q^{-10} -6 q^{-12} -10 q^{-14} +14 q^{-16} -2 q^{-18} -7 q^{-20} +5 q^{-22} -2 q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ q^{23}-q^{21}+q^{19}-2 q^{17}+q^{15}-2 q^{13}+3 q^{11}+2 q^7-2 q^{-1} +2 q^{-3} -2 q^{-5} +2 q^{-7} - q^{-9} +2 q^{-11} - q^{-13} + q^{-15} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{40}-2 q^{38}+5 q^{36}-8 q^{34}+11 q^{32}-16 q^{30}+18 q^{28}-20 q^{26}+20 q^{24}-17 q^{22}+12 q^{20}-4 q^{18}-5 q^{16}+16 q^{14}-25 q^{12}+34 q^{10}-38 q^8+40 q^6-37 q^4+32 q^2-24+14 q^{-2} -3 q^{-4} -6 q^{-6} +13 q^{-8} -18 q^{-10} +20 q^{-12} -20 q^{-14} +18 q^{-16} -14 q^{-18} +11 q^{-20} -7 q^{-22} +4 q^{-24} -2 q^{-26} + q^{-28} }[/math]
1,0	[math]\displaystyle{ q^{66}-2 q^{62}-2 q^{60}+3 q^{58}+5 q^{56}-3 q^{54}-9 q^{52}-2 q^{50}+13 q^{48}+10 q^{46}-10 q^{44}-17 q^{42}+2 q^{40}+21 q^{38}+9 q^{36}-18 q^{34}-19 q^{32}+7 q^{30}+21 q^{28}+3 q^{26}-18 q^{24}-8 q^{22}+13 q^{20}+12 q^{18}-8 q^{16}-10 q^{14}+7 q^{12}+12 q^{10}-5 q^8-14 q^6+q^4+15 q^2+3-16 q^{-2} -9 q^{-4} +15 q^{-6} +15 q^{-8} -9 q^{-10} -20 q^{-12} +21 q^{-16} +11 q^{-18} -13 q^{-20} -17 q^{-22} +3 q^{-24} +16 q^{-26} +6 q^{-28} -8 q^{-30} -9 q^{-32} + q^{-34} +6 q^{-36} +2 q^{-38} -2 q^{-40} -2 q^{-42} + q^{-46} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{54}-2 q^{52}+3 q^{50}-5 q^{48}+7 q^{46}-10 q^{44}+11 q^{42}-13 q^{40}+16 q^{38}-16 q^{36}+15 q^{34}-14 q^{32}+14 q^{30}-10 q^{28}+4 q^{26}-3 q^{22}+11 q^{20}-17 q^{18}+23 q^{16}-23 q^{14}+30 q^{12}-32 q^{10}+29 q^8-28 q^6+27 q^4-24 q^2+15-13 q^{-2} +8 q^{-4} + q^{-6} -6 q^{-8} +7 q^{-10} -10 q^{-12} +18 q^{-14} -14 q^{-16} +14 q^{-18} -15 q^{-20} +16 q^{-22} -10 q^{-24} +8 q^{-26} -9 q^{-28} +6 q^{-30} -3 q^{-32} +2 q^{-34} -2 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{94}-2 q^{92}+5 q^{90}-9 q^{88}+9 q^{86}-8 q^{84}-q^{82}+19 q^{80}-35 q^{78}+49 q^{76}-47 q^{74}+23 q^{72}+14 q^{70}-62 q^{68}+99 q^{66}-108 q^{64}+80 q^{62}-18 q^{60}-52 q^{58}+110 q^{56}-129 q^{54}+105 q^{52}-46 q^{50}-25 q^{48}+76 q^{46}-97 q^{44}+68 q^{42}-6 q^{40}-48 q^{38}+83 q^{36}-75 q^{34}+24 q^{32}+46 q^{30}-114 q^{28}+146 q^{26}-129 q^{24}+62 q^{22}+40 q^{20}-129 q^{18}+182 q^{16}-171 q^{14}+109 q^{12}-16 q^{10}-75 q^8+126 q^6-128 q^4+84 q^2-11-48 q^{-2} +72 q^{-4} -55 q^{-6} +4 q^{-8} +48 q^{-10} -84 q^{-12} +83 q^{-14} -50 q^{-16} -6 q^{-18} +65 q^{-20} -104 q^{-22} +113 q^{-24} -83 q^{-26} +37 q^{-28} +14 q^{-30} -58 q^{-32} +78 q^{-34} -76 q^{-36} +58 q^{-38} -25 q^{-40} -3 q^{-42} +23 q^{-44} -32 q^{-46} +30 q^{-48} -21 q^{-50} +12 q^{-52} -2 q^{-54} -4 q^{-56} +5 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/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 32"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^3+8 t^2-15 t+19-15 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^6-4 z^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{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 69, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-3 q^3+6 q^2-9 q+11-11 q^{-1} +11 q^{-2} -8 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+6 a^2 z^8+3 z^8+3 a^5 z^7+5 a^3 z^7+7 a z^7+5 z^7 a^{-1} +a^6 z^6-7 a^4 z^6-10 a^2 z^6+5 z^6 a^{-2} +3 z^6-10 a^5 z^5-18 a^3 z^5-15 a z^5-4 z^5 a^{-1} +3 z^5 a^{-3} -3 a^6 z^4+3 a^4 z^4+2 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} -11 z^4+9 a^5 z^3+13 a^3 z^3+7 a z^3-3 z^3 a^{-3} +2 a^6 z^2+4 z^2 a^{-2} -z^2 a^{-4} +7 z^2-a^5 z-2 a^3 z-a z+z a^{-1} +z a^{-3} -a^2- a^{-2} -1 }[/math]

Vassiliev invariants

V2 and V3:

(-1, 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{ -4 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{130}{3} }[/math] [math]\displaystyle{ \frac{110}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{520}{3} }[/math] [math]\displaystyle{ -\frac{440}{3} }[/math] [math]\displaystyle{ -\frac{3631}{30} }[/math] [math]\displaystyle{ \frac{1514}{5} }[/math] [math]\displaystyle{ -\frac{22382}{45} }[/math] [math]\displaystyle{ \frac{1615}{18} }[/math] [math]\displaystyle{ -\frac{4111}{30} }[/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 32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   2   -2 5                 4 1   3 3               5 2     -3 1             6 4       2 -1           6 6         0 -3         5 5           0 -5       3 6             3 -7     2 5               -3 -9   1 3                 2 -11   2                   -2 -13 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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, 32]]
Out[2]=	10
In[3]:=	PD[Knot[10, 32]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[15, 20, 16, 1], X[7, 17, 8, 16], X[19, 7, 20, 6], X[9, 19, 10, 18], X[17, 9, 18, 8], X[13, 10, 14, 11], X[11, 2, 12, 3]]
In[4]:=	GaussCode[Knot[10, 32]]
Out[4]=	GaussCode[-1, 10, -2, 1, -3, 6, -5, 8, -7, 9, -10, 2, -9, 3, -4, 5, -8, 7, -6, 4]
In[5]:=	BR[Knot[10, 32]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, -2, -2, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[10, 32]][t]
Out[6]=	2 8 15 2 3 19 - -- + -- - -- - 15 t + 8 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 32]][z]
Out[7]=	2 4 6 1 - z - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 32]}
In[9]:=	{KnotDet[Knot[10, 32]], KnotSignature[Knot[10, 32]]}
Out[9]=	{69, 0}
In[10]:=	J=Jones[Knot[10, 32]][q]
Out[10]=	-6 3 5 8 11 11 2 3 4 11 + q - -- + -- - -- + -- - -- - 9 q + 6 q - 3 q + q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 32]}
In[12]:=	A2Invariant[Knot[10, 32]][q]
Out[12]=	-18 -16 2 3 -4 -2 2 4 6 8 10 -2 + q - q - --- + -- + q + q + 2 q - 2 q + q + q - q + 10 8 q q 12 q
In[13]:=	Kauffman[Knot[10, 32]][a, z]
Out[13]=	2 2 -2 2 z z 3 5 2 z 4 z -1 - a - a + -- + - - a z - 2 a z - a z + 7 z - -- + ---- + 3 a 4 2 a a a 3 4 4 6 2 3 z 3 3 3 5 3 4 z 6 z 2 a z - ---- + 7 a z + 13 a z + 9 a z - 11 z + -- - ---- + 3 4 2 a a a 5 5 2 4 4 4 6 4 3 z 4 z 5 3 5 2 a z + 3 a z - 3 a z + ---- - ---- - 15 a z - 18 a z - 3 a a 6 7 5 5 6 5 z 2 6 4 6 6 6 5 z 7 10 a z + 3 z + ---- - 10 a z - 7 a z + a z + ---- + 7 a z + 2 a a 3 7 5 7 8 2 8 4 8 9 3 9 5 a z + 3 a z + 3 z + 6 a z + 3 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 32]], Vassiliev[3][Knot[10, 32]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 32]][q, t]
Out[15]=	6 1 2 1 3 2 5 3 - + 6 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 6 5 5 6 3 3 2 5 2 ----- + ----- + ---- + --- + 4 q t + 5 q t + 2 q t + 4 q t + 5 2 3 2 3 q t q t q t q t 5 3 7 3 9 4 q t + 2 q t + q t