10 71

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+7 t^2-18 t+25-18 t^{-1} +7 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 77, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-12 q+13-12 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -3 z^4-a^4 z^2+4 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} -5 z^2-a^4+3 a^2+3 a^{-2} - a^{-4} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^9+z^9 a^{-1} +3 a^2 z^8+3 z^8 a^{-2} +6 z^8+4 a^3 z^7+8 a z^7+8 z^7 a^{-1} +4 z^7 a^{-3} +3 a^4 z^6+2 a^2 z^6+2 z^6 a^{-2} +3 z^6 a^{-4} -2 z^6+a^5 z^5-5 a^3 z^5-15 a z^5-15 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-12 a^2 z^4-12 z^4 a^{-2} -6 z^4 a^{-4} -12 z^4-2 a^5 z^3+7 a z^3+7 z^3 a^{-1} -2 z^3 a^{-5} +4 a^4 z^2+10 a^2 z^2+10 z^2 a^{-2} +4 z^2 a^{-4} +12 z^2+a^5 z+a^3 z-a z-z a^{-1} +z a^{-3} +z a^{-5} -a^4-3 a^2-3 a^{-2} - a^{-4} -3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{12}-2 q^{10}+3 q^8+q^6-q^4+2 q^2-3+2 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -2 q^{-10} + q^{-12} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-7 q^{70}-2 q^{68}+16 q^{66}-32 q^{64}+47 q^{62}-52 q^{60}+36 q^{58}-3 q^{56}-48 q^{54}+102 q^{52}-137 q^{50}+132 q^{48}-84 q^{46}-6 q^{44}+105 q^{42}-179 q^{40}+206 q^{38}-155 q^{36}+56 q^{34}+61 q^{32}-147 q^{30}+164 q^{28}-107 q^{26}+10 q^{24}+90 q^{22}-135 q^{20}+110 q^{18}-14 q^{16}-110 q^{14}+208 q^{12}-235 q^{10}+166 q^8-34 q^6-128 q^4+254 q^2-299+253 q^{-2} -128 q^{-4} -32 q^{-6} +166 q^{-8} -233 q^{-10} +206 q^{-12} -108 q^{-14} -12 q^{-16} +109 q^{-18} -135 q^{-20} +90 q^{-22} +10 q^{-24} -107 q^{-26} +164 q^{-28} -150 q^{-30} +63 q^{-32} +55 q^{-34} -156 q^{-36} +206 q^{-38} -180 q^{-40} +106 q^{-42} -6 q^{-44} -84 q^{-46} +132 q^{-48} -136 q^{-50} +102 q^{-52} -47 q^{-54} -4 q^{-56} +36 q^{-58} -51 q^{-60} +46 q^{-62} -32 q^{-64} +16 q^{-66} -2 q^{-68} -7 q^{-70} +8 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_71.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-3 q^7+4 q^5-2 q^3+q+ q^{-1} -2 q^{-3} +4 q^{-5} -3 q^{-7} +2 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+7 q^{26}-7 q^{24}-7 q^{22}+21 q^{20}-9 q^{18}-22 q^{16}+30 q^{14}-30 q^{10}+21 q^8+11 q^6-20 q^4+q^2+15+ q^{-2} -20 q^{-4} +11 q^{-6} +21 q^{-8} -30 q^{-10} +30 q^{-14} -22 q^{-16} -9 q^{-18} +21 q^{-20} -7 q^{-22} -7 q^{-24} +7 q^{-26} - q^{-28} -2 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+q^{59}-3 q^{57}-4 q^{55}+7 q^{53}+10 q^{51}-15 q^{49}-21 q^{47}+21 q^{45}+44 q^{43}-23 q^{41}-77 q^{39}+16 q^{37}+115 q^{35}+8 q^{33}-148 q^{31}-54 q^{29}+171 q^{27}+99 q^{25}-165 q^{23}-144 q^{21}+138 q^{19}+175 q^{17}-96 q^{15}-182 q^{13}+46 q^{11}+168 q^9+7 q^7-138 q^5-56 q^3+101 q+101 q^{-1} -56 q^{-3} -138 q^{-5} +7 q^{-7} +167 q^{-9} +46 q^{-11} -180 q^{-13} -95 q^{-15} +174 q^{-17} +137 q^{-19} -144 q^{-21} -166 q^{-23} +99 q^{-25} +172 q^{-27} -54 q^{-29} -150 q^{-31} +8 q^{-33} +119 q^{-35} +17 q^{-37} -79 q^{-39} -25 q^{-41} +44 q^{-43} +22 q^{-45} -21 q^{-47} -15 q^{-49} +10 q^{-51} +7 q^{-53} -4 q^{-55} -3 q^{-57} + q^{-59} +2 q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}-q^{100}+3 q^{98}+4 q^{94}-10 q^{92}-5 q^{90}+16 q^{88}+6 q^{86}+11 q^{84}-45 q^{82}-35 q^{80}+50 q^{78}+61 q^{76}+63 q^{74}-128 q^{72}-170 q^{70}+41 q^{68}+205 q^{66}+292 q^{64}-168 q^{62}-477 q^{60}-208 q^{58}+306 q^{56}+779 q^{54}+104 q^{52}-753 q^{50}-794 q^{48}+37 q^{46}+1242 q^{44}+765 q^{42}-599 q^{40}-1349 q^{38}-638 q^{36}+1194 q^{34}+1370 q^{32}+24 q^{30}-1377 q^{28}-1238 q^{26}+621 q^{24}+1436 q^{22}+642 q^{20}-871 q^{18}-1332 q^{16}-66 q^{14}+999 q^{12}+932 q^{10}-205 q^8-1027 q^6-598 q^4+403 q^2+976+409 q^{-2} -588 q^{-4} -1021 q^{-6} -209 q^{-8} +915 q^{-10} +983 q^{-12} -62 q^{-14} -1311 q^{-16} -851 q^{-18} +637 q^{-20} +1410 q^{-22} +603 q^{-24} -1234 q^{-26} -1355 q^{-28} +44 q^{-30} +1375 q^{-32} +1184 q^{-34} -661 q^{-36} -1371 q^{-38} -596 q^{-40} +798 q^{-42} +1277 q^{-44} +35 q^{-46} -841 q^{-48} -795 q^{-50} +113 q^{-52} +829 q^{-54} +344 q^{-56} -223 q^{-58} -520 q^{-60} -192 q^{-62} +306 q^{-64} +235 q^{-66} +53 q^{-68} -179 q^{-70} -143 q^{-72} +57 q^{-74} +65 q^{-76} +55 q^{-78} -33 q^{-80} -46 q^{-82} +10 q^{-84} +6 q^{-86} +16 q^{-88} -5 q^{-90} -10 q^{-92} +4 q^{-94} +3 q^{-98} - q^{-100} -2 q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+2 q^{153}+q^{151}-3 q^{149}-q^{143}+5 q^{141}+4 q^{139}-12 q^{137}-9 q^{135}+7 q^{133}+17 q^{131}+24 q^{129}+2 q^{127}-49 q^{125}-75 q^{123}-12 q^{121}+101 q^{119}+158 q^{117}+78 q^{115}-145 q^{113}-337 q^{111}-254 q^{109}+171 q^{107}+604 q^{105}+589 q^{103}-39 q^{101}-912 q^{99}-1202 q^{97}-373 q^{95}+1165 q^{93}+2062 q^{91}+1214 q^{89}-1099 q^{87}-3064 q^{85}-2613 q^{83}+473 q^{81}+3942 q^{79}+4493 q^{77}+902 q^{75}-4300 q^{73}-6533 q^{71}-3114 q^{69}+3789 q^{67}+8352 q^{65}+5868 q^{63}-2305 q^{61}-9333 q^{59}-8700 q^{57}-148 q^{55}+9257 q^{53}+11041 q^{51}+3042 q^{49}-8010 q^{47}-12339 q^{45}-5899 q^{43}+5830 q^{41}+12446 q^{39}+8135 q^{37}-3201 q^{35}-11407 q^{33}-9408 q^{31}+579 q^{29}+9517 q^{27}+9738 q^{25}+1643 q^{23}-7232 q^{21}-9264 q^{19}-3313 q^{17}+4893 q^{15}+8320 q^{13}+4522 q^{11}-2747 q^9-7229 q^7-5426 q^5+827 q^3+6187 q+6247 q^{-1} +985 q^{-3} -5254 q^{-5} -7142 q^{-7} -2811 q^{-9} +4318 q^{-11} +8093 q^{-13} +4802 q^{-15} -3177 q^{-17} -8965 q^{-19} -6968 q^{-21} +1664 q^{-23} +9472 q^{-25} +9135 q^{-27} +357 q^{-29} -9294 q^{-31} -11025 q^{-33} -2813 q^{-35} +8255 q^{-37} +12209 q^{-39} +5389 q^{-41} -6250 q^{-43} -12367 q^{-45} -7686 q^{-47} +3530 q^{-49} +11371 q^{-51} +9196 q^{-53} -591 q^{-55} -9250 q^{-57} -9606 q^{-59} -2079 q^{-61} +6469 q^{-63} +8899 q^{-65} +3873 q^{-67} -3574 q^{-69} -7197 q^{-71} -4670 q^{-73} +1100 q^{-75} +5092 q^{-77} +4462 q^{-79} +541 q^{-81} -3006 q^{-83} -3577 q^{-85} -1341 q^{-87} +1376 q^{-89} +2443 q^{-91} +1448 q^{-93} -362 q^{-95} -1411 q^{-97} -1141 q^{-99} -124 q^{-101} +664 q^{-103} +741 q^{-105} +256 q^{-107} -257 q^{-109} -396 q^{-111} -201 q^{-113} +61 q^{-115} +174 q^{-117} +125 q^{-119} + q^{-121} -75 q^{-123} -56 q^{-125} -3 q^{-127} +23 q^{-129} +18 q^{-131} +8 q^{-133} -9 q^{-135} -12 q^{-137} +4 q^{-139} +5 q^{-141} - q^{-143} -3 q^{-149} + q^{-151} +2 q^{-153} - q^{-155} }[/math]

A2 Invariants.

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

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+8 q^{28}-13 q^{26}+18 q^{24}-23 q^{22}+27 q^{20}-27 q^{18}+25 q^{16}-17 q^{14}+9 q^{12}+5 q^{10}-18 q^8+32 q^6-43 q^4+50 q^2-54+50 q^{-2} -43 q^{-4} +32 q^{-6} -18 q^{-8} +5 q^{-10} +9 q^{-12} -17 q^{-14} +25 q^{-16} -27 q^{-18} +27 q^{-20} -23 q^{-22} +18 q^{-24} -13 q^{-26} +8 q^{-28} -5 q^{-30} +2 q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+6 q^{46}-q^{44}-11 q^{42}-7 q^{40}+11 q^{38}+17 q^{36}-4 q^{34}-25 q^{32}-10 q^{30}+23 q^{28}+24 q^{26}-12 q^{24}-29 q^{22}-2 q^{20}+28 q^{18}+13 q^{16}-19 q^{14}-17 q^{12}+12 q^{10}+18 q^8-6 q^6-18 q^4+2 q^2+19+2 q^{-2} -18 q^{-4} -6 q^{-6} +18 q^{-8} +12 q^{-10} -17 q^{-12} -19 q^{-14} +13 q^{-16} +28 q^{-18} -2 q^{-20} -29 q^{-22} -12 q^{-24} +24 q^{-26} +23 q^{-28} -10 q^{-30} -25 q^{-32} -4 q^{-34} +17 q^{-36} +11 q^{-38} -7 q^{-40} -11 q^{-42} - q^{-44} +6 q^{-46} +3 q^{-48} -2 q^{-50} -2 q^{-52} + q^{-56} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-7 q^{70}-2 q^{68}+16 q^{66}-32 q^{64}+47 q^{62}-52 q^{60}+36 q^{58}-3 q^{56}-48 q^{54}+102 q^{52}-137 q^{50}+132 q^{48}-84 q^{46}-6 q^{44}+105 q^{42}-179 q^{40}+206 q^{38}-155 q^{36}+56 q^{34}+61 q^{32}-147 q^{30}+164 q^{28}-107 q^{26}+10 q^{24}+90 q^{22}-135 q^{20}+110 q^{18}-14 q^{16}-110 q^{14}+208 q^{12}-235 q^{10}+166 q^8-34 q^6-128 q^4+254 q^2-299+253 q^{-2} -128 q^{-4} -32 q^{-6} +166 q^{-8} -233 q^{-10} +206 q^{-12} -108 q^{-14} -12 q^{-16} +109 q^{-18} -135 q^{-20} +90 q^{-22} +10 q^{-24} -107 q^{-26} +164 q^{-28} -150 q^{-30} +63 q^{-32} +55 q^{-34} -156 q^{-36} +206 q^{-38} -180 q^{-40} +106 q^{-42} -6 q^{-44} -84 q^{-46} +132 q^{-48} -136 q^{-50} +102 q^{-52} -47 q^{-54} -4 q^{-56} +36 q^{-58} -51 q^{-60} +46 q^{-62} -32 q^{-64} +16 q^{-66} -2 q^{-68} -7 q^{-70} +8 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} }[/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 71"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+7 t^2-18 t+25-18 t^{-1} +7 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 77, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-12 q+13-12 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -3 z^4-a^4 z^2+4 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} -5 z^2-a^4+3 a^2+3 a^{-2} - a^{-4} -3 }[/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} +3 a^2 z^8+3 z^8 a^{-2} +6 z^8+4 a^3 z^7+8 a z^7+8 z^7 a^{-1} +4 z^7 a^{-3} +3 a^4 z^6+2 a^2 z^6+2 z^6 a^{-2} +3 z^6 a^{-4} -2 z^6+a^5 z^5-5 a^3 z^5-15 a z^5-15 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-12 a^2 z^4-12 z^4 a^{-2} -6 z^4 a^{-4} -12 z^4-2 a^5 z^3+7 a z^3+7 z^3 a^{-1} -2 z^3 a^{-5} +4 a^4 z^2+10 a^2 z^2+10 z^2 a^{-2} +4 z^2 a^{-4} +12 z^2+a^5 z+a^3 z-a z-z a^{-1} +z a^{-3} +z a^{-5} -a^4-3 a^2-3 a^{-2} - a^{-4} -3 }[/math]