10 114

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+10 t^2-21 t+27-21 t^{-1} +10 t^{-2} -2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^6-2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 93, 0 }
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+8 q^2-12 q+15-15 q^{-1} +15 q^{-2} -11 q^{-3} +7 q^{-4} -4 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^6-z^6+a^4 z^4-2 a^2 z^4+z^4 a^{-2} -2 z^4+a^4 z^2+z^2 a^{-2} -z^2-a^4+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 a^3 z^9+3 a z^9+6 a^4 z^8+14 a^2 z^8+8 z^8+4 a^5 z^7+2 a^3 z^7+8 a z^7+10 z^7 a^{-1} +a^6 z^6-17 a^4 z^6-35 a^2 z^6+8 z^6 a^{-2} -9 z^6-11 a^5 z^5-21 a^3 z^5-27 a z^5-13 z^5 a^{-1} +4 z^5 a^{-3} -2 a^6 z^4+14 a^4 z^4+26 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +z^4+7 a^5 z^3+18 a^3 z^3+18 a z^3+5 z^3 a^{-1} -2 z^3 a^{-3} -3 a^4 z^2-5 a^2 z^2+2 z^2 a^{-2} -2 a^3 z-3 a z-z a^{-1} -a^4-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{18}-2 q^{16}-3 q^{10}+4 q^8+2 q^4+2 q^2-2+3 q^{-2} -3 q^{-4} + q^{-6} + q^{-8} -2 q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-3 q^{92}+7 q^{90}-14 q^{88}+17 q^{86}-18 q^{84}+7 q^{82}+23 q^{80}-59 q^{78}+102 q^{76}-118 q^{74}+87 q^{72}-8 q^{70}-116 q^{68}+233 q^{66}-288 q^{64}+240 q^{62}-90 q^{60}-114 q^{58}+292 q^{56}-369 q^{54}+304 q^{52}-124 q^{50}-102 q^{48}+262 q^{46}-301 q^{44}+192 q^{42}+8 q^{40}-188 q^{38}+283 q^{36}-240 q^{34}+79 q^{32}+135 q^{30}-323 q^{28}+407 q^{26}-345 q^{24}+160 q^{22}+105 q^{20}-340 q^{18}+471 q^{16}-440 q^{14}+265 q^{12}-9 q^{10}-238 q^8+371 q^6-346 q^4+186 q^2+42-216 q^{-2} +264 q^{-4} -170 q^{-6} -16 q^{-8} +194 q^{-10} -288 q^{-12} +259 q^{-14} -123 q^{-16} -57 q^{-18} +212 q^{-20} -287 q^{-22} +265 q^{-24} -164 q^{-26} +35 q^{-28} +77 q^{-30} -152 q^{-32} +168 q^{-34} -142 q^{-36} +92 q^{-38} -29 q^{-40} -22 q^{-42} +51 q^{-44} -63 q^{-46} +54 q^{-48} -34 q^{-50} +16 q^{-52} + q^{-54} -8 q^{-56} +10 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_114.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-3 q^{11}+3 q^9-4 q^7+4 q^5+3 q^{-1} -4 q^{-3} +4 q^{-5} -3 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{38}-3 q^{36}-q^{34}+12 q^{32}-8 q^{30}-17 q^{28}+27 q^{26}+4 q^{24}-39 q^{22}+24 q^{20}+24 q^{18}-43 q^{16}+6 q^{14}+33 q^{12}-23 q^{10}-14 q^8+24 q^6+9 q^4-26 q^2+38 q^{-2} -24 q^{-4} -26 q^{-6} +43 q^{-8} -8 q^{-10} -30 q^{-12} +25 q^{-14} +2 q^{-16} -14 q^{-18} +8 q^{-20} + q^{-22} -3 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{75}-3 q^{73}-q^{71}+8 q^{69}+7 q^{67}-16 q^{65}-30 q^{63}+23 q^{61}+66 q^{59}-2 q^{57}-112 q^{55}-54 q^{53}+140 q^{51}+142 q^{49}-124 q^{47}-235 q^{45}+52 q^{43}+303 q^{41}+54 q^{39}-318 q^{37}-163 q^{35}+279 q^{33}+257 q^{31}-217 q^{29}-302 q^{27}+125 q^{25}+321 q^{23}-50 q^{21}-307 q^{19}-25 q^{17}+276 q^{15}+96 q^{13}-226 q^{11}-167 q^9+162 q^7+241 q^5-70 q^3-291 q-46 q^{-1} +317 q^{-3} +160 q^{-5} -286 q^{-7} -261 q^{-9} +216 q^{-11} +304 q^{-13} -116 q^{-15} -291 q^{-17} +27 q^{-19} +230 q^{-21} +25 q^{-23} -151 q^{-25} -37 q^{-27} +81 q^{-29} +30 q^{-31} -41 q^{-33} -15 q^{-35} +23 q^{-37} +2 q^{-39} -9 q^{-41} -2 q^{-43} +5 q^{-45} + q^{-47} -3 q^{-49} + q^{-51} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{18}-2 q^{16}-3 q^{10}+4 q^8+2 q^4+2 q^2-2+3 q^{-2} -3 q^{-4} + q^{-6} + q^{-8} -2 q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{52}-6 q^{50}+20 q^{48}-52 q^{46}+119 q^{44}-240 q^{42}+424 q^{40}-670 q^{38}+971 q^{36}-1274 q^{34}+1510 q^{32}-1624 q^{30}+1560 q^{28}-1284 q^{26}+786 q^{24}-128 q^{22}-620 q^{20}+1380 q^{18}-2070 q^{16}+2604 q^{14}-2922 q^{12}+2992 q^{10}-2792 q^8+2358 q^6-1732 q^4+1018 q^2-276-380 q^{-2} +881 q^{-4} -1196 q^{-6} +1332 q^{-8} -1326 q^{-10} +1202 q^{-12} -1022 q^{-14} +830 q^{-16} -640 q^{-18} +468 q^{-20} -328 q^{-22} +220 q^{-24} -134 q^{-26} +73 q^{-28} -38 q^{-30} +18 q^{-32} -6 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{48}-2 q^{46}-2 q^{44}+5 q^{42}+3 q^{40}-4 q^{38}-7 q^{36}+7 q^{34}+12 q^{32}-13 q^{30}-12 q^{28}+10 q^{26}+9 q^{24}-12 q^{22}-12 q^{20}+16 q^{18}+8 q^{16}-14 q^{14}+11 q^{10}-6 q^8-q^6+14 q^4-2 q^2-7+8 q^{-2} +13 q^{-4} -17 q^{-6} -15 q^{-8} +19 q^{-10} +6 q^{-12} -18 q^{-14} -2 q^{-16} +14 q^{-18} +3 q^{-20} -9 q^{-22} -2 q^{-24} +6 q^{-26} - q^{-28} -2 q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{40}-3 q^{38}+q^{36}+5 q^{34}-11 q^{32}+10 q^{30}+9 q^{28}-24 q^{26}+17 q^{24}+7 q^{22}-34 q^{20}+17 q^{18}+12 q^{16}-26 q^{14}+13 q^{12}+17 q^{10}-8 q^8-3 q^6+5 q^4+12 q^2-15-9 q^{-2} +31 q^{-4} -19 q^{-6} -16 q^{-8} +33 q^{-10} -14 q^{-12} -16 q^{-14} +22 q^{-16} -5 q^{-18} -10 q^{-20} +9 q^{-22} -3 q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ q^{23}-2 q^{21}+q^{19}-3 q^{17}+q^{15}-3 q^{13}+4 q^{11}+3 q^7+q^5+q^3+q-2 q^{-1} +3 q^{-3} -3 q^{-5} +2 q^{-7} -2 q^{-9} +2 q^{-11} -2 q^{-13} + q^{-15} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{50}-2 q^{48}-2 q^{46}+6 q^{44}-3 q^{42}-8 q^{40}+12 q^{38}+9 q^{36}-15 q^{34}-2 q^{32}+19 q^{30}-7 q^{28}-31 q^{26}+3 q^{24}+23 q^{22}-21 q^{20}-16 q^{18}+36 q^{16}+12 q^{14}-21 q^{12}+15 q^{10}+20 q^8-19 q^6-12 q^4+20 q^2+1-25 q^{-2} +10 q^{-4} +26 q^{-6} -18 q^{-8} -15 q^{-10} +20 q^{-12} +6 q^{-14} -18 q^{-16} -3 q^{-18} +12 q^{-20} +2 q^{-22} -8 q^{-24} +6 q^{-28} -2 q^{-30} -2 q^{-32} + q^{-34} }[/math]
1,0,0,0	[math]\displaystyle{ q^{28}-2 q^{26}+q^{24}-2 q^{22}-2 q^{20}+q^{18}-3 q^{16}+4 q^{14}+3 q^{10}+2 q^8+q^6+q^4+1-2 q^{-2} +3 q^{-4} -3 q^{-6} +2 q^{-8} - q^{-10} - q^{-12} +2 q^{-14} -2 q^{-16} + q^{-18} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{40}-3 q^{38}+7 q^{36}-13 q^{34}+21 q^{32}-30 q^{30}+35 q^{28}-40 q^{26}+39 q^{24}-35 q^{22}+24 q^{20}-9 q^{18}-10 q^{16}+32 q^{14}-49 q^{12}+67 q^{10}-74 q^8+79 q^6-71 q^4+60 q^2-43+23 q^{-2} -3 q^{-4} -15 q^{-6} +28 q^{-8} -37 q^{-10} +40 q^{-12} -38 q^{-14} +32 q^{-16} -25 q^{-18} +18 q^{-20} -11 q^{-22} +6 q^{-24} -3 q^{-26} + q^{-28} }[/math]
1,0	[math]\displaystyle{ q^{66}-3 q^{62}-3 q^{60}+4 q^{58}+9 q^{56}-2 q^{54}-16 q^{52}-6 q^{50}+23 q^{48}+21 q^{46}-17 q^{44}-35 q^{42}+q^{40}+40 q^{38}+18 q^{36}-36 q^{34}-37 q^{32}+15 q^{30}+40 q^{28}+2 q^{26}-36 q^{24}-13 q^{22}+27 q^{20}+22 q^{18}-17 q^{16}-19 q^{14}+15 q^{12}+26 q^{10}-9 q^8-28 q^6+3 q^4+32 q^2+6-33 q^{-2} -18 q^{-4} +30 q^{-6} +30 q^{-8} -20 q^{-10} -40 q^{-12} +2 q^{-14} +40 q^{-16} +17 q^{-18} -27 q^{-20} -29 q^{-22} +8 q^{-24} +27 q^{-26} +8 q^{-28} -15 q^{-30} -14 q^{-32} +3 q^{-34} +10 q^{-36} +3 q^{-38} -3 q^{-40} -3 q^{-42} + q^{-46} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{54}-3 q^{52}+4 q^{50}-7 q^{48}+12 q^{46}-17 q^{44}+22 q^{42}-25 q^{40}+31 q^{38}-32 q^{36}+30 q^{34}-31 q^{32}+26 q^{30}-23 q^{28}+7 q^{26}-2 q^{24}-8 q^{22}+22 q^{20}-33 q^{18}+46 q^{16}-45 q^{14}+60 q^{12}-60 q^{10}+59 q^8-54 q^6+53 q^4-43 q^2+29-21 q^{-2} +13 q^{-4} +4 q^{-6} -14 q^{-8} +17 q^{-10} -25 q^{-12} +33 q^{-14} -31 q^{-16} +26 q^{-18} -29 q^{-20} +27 q^{-22} -18 q^{-24} +14 q^{-26} -14 q^{-28} +10 q^{-30} -4 q^{-32} +3 q^{-34} -3 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{94}-3 q^{92}+7 q^{90}-14 q^{88}+17 q^{86}-18 q^{84}+7 q^{82}+23 q^{80}-59 q^{78}+102 q^{76}-118 q^{74}+87 q^{72}-8 q^{70}-116 q^{68}+233 q^{66}-288 q^{64}+240 q^{62}-90 q^{60}-114 q^{58}+292 q^{56}-369 q^{54}+304 q^{52}-124 q^{50}-102 q^{48}+262 q^{46}-301 q^{44}+192 q^{42}+8 q^{40}-188 q^{38}+283 q^{36}-240 q^{34}+79 q^{32}+135 q^{30}-323 q^{28}+407 q^{26}-345 q^{24}+160 q^{22}+105 q^{20}-340 q^{18}+471 q^{16}-440 q^{14}+265 q^{12}-9 q^{10}-238 q^8+371 q^6-346 q^4+186 q^2+42-216 q^{-2} +264 q^{-4} -170 q^{-6} -16 q^{-8} +194 q^{-10} -288 q^{-12} +259 q^{-14} -123 q^{-16} -57 q^{-18} +212 q^{-20} -287 q^{-22} +265 q^{-24} -164 q^{-26} +35 q^{-28} +77 q^{-30} -152 q^{-32} +168 q^{-34} -142 q^{-36} +92 q^{-38} -29 q^{-40} -22 q^{-42} +51 q^{-44} -63 q^{-46} +54 q^{-48} -34 q^{-50} +16 q^{-52} + q^{-54} -8 q^{-56} +10 q^{-58} -10 q^{-60} +6 q^{-62} -3 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["10 114"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^3+10 t^2-21 t+27-21 t^{-1} +10 t^{-2} -2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 93, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-4 q^3+8 q^2-12 q+15-15 q^{-1} +15 q^{-2} -11 q^{-3} +7 q^{-4} -4 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^4-2 a^2 z^4+z^4 a^{-2} -2 z^4+a^4 z^2+z^2 a^{-2} -z^2-a^4+2 a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 3 a^3 z^9+3 a z^9+6 a^4 z^8+14 a^2 z^8+8 z^8+4 a^5 z^7+2 a^3 z^7+8 a z^7+10 z^7 a^{-1} +a^6 z^6-17 a^4 z^6-35 a^2 z^6+8 z^6 a^{-2} -9 z^6-11 a^5 z^5-21 a^3 z^5-27 a z^5-13 z^5 a^{-1} +4 z^5 a^{-3} -2 a^6 z^4+14 a^4 z^4+26 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +z^4+7 a^5 z^3+18 a^3 z^3+18 a z^3+5 z^3 a^{-1} -2 z^3 a^{-3} -3 a^4 z^2-5 a^2 z^2+2 z^2 a^{-2} -2 a^3 z-3 a z-z a^{-1} -a^4-2 a^2 }[/math]