10 102

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+8 t^2-16 t+21-16 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^6-4 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 73, 0 }
Jones polynomial	[math]\displaystyle{ q^6-3 q^5+6 q^4-9 q^3+11 q^2-12 q+12-9 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -3 z^2+a^2- a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+6 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +5 a^2 z^6-24 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -7 z^6+3 a^3 z^5-9 a z^5-17 z^5 a^{-1} -14 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-6 a^2 z^4+21 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} +3 z^4-3 a^3 z^3+7 a z^3+16 z^3 a^{-1} +13 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+3 a^2 z^2-8 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -2 z a^{-5} -a^2+ a^{-2} + a^{-4} }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}-q^{10}+q^8+q^6-2 q^4+3 q^2-1+ q^{-2} -2 q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+11 q^{52}-18 q^{50}+26 q^{48}-28 q^{46}+19 q^{44}-4 q^{42}-19 q^{40}+42 q^{38}-59 q^{36}+68 q^{34}-61 q^{32}+33 q^{30}+15 q^{28}-64 q^{26}+110 q^{24}-123 q^{22}+100 q^{20}-42 q^{18}-40 q^{16}+108 q^{14}-134 q^{12}+108 q^{10}-29 q^8-57 q^6+110 q^4-105 q^2+39+57 q^{-2} -141 q^{-4} +164 q^{-6} -116 q^{-8} +14 q^{-10} +107 q^{-12} -193 q^{-14} +216 q^{-16} -165 q^{-18} +60 q^{-20} +55 q^{-22} -151 q^{-24} +192 q^{-26} -166 q^{-28} +88 q^{-30} +14 q^{-32} -100 q^{-34} +135 q^{-36} -108 q^{-38} +29 q^{-40} +61 q^{-42} -125 q^{-44} +126 q^{-46} -65 q^{-48} -34 q^{-50} +131 q^{-52} -171 q^{-54} +148 q^{-56} -68 q^{-58} -33 q^{-60} +111 q^{-62} -142 q^{-64} +125 q^{-66} -69 q^{-68} +6 q^{-70} +42 q^{-72} -63 q^{-74} +57 q^{-76} -36 q^{-78} +16 q^{-80} + q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_102.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-2 q^7+3 q^5-3 q^3+3 q- q^{-3} +2 q^{-5} -3 q^{-7} +3 q^{-9} -2 q^{-11} + q^{-13} }[/math]
2	[math]\displaystyle{ q^{26}-2 q^{24}+5 q^{20}-7 q^{18}+q^{16}+13 q^{14}-17 q^{12}-3 q^{10}+25 q^8-17 q^6-13 q^4+24 q^2-2-15 q^{-2} +6 q^{-4} +13 q^{-6} -9 q^{-8} -12 q^{-10} +20 q^{-12} + q^{-14} -24 q^{-16} +16 q^{-18} +13 q^{-20} -24 q^{-22} +4 q^{-24} +17 q^{-26} -12 q^{-28} -5 q^{-30} +8 q^{-32} - q^{-34} -2 q^{-36} + q^{-38} }[/math]
3	[math]\displaystyle{ q^{51}-2 q^{49}+2 q^{45}+q^{43}-3 q^{41}-q^{39}+6 q^{37}-5 q^{35}-10 q^{33}+14 q^{31}+24 q^{29}-23 q^{27}-53 q^{25}+28 q^{23}+91 q^{21}-13 q^{19}-129 q^{17}-24 q^{15}+148 q^{13}+70 q^{11}-140 q^9-107 q^7+99 q^5+134 q^3-46 q-131 q^{-1} -6 q^{-3} +111 q^{-5} +57 q^{-7} -87 q^{-9} -90 q^{-11} +58 q^{-13} +118 q^{-15} -35 q^{-17} -136 q^{-19} +5 q^{-21} +148 q^{-23} +29 q^{-25} -150 q^{-27} -67 q^{-29} +135 q^{-31} +105 q^{-33} -98 q^{-35} -133 q^{-37} +47 q^{-39} +140 q^{-41} +4 q^{-43} -116 q^{-45} -47 q^{-47} +77 q^{-49} +65 q^{-51} -36 q^{-53} -56 q^{-55} +3 q^{-57} +36 q^{-59} +10 q^{-61} -17 q^{-63} -8 q^{-65} +5 q^{-67} +4 q^{-69} - q^{-71} -2 q^{-73} + q^{-75} }[/math]
4	[math]\displaystyle{ q^{84}-2 q^{82}+2 q^{78}-2 q^{76}+5 q^{74}-5 q^{72}-q^{70}+q^{68}-11 q^{66}+21 q^{64}+5 q^{62}+5 q^{60}-18 q^{58}-67 q^{56}+25 q^{54}+70 q^{52}+104 q^{50}-17 q^{48}-250 q^{46}-139 q^{44}+118 q^{42}+426 q^{40}+261 q^{38}-418 q^{36}-636 q^{34}-223 q^{32}+708 q^{30}+945 q^{28}-70 q^{26}-1017 q^{24}-1010 q^{22}+343 q^{20}+1386 q^{18}+739 q^{16}-626 q^{14}-1428 q^{12}-481 q^{10}+947 q^8+1150 q^6+222 q^4-987 q^2-928+87 q^{-2} +845 q^{-4} +724 q^{-6} -234 q^{-8} -813 q^{-10} -493 q^{-12} +369 q^{-14} +818 q^{-16} +262 q^{-18} -620 q^{-20} -779 q^{-22} +102 q^{-24} +874 q^{-26} +588 q^{-28} -530 q^{-30} -1064 q^{-32} -171 q^{-34} +923 q^{-36} +1002 q^{-38} -243 q^{-40} -1256 q^{-42} -685 q^{-44} +591 q^{-46} +1318 q^{-48} +423 q^{-50} -934 q^{-52} -1115 q^{-54} -211 q^{-56} +1022 q^{-58} +978 q^{-60} -69 q^{-62} -868 q^{-64} -845 q^{-66} +174 q^{-68} +789 q^{-70} +566 q^{-72} -105 q^{-74} -684 q^{-76} -390 q^{-78} +129 q^{-80} +441 q^{-82} +324 q^{-84} -144 q^{-86} -277 q^{-88} -183 q^{-90} +58 q^{-92} +199 q^{-94} +72 q^{-96} -25 q^{-98} -93 q^{-100} -47 q^{-102} +32 q^{-104} +26 q^{-106} +19 q^{-108} -11 q^{-110} -14 q^{-112} +2 q^{-114} + q^{-116} +4 q^{-118} - q^{-120} -2 q^{-122} + q^{-124} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}-q^{10}+q^8+q^6-2 q^4+3 q^2-1+ q^{-2} -2 q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }[/math]
1,1	[math]\displaystyle{ q^{36}-4 q^{34}+10 q^{32}-20 q^{30}+36 q^{28}-58 q^{26}+90 q^{24}-128 q^{22}+171 q^{20}-218 q^{18}+278 q^{16}-348 q^{14}+407 q^{12}-458 q^{10}+496 q^8-480 q^6+399 q^4-242 q^2+24+242 q^{-2} -534 q^{-4} +798 q^{-6} -1008 q^{-8} +1140 q^{-10} -1165 q^{-12} +1098 q^{-14} -934 q^{-16} +698 q^{-18} -408 q^{-20} +96 q^{-22} +188 q^{-24} -420 q^{-26} +583 q^{-28} -656 q^{-30} +638 q^{-32} -556 q^{-34} +445 q^{-36} -324 q^{-38} +208 q^{-40} -122 q^{-42} +66 q^{-44} -30 q^{-46} +12 q^{-48} -4 q^{-50} + q^{-52} }[/math]
2,0	[math]\displaystyle{ q^{32}-q^{30}-q^{28}+3 q^{26}-4 q^{22}+2 q^{20}+7 q^{18}-2 q^{16}-10 q^{14}+4 q^{12}+12 q^{10}-9 q^8-9 q^6+9 q^4+4 q^2-5-3 q^{-2} +8 q^{-4} -2 q^{-6} -4 q^{-8} +7 q^{-10} + q^{-12} -7 q^{-14} +4 q^{-16} +9 q^{-18} -8 q^{-20} -5 q^{-22} +6 q^{-24} +6 q^{-26} -7 q^{-28} -7 q^{-30} +8 q^{-32} +3 q^{-34} -5 q^{-36} -3 q^{-38} +2 q^{-40} +3 q^{-42} - q^{-44} - q^{-46} + q^{-48} }[/math]

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-7 q^{22}+10 q^{20}-14 q^{18}+19 q^{16}-21 q^{14}+24 q^{12}-22 q^{10}+18 q^8-10 q^6+12 q^2-24+34 q^{-2} -42 q^{-4} +46 q^{-6} -46 q^{-8} +41 q^{-10} -32 q^{-12} +22 q^{-14} -9 q^{-16} -3 q^{-18} +13 q^{-20} -20 q^{-22} +23 q^{-24} -24 q^{-26} +23 q^{-28} -19 q^{-30} +14 q^{-32} -9 q^{-34} +5 q^{-36} -2 q^{-38} + q^{-40} }[/math]
1,0	[math]\displaystyle{ q^{46}-2 q^{42}-2 q^{40}+2 q^{38}+6 q^{36}+q^{34}-8 q^{32}-8 q^{30}+5 q^{28}+16 q^{26}+4 q^{24}-16 q^{22}-16 q^{20}+9 q^{18}+24 q^{16}+3 q^{14}-23 q^{12}-13 q^{10}+17 q^8+20 q^6-9 q^4-20 q^2+3+20 q^{-2} +3 q^{-4} -17 q^{-6} -7 q^{-8} +13 q^{-10} +9 q^{-12} -12 q^{-14} -12 q^{-16} +10 q^{-18} +15 q^{-20} -7 q^{-22} -21 q^{-24} +24 q^{-28} +12 q^{-30} -20 q^{-32} -22 q^{-34} +11 q^{-36} +26 q^{-38} +3 q^{-40} -21 q^{-42} -12 q^{-44} +13 q^{-46} +14 q^{-48} -4 q^{-50} -11 q^{-52} -2 q^{-54} +6 q^{-56} +3 q^{-58} -2 q^{-60} -2 q^{-62} + q^{-66} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{38}-2 q^{36}+2 q^{34}-3 q^{32}+6 q^{30}-8 q^{28}+8 q^{26}-10 q^{24}+16 q^{22}-16 q^{20}+15 q^{18}-17 q^{16}+21 q^{14}-15 q^{12}+11 q^{10}-9 q^8+5 q^6+7 q^4-10 q^2+17-24 q^{-2} +31 q^{-4} -32 q^{-6} +33 q^{-8} -39 q^{-10} +33 q^{-12} -32 q^{-14} +26 q^{-16} -24 q^{-18} +14 q^{-20} -7 q^{-22} +3 q^{-24} +5 q^{-26} -9 q^{-28} +18 q^{-30} -16 q^{-32} +19 q^{-34} -19 q^{-36} +20 q^{-38} -17 q^{-40} +13 q^{-42} -12 q^{-44} +8 q^{-46} -5 q^{-48} +3 q^{-50} -2 q^{-52} + q^{-54} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+11 q^{52}-18 q^{50}+26 q^{48}-28 q^{46}+19 q^{44}-4 q^{42}-19 q^{40}+42 q^{38}-59 q^{36}+68 q^{34}-61 q^{32}+33 q^{30}+15 q^{28}-64 q^{26}+110 q^{24}-123 q^{22}+100 q^{20}-42 q^{18}-40 q^{16}+108 q^{14}-134 q^{12}+108 q^{10}-29 q^8-57 q^6+110 q^4-105 q^2+39+57 q^{-2} -141 q^{-4} +164 q^{-6} -116 q^{-8} +14 q^{-10} +107 q^{-12} -193 q^{-14} +216 q^{-16} -165 q^{-18} +60 q^{-20} +55 q^{-22} -151 q^{-24} +192 q^{-26} -166 q^{-28} +88 q^{-30} +14 q^{-32} -100 q^{-34} +135 q^{-36} -108 q^{-38} +29 q^{-40} +61 q^{-42} -125 q^{-44} +126 q^{-46} -65 q^{-48} -34 q^{-50} +131 q^{-52} -171 q^{-54} +148 q^{-56} -68 q^{-58} -33 q^{-60} +111 q^{-62} -142 q^{-64} +125 q^{-66} -69 q^{-68} +6 q^{-70} +42 q^{-72} -63 q^{-74} +57 q^{-76} -36 q^{-78} +16 q^{-80} + q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }[/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["10 102"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^3+8 t^2-16 t+21-16 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^6-4 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]=

{ 73, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-3 q^5+6 q^4-9 q^3+11 q^2-12 q+12-9 q^{-1} +6 q^{-2} -3 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^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -3 z^2+a^2- a^{-2} + a^{-4} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+6 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +5 a^2 z^6-24 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -7 z^6+3 a^3 z^5-9 a z^5-17 z^5 a^{-1} -14 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-6 a^2 z^4+21 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} +3 z^4-3 a^3 z^3+7 a z^3+16 z^3 a^{-1} +13 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+3 a^2 z^2-8 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -2 z a^{-5} -a^2+ a^{-2} + a^{-4} }[/math]