10 15

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]

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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 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]