10 15

10_14

10_16

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10 15 Quick Notes

10 15 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_3,12,4,13 X_9,14,10,15 X_13,10,14,11 X_15,1,16,20 X_5,17,6,16 X_7,19,8,18 X_17,7,18,6 X_19,9,20,8 X_11,2,12,3
Gauss code	-1, 10, -2, 1, -6, 8, -7, 9, -3, 4, -10, 2, -4, 3, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code	4 12 16 18 14 2 10 20 6 8
Conway Notation	[4132]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-4][-8]
Hyperbolic Volume	8.97345
A-Polynomial	See Data:10 15/A-polynomial

[edit Notes for 10 15'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	2

[edit Notes for 10 15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-6 t^2+9 t-9+9 t^{-1} -6 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+6 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 43, 2 }
Jones polynomial	[math]\displaystyle{ -q^6+2 q^5-4 q^4+6 q^3-6 q^2+7 q-6+5 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +z^6-a^2 z^4+4 z^4 a^{-2} -z^4 a^{-4} +4 z^4-3 a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} +4 z^2-a^2+3 a^{-2} -2 a^{-4} +1 }[/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+a^3 z^7-2 a z^7+3 z^7 a^{-3} -10 a^2 z^6-z^6 a^{-2} +4 z^6 a^{-4} -15 z^6-5 a^3 z^5-4 a z^5-5 z^5 a^{-1} -3 z^5 a^{-3} +3 z^5 a^{-5} +15 a^2 z^4-8 z^4 a^{-2} -7 z^4 a^{-4} +2 z^4 a^{-6} +16 z^4+7 a^3 z^3+8 a z^3+z^3 a^{-1} -3 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} -7 a^2 z^2+8 z^2 a^{-2} +7 z^2 a^{-4} -z^2 a^{-6} -7 z^2-2 a^3 z-3 a z+3 z a^{-3} +z a^{-5} -z a^{-7} +a^2-3 a^{-2} -2 a^{-4} +1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{12}+q^4-q^2+1+ q^{-2} + q^{-4} +3 q^{-6} + q^{-10} - q^{-12} - q^{-14} - q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{60}-q^{58}+3 q^{56}-5 q^{54}+4 q^{52}-3 q^{50}-2 q^{48}+9 q^{46}-15 q^{44}+17 q^{42}-14 q^{40}+q^{38}+10 q^{36}-22 q^{34}+25 q^{32}-20 q^{30}+8 q^{28}+7 q^{26}-17 q^{24}+21 q^{22}-16 q^{20}+5 q^{18}+6 q^{16}-11 q^{14}+10 q^{12}-5 q^{10}-2 q^8+12 q^6-14 q^4+15 q^2-8-7 q^{-2} +18 q^{-4} -26 q^{-6} +27 q^{-8} -16 q^{-10} +3 q^{-12} +15 q^{-14} -23 q^{-16} +29 q^{-18} -19 q^{-20} +6 q^{-22} +8 q^{-24} -13 q^{-26} +15 q^{-28} -6 q^{-30} + q^{-32} +7 q^{-34} -6 q^{-36} +5 q^{-38} -7 q^{-42} +9 q^{-44} -9 q^{-46} +7 q^{-48} -2 q^{-50} -4 q^{-52} +7 q^{-54} -12 q^{-56} +14 q^{-58} -13 q^{-60} +5 q^{-62} -9 q^{-66} +11 q^{-68} -13 q^{-70} +11 q^{-72} -6 q^{-74} + q^{-76} +3 q^{-78} -7 q^{-80} +6 q^{-82} -5 q^{-84} +4 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 10_15.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^9+q^7-q^5+2 q^3-q+ q^{-1} + q^{-3} +2 q^{-7} -2 q^{-9} + q^{-11} - q^{-13} }[/math]
2	[math]\displaystyle{ q^{28}-q^{26}-2 q^{24}+3 q^{22}+q^{20}-5 q^{18}+2 q^{16}+4 q^{14}-6 q^{12}+5 q^8-4 q^6-q^4+6 q^2-3 q^{-2} +3 q^{-4} +3 q^{-6} -3 q^{-8} -2 q^{-10} +4 q^{-12} + q^{-14} -5 q^{-16} +4 q^{-18} + q^{-20} -5 q^{-22} +4 q^{-24} -4 q^{-28} +2 q^{-30} - q^{-34} + q^{-36} }[/math]
3	[math]\displaystyle{ -q^{57}+q^{55}+2 q^{53}-4 q^{49}-3 q^{47}+5 q^{45}+7 q^{43}-3 q^{41}-10 q^{39}-2 q^{37}+12 q^{35}+8 q^{33}-11 q^{31}-12 q^{29}+5 q^{27}+15 q^{25}-q^{23}-15 q^{21}-6 q^{19}+11 q^{17}+8 q^{15}-6 q^{13}-9 q^{11}+5 q^9+11 q^7-11 q^3+12 q^{-1} +4 q^{-3} -13 q^{-5} -6 q^{-7} +13 q^{-9} +10 q^{-11} -8 q^{-13} -11 q^{-15} + q^{-17} +11 q^{-19} +5 q^{-21} -8 q^{-23} -10 q^{-25} +2 q^{-27} +13 q^{-29} -8 q^{-33} -4 q^{-35} +7 q^{-37} + q^{-39} -2 q^{-41} -3 q^{-47} +3 q^{-51} -3 q^{-55} +2 q^{-59} + q^{-67} - q^{-69} }[/math]
4	[math]\displaystyle{ q^{96}-q^{94}-2 q^{92}+q^{88}+6 q^{86}+q^{84}-5 q^{82}-7 q^{80}-8 q^{78}+11 q^{76}+14 q^{74}+6 q^{72}-7 q^{70}-27 q^{68}-7 q^{66}+12 q^{64}+28 q^{62}+21 q^{60}-25 q^{58}-31 q^{56}-20 q^{54}+20 q^{52}+48 q^{50}+12 q^{48}-18 q^{46}-49 q^{44}-21 q^{42}+34 q^{40}+36 q^{38}+24 q^{36}-33 q^{34}-48 q^{32}-9 q^{30}+20 q^{28}+53 q^{26}+11 q^{24}-34 q^{22}-37 q^{20}-18 q^{18}+44 q^{16}+40 q^{14}-4 q^{12}-37 q^{10}-37 q^8+23 q^6+43 q^4+11 q^2-31-35 q^{-2} +18 q^{-4} +44 q^{-6} +11 q^{-8} -38 q^{-10} -36 q^{-12} +12 q^{-14} +48 q^{-16} +25 q^{-18} -31 q^{-20} -46 q^{-22} -19 q^{-24} +34 q^{-26} +47 q^{-28} +11 q^{-30} -24 q^{-32} -54 q^{-34} -19 q^{-36} +34 q^{-38} +54 q^{-40} +29 q^{-42} -51 q^{-44} -59 q^{-46} -7 q^{-48} +48 q^{-50} +64 q^{-52} -17 q^{-54} -53 q^{-56} -32 q^{-58} +13 q^{-60} +55 q^{-62} +6 q^{-64} -21 q^{-66} -25 q^{-68} -10 q^{-70} +27 q^{-72} +6 q^{-74} + q^{-76} -7 q^{-78} -12 q^{-80} +7 q^{-82} - q^{-84} +6 q^{-86} +2 q^{-88} -6 q^{-90} +3 q^{-92} -3 q^{-94} +2 q^{-96} +2 q^{-98} -2 q^{-100} +2 q^{-102} -2 q^{-104} - q^{-110} + q^{-112} }[/math]
5	[math]\displaystyle{ -q^{145}+q^{143}+2 q^{141}-q^{137}-3 q^{135}-4 q^{133}-q^{131}+7 q^{129}+9 q^{127}+5 q^{125}-2 q^{123}-14 q^{121}-19 q^{119}-8 q^{117}+14 q^{115}+27 q^{113}+26 q^{111}+6 q^{109}-28 q^{107}-46 q^{105}-34 q^{103}+8 q^{101}+49 q^{99}+64 q^{97}+35 q^{95}-30 q^{93}-80 q^{91}-78 q^{89}-17 q^{87}+62 q^{85}+106 q^{83}+78 q^{81}-13 q^{79}-101 q^{77}-123 q^{75}-52 q^{73}+55 q^{71}+130 q^{69}+119 q^{67}+17 q^{65}-99 q^{63}-146 q^{61}-90 q^{59}+24 q^{57}+129 q^{55}+149 q^{53}+56 q^{51}-76 q^{49}-158 q^{47}-133 q^{45}-11 q^{43}+131 q^{41}+182 q^{39}+93 q^{37}-70 q^{35}-194 q^{33}-166 q^{31}-7 q^{29}+169 q^{27}+215 q^{25}+81 q^{23}-125 q^{21}-224 q^{19}-136 q^{17}+67 q^{15}+215 q^{13}+174 q^{11}-22 q^9-182 q^7-174 q^5-14 q^3+150 q+163 q^{-1} +21 q^{-3} -124 q^{-5} -137 q^{-7} -14 q^{-9} +117 q^{-11} +122 q^{-13} - q^{-15} -129 q^{-17} -122 q^{-19} +11 q^{-21} +144 q^{-23} +143 q^{-25} +6 q^{-27} -146 q^{-29} -172 q^{-31} -52 q^{-33} +114 q^{-35} +190 q^{-37} +115 q^{-39} -41 q^{-41} -172 q^{-43} -187 q^{-45} -62 q^{-47} +116 q^{-49} +226 q^{-51} +168 q^{-53} -21 q^{-55} -213 q^{-57} -260 q^{-59} -88 q^{-61} +173 q^{-63} +297 q^{-65} +175 q^{-67} -88 q^{-69} -288 q^{-71} -245 q^{-73} +20 q^{-75} +249 q^{-77} +253 q^{-79} +45 q^{-81} -188 q^{-83} -244 q^{-85} -79 q^{-87} +133 q^{-89} +201 q^{-91} +92 q^{-93} -85 q^{-95} -157 q^{-97} -81 q^{-99} +43 q^{-101} +111 q^{-103} +70 q^{-105} -20 q^{-107} -71 q^{-109} -50 q^{-111} +2 q^{-113} +38 q^{-115} +37 q^{-117} +7 q^{-119} -19 q^{-121} -23 q^{-123} -7 q^{-125} +6 q^{-127} +12 q^{-129} +8 q^{-131} -7 q^{-135} -4 q^{-137} +2 q^{-143} +3 q^{-145} -2 q^{-147} - q^{-149} -2 q^{-153} +2 q^{-157} + q^{-163} - q^{-165} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{12}+q^4-q^2+1+ q^{-2} + q^{-4} +3 q^{-6} + q^{-10} - q^{-12} - q^{-14} - q^{-18} }[/math]
1,1	[math]\displaystyle{ q^{36}-2 q^{34}+6 q^{32}-14 q^{30}+23 q^{28}-36 q^{26}+52 q^{24}-64 q^{22}+69 q^{20}-70 q^{18}+62 q^{16}-46 q^{14}+20 q^{12}+4 q^{10}-32 q^8+54 q^6-79 q^4+98 q^2-102+112 q^{-2} -97 q^{-4} +92 q^{-6} -68 q^{-8} +50 q^{-10} -27 q^{-12} +10 q^{-14} +2 q^{-16} -12 q^{-18} +20 q^{-20} -28 q^{-22} +28 q^{-24} -34 q^{-26} +36 q^{-28} -38 q^{-30} +34 q^{-32} -30 q^{-34} +26 q^{-36} -22 q^{-38} +16 q^{-40} -12 q^{-42} +9 q^{-44} -6 q^{-46} +4 q^{-48} -2 q^{-50} + q^{-52} }[/math]
2,0	[math]\displaystyle{ q^{34}-q^{30}-q^{28}+q^{26}+q^{24}-2 q^{22}-q^{20}+2 q^{18}+2 q^{16}-3 q^{14}-3 q^{12}-2 q^6+4 q^2+1+2 q^{-2} +4 q^{-4} + q^{-6} +2 q^{-10} +4 q^{-12} -2 q^{-14} +4 q^{-18} +2 q^{-20} -5 q^{-22} -2 q^{-24} + q^{-26} -2 q^{-28} -2 q^{-30} - q^{-32} - q^{-36} + q^{-40} + q^{-46} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{26}-q^{24}+q^{22}-3 q^{18}+2 q^{16}-q^{14}-3 q^{12}+2 q^{10}-q^8-4 q^6+4 q^4-3+6 q^{-2} +3 q^{-4} +4 q^{-8} +5 q^{-10} +3 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} -6 q^{-20} -2 q^{-22} +3 q^{-24} -6 q^{-26} -2 q^{-28} +5 q^{-30} -3 q^{-32} -2 q^{-34} +3 q^{-36} - q^{-40} + q^{-42} }[/math]
1,0,0	[math]\displaystyle{ -q^{15}-q^{11}+q^9-q^7+q^5-q^3+q+2 q^{-3} +2 q^{-5} +2 q^{-7} +3 q^{-9} + q^{-13} -2 q^{-15} -2 q^{-19} - q^{-23} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight

Invariant

0,1

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

1,0

[math]\displaystyle{ q^{44}-q^{40}-q^{38}+2 q^{36}+2 q^{34}-3 q^{32}-4 q^{30}+q^{28}+5 q^{26}+q^{24}-6 q^{22}-4 q^{20}+4 q^{18}+6 q^{16}-q^{14}-8 q^{12}-3 q^{10}+5 q^8+6 q^6-2 q^4-6 q^2+6 q^{-2} +3 q^{-4} -2 q^{-6} - q^{-8} +5 q^{-10} +3 q^{-12} -2 q^{-14} -2 q^{-16} +5 q^{-18} +5 q^{-20} - q^{-22} -5 q^{-24} +5 q^{-28} +2 q^{-30} -5 q^{-32} -5 q^{-34} + q^{-36} +5 q^{-38} -6 q^{-42} -5 q^{-44} +2 q^{-46} +6 q^{-48} -4 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^{34}-q^{32}+2 q^{30}-3 q^{28}+4 q^{26}-5 q^{24}+4 q^{22}-6 q^{20}+5 q^{18}-7 q^{16}+3 q^{14}-5 q^{12}+3 q^{10}-2 q^8-q^6+2 q^4-2 q^2+7-6 q^{-2} +10 q^{-4} -7 q^{-6} +13 q^{-8} -7 q^{-10} +14 q^{-12} -5 q^{-14} +11 q^{-16} -4 q^{-18} +6 q^{-20} -3 q^{-22} -2 q^{-26} -5 q^{-28} -5 q^{-32} +3 q^{-34} -6 q^{-36} +3 q^{-38} -5 q^{-40} +6 q^{-42} -4 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^{60}-q^{58}+3 q^{56}-5 q^{54}+4 q^{52}-3 q^{50}-2 q^{48}+9 q^{46}-15 q^{44}+17 q^{42}-14 q^{40}+q^{38}+10 q^{36}-22 q^{34}+25 q^{32}-20 q^{30}+8 q^{28}+7 q^{26}-17 q^{24}+21 q^{22}-16 q^{20}+5 q^{18}+6 q^{16}-11 q^{14}+10 q^{12}-5 q^{10}-2 q^8+12 q^6-14 q^4+15 q^2-8-7 q^{-2} +18 q^{-4} -26 q^{-6} +27 q^{-8} -16 q^{-10} +3 q^{-12} +15 q^{-14} -23 q^{-16} +29 q^{-18} -19 q^{-20} +6 q^{-22} +8 q^{-24} -13 q^{-26} +15 q^{-28} -6 q^{-30} + q^{-32} +7 q^{-34} -6 q^{-36} +5 q^{-38} -7 q^{-42} +9 q^{-44} -9 q^{-46} +7 q^{-48} -2 q^{-50} -4 q^{-52} +7 q^{-54} -12 q^{-56} +14 q^{-58} -13 q^{-60} +5 q^{-62} -9 q^{-66} +11 q^{-68} -13 q^{-70} +11 q^{-72} -6 q^{-74} + q^{-76} +3 q^{-78} -7 q^{-80} +6 q^{-82} -5 q^{-84} +4 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.

In[3]:= K = Knot["10 15"];

In[4]:= Alexander[K][t]

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

Out[4]= [math]\displaystyle{ 2 t^3-6 t^2+9 t-9+9 t^{-1} -6 t^{-2} +2 t^{-3} }[/math]

In[5]:= Conway[K][z]

Out[5]= [math]\displaystyle{ 2 z^6+6 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]= { 43, 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+6 q^3-6 q^2+7 q-6+5 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} }[/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} +z^6-a^2 z^4+4 z^4 a^{-2} -z^4 a^{-4} +4 z^4-3 a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} +4 z^2-a^2+3 a^{-2} -2 a^{-4} +1 }[/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+a^3 z^7-2 a z^7+3 z^7 a^{-3} -10 a^2 z^6-z^6 a^{-2} +4 z^6 a^{-4} -15 z^6-5 a^3 z^5-4 a z^5-5 z^5 a^{-1} -3 z^5 a^{-3} +3 z^5 a^{-5} +15 a^2 z^4-8 z^4 a^{-2} -7 z^4 a^{-4} +2 z^4 a^{-6} +16 z^4+7 a^3 z^3+8 a z^3+z^3 a^{-1} -3 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} -7 a^2 z^2+8 z^2 a^{-2} +7 z^2 a^{-4} -z^2 a^{-6} -7 z^2-2 a^3 z-3 a z+3 z a^{-3} +z a^{-5} -z a^{-7} +a^2-3 a^{-2} -2 a^{-4} +1 }[/math]

Vassiliev invariants

V₂ and V₃:

(3, 2)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

[math]\displaystyle{ 12 }[/math]

[math]\displaystyle{ 16 }[/math]

[math]\displaystyle{ 72 }[/math]

[math]\displaystyle{ 78 }[/math]

[math]\displaystyle{ -14 }[/math]

[math]\displaystyle{ 192 }[/math]

[math]\displaystyle{ \frac{736}{3} }[/math]

[math]\displaystyle{ \frac{160}{3} }[/math]

[math]\displaystyle{ -16 }[/math]

[math]\displaystyle{ 288 }[/math]

[math]\displaystyle{ 128 }[/math]

[math]\displaystyle{ 936 }[/math]

[math]\displaystyle{ -168 }[/math]

[math]\displaystyle{ \frac{11711}{10} }[/math]

[math]\displaystyle{ \frac{7334}{15} }[/math]

[math]\displaystyle{ -\frac{4738}{15} }[/math]

[math]\displaystyle{ \frac{257}{6} }[/math]

[math]\displaystyle{ -\frac{1729}{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 10 15. 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

13

1

-1

11

1

9

3

1

-2

7

3

1

2

5

3

0

3

4

3

1

3

4

1

-1

2

3

-1

-3

1

3

2

-5

1

2

-1

-7

1

-9

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, 15]]
Out[2]=	10
In[3]:=	PD[Knot[10, 15]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[9, 14, 10, 15], X[13, 10, 14, 11], X[15, 1, 16, 20], X[5, 17, 6, 16], X[7, 19, 8, 18], X[17, 7, 18, 6], X[19, 9, 20, 8], X[11, 2, 12, 3]]
In[4]:=	GaussCode[Knot[10, 15]]
Out[4]=	GaussCode[-1, 10, -2, 1, -6, 8, -7, 9, -3, 4, -10, 2, -4, 3, -5, 6, -8, 7, -9, 5]
In[5]:=	BR[Knot[10, 15]]
Out[5]=	BR[4, {1, 1, 1, 1, -2, 1, -2, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[10, 15]][t]
Out[6]=	2 6 9 2 3 -9 + -- - -- + - + 9 t - 6 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 15]][z]
Out[7]=	2 4 6 1 + 3 z + 6 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 15]}
In[9]:=	{KnotDet[Knot[10, 15]], KnotSignature[Knot[10, 15]]}
Out[9]=	{43, 2}
In[10]:=	J=Jones[Knot[10, 15]][q]
Out[10]=	-4 2 3 5 2 3 4 5 6 -6 - q + -- - -- + - + 7 q - 6 q + 6 q - 4 q + 2 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 15]}
In[12]:=	A2Invariant[Knot[10, 15]][q]
Out[12]=	-12 -4 -2 2 4 6 10 12 14 18 1 - q + q - q + q + q + 3 q + q - q - q - q
In[13]:=	Kauffman[Knot[10, 15]][a, z]
Out[13]=	2 2 2 3 2 z z 3 z 3 2 z 7 z 1 - -- - -- + a - -- + -- + --- - 3 a z - 2 a z - 7 z - -- + ---- + 4 2 7 5 3 6 4 a a a a a a a 2 3 3 3 3 8 z 2 2 z 2 z 3 z z 3 3 3 4 ---- - 7 a z + -- - ---- - ---- + -- + 8 a z + 7 a z + 16 z + 2 7 5 3 a a a a a 4 4 4 5 5 5 2 z 7 z 8 z 2 4 3 z 3 z 5 z 5 ---- - ---- - ---- + 15 a z + ---- - ---- - ---- - 4 a z - 6 4 2 5 3 a a a a a a 6 6 7 3 5 6 4 z z 2 6 3 z 7 3 7 5 a z - 15 z + ---- - -- - 10 a z + ---- - 2 a z + a z + 4 2 3 a a a 8 9 8 2 z 2 8 z 9 4 z + ---- + 2 a z + -- + a z 2 a a
In[14]:=	{Vassiliev[2][Knot[10, 15]], Vassiliev[3][Knot[10, 15]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[10, 15]][q, t]
Out[15]=	3 1 1 1 2 1 3 2 4 q + 4 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t 3 3 q 3 5 5 2 7 2 7 3 9 3 --- + --- + 3 q t + 3 q t + 3 q t + 3 q t + q t + 3 q t + q t t 9 4 11 4 13 5 q t + q t + q t

10 15

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