10 70

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-7 t^2+16 t-19+16 t^{-1} -7 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6-z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 67, 2 }
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+6 q^5-9 q^4+11 q^3-11 q^2+10 q-8+5 q^{-1} -2 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +3 z^4 a^{-2} -2 z^4 a^{-4} -2 z^4+a^2 z^2+4 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -5 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+2 a z^7+3 z^7 a^{-1} +6 z^7 a^{-3} +5 z^7 a^{-5} +a^2 z^6-5 z^6 a^{-2} +3 z^6 a^{-4} +5 z^6 a^{-6} -2 z^6-6 a z^5-10 z^5 a^{-1} -11 z^5 a^{-3} -4 z^5 a^{-5} +3 z^5 a^{-7} -4 a^2 z^4-8 z^4 a^{-2} -12 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} -7 z^4+5 a z^3+4 z^3 a^{-1} +2 z^3 a^{-3} -3 z^3 a^{-7} +5 a^2 z^2+9 z^2 a^{-2} +9 z^2 a^{-4} +4 z^2 a^{-6} -z^2 a^{-8} +10 z^2-a z+z a^{-3} +z a^{-5} +z a^{-7} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{10}+q^8+2 q^4-2 q^2-1+ q^{-2} -2 q^{-4} +3 q^{-6} - q^{-8} + q^{-10} -2 q^{-14} +2 q^{-16} - q^{-18} + q^{-22} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-q^{44}+4 q^{42}-5 q^{40}+6 q^{38}-5 q^{36}+12 q^{32}-23 q^{30}+36 q^{28}-37 q^{26}+25 q^{24}+5 q^{22}-42 q^{20}+80 q^{18}-97 q^{16}+85 q^{14}-40 q^{12}-32 q^{10}+96 q^8-132 q^6+124 q^4-70 q^2-6+70 q^{-2} -108 q^{-4} +90 q^{-6} -38 q^{-8} -31 q^{-10} +81 q^{-12} -86 q^{-14} +46 q^{-16} +27 q^{-18} -97 q^{-20} +140 q^{-22} -132 q^{-24} +74 q^{-26} +17 q^{-28} -112 q^{-30} +178 q^{-32} -181 q^{-34} +132 q^{-36} -36 q^{-38} -62 q^{-40} +129 q^{-42} -147 q^{-44} +107 q^{-46} -35 q^{-48} -39 q^{-50} +81 q^{-52} -76 q^{-54} +28 q^{-56} +37 q^{-58} -86 q^{-60} +94 q^{-62} -64 q^{-64} + q^{-66} +61 q^{-68} -107 q^{-70} +119 q^{-72} -89 q^{-74} +41 q^{-76} +16 q^{-78} -61 q^{-80} +80 q^{-82} -76 q^{-84} +57 q^{-86} -25 q^{-88} -3 q^{-90} +23 q^{-92} -32 q^{-94} +30 q^{-96} -21 q^{-98} +12 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_70.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^7-q^5+3 q^3-3 q+2 q^{-1} - q^{-3} +2 q^{-7} -3 q^{-9} +3 q^{-11} -2 q^{-13} + q^{-15} }[/math]
2	[math]\displaystyle{ q^{22}-q^{20}-q^{18}+5 q^{16}-3 q^{14}-8 q^{12}+13 q^{10}+q^8-19 q^6+15 q^4+11 q^2-23+7 q^{-2} +16 q^{-4} -15 q^{-6} -4 q^{-8} +12 q^{-10} + q^{-12} -12 q^{-14} + q^{-16} +18 q^{-18} -14 q^{-20} -11 q^{-22} +25 q^{-24} -8 q^{-26} -15 q^{-28} +16 q^{-30} -2 q^{-32} -8 q^{-34} +6 q^{-36} -2 q^{-40} + q^{-42} }[/math]
3	[math]\displaystyle{ q^{45}-q^{43}-q^{41}+q^{39}+4 q^{37}-3 q^{35}-9 q^{33}+2 q^{31}+19 q^{29}+3 q^{27}-30 q^{25}-18 q^{23}+42 q^{21}+40 q^{19}-41 q^{17}-69 q^{15}+28 q^{13}+97 q^{11}-6 q^9-110 q^7-28 q^5+113 q^3+60 q-102 q^{-1} -84 q^{-3} +82 q^{-5} +98 q^{-7} -54 q^{-9} -103 q^{-11} +28 q^{-13} +100 q^{-15} -86 q^{-19} -31 q^{-21} +67 q^{-23} +61 q^{-25} -41 q^{-27} -89 q^{-29} +8 q^{-31} +108 q^{-33} +27 q^{-35} -113 q^{-37} -61 q^{-39} +106 q^{-41} +82 q^{-43} -84 q^{-45} -87 q^{-47} +55 q^{-49} +81 q^{-51} -32 q^{-53} -63 q^{-55} +16 q^{-57} +41 q^{-59} -6 q^{-61} -25 q^{-63} +3 q^{-65} +15 q^{-67} -3 q^{-69} -7 q^{-71} + q^{-73} +3 q^{-75} -2 q^{-79} + q^{-81} }[/math]
4	[math]\displaystyle{ q^{76}-q^{74}-q^{72}+q^{70}+4 q^{66}-5 q^{64}-7 q^{62}+4 q^{60}+6 q^{58}+21 q^{56}-11 q^{54}-36 q^{52}-15 q^{50}+14 q^{48}+86 q^{46}+26 q^{44}-76 q^{42}-110 q^{40}-60 q^{38}+172 q^{36}+182 q^{34}+3 q^{32}-223 q^{30}-308 q^{28}+93 q^{26}+359 q^{24}+306 q^{22}-122 q^{20}-577 q^{18}-250 q^{16}+281 q^{14}+626 q^{12}+267 q^{10}-557 q^8-618 q^6-95 q^4+668 q^2+660-241 q^{-2} -716 q^{-4} -472 q^{-6} +434 q^{-8} +789 q^{-10} +107 q^{-12} -569 q^{-14} -632 q^{-16} +156 q^{-18} +686 q^{-20} +315 q^{-22} -348 q^{-24} -609 q^{-26} -79 q^{-28} +480 q^{-30} +452 q^{-32} -90 q^{-34} -511 q^{-36} -340 q^{-38} +182 q^{-40} +565 q^{-42} +267 q^{-44} -296 q^{-46} -611 q^{-48} -251 q^{-50} +536 q^{-52} +630 q^{-54} +92 q^{-56} -682 q^{-58} -678 q^{-60} +251 q^{-62} +738 q^{-64} +489 q^{-66} -434 q^{-68} -794 q^{-70} -100 q^{-72} +485 q^{-74} +591 q^{-76} -79 q^{-78} -545 q^{-80} -233 q^{-82} +144 q^{-84} +391 q^{-86} +80 q^{-88} -229 q^{-90} -138 q^{-92} -15 q^{-94} +157 q^{-96} +57 q^{-98} -68 q^{-100} -34 q^{-102} -24 q^{-104} +48 q^{-106} +14 q^{-108} -23 q^{-110} -8 q^{-114} +14 q^{-116} +2 q^{-118} -8 q^{-120} +2 q^{-122} -2 q^{-124} +3 q^{-126} -2 q^{-130} + q^{-132} }[/math]
5	[math]\displaystyle{ q^{115}-q^{113}-q^{111}+q^{109}+2 q^{103}-3 q^{101}-6 q^{99}+4 q^{97}+9 q^{95}+6 q^{93}+3 q^{91}-16 q^{89}-32 q^{87}-12 q^{85}+33 q^{83}+62 q^{81}+51 q^{79}-23 q^{77}-122 q^{75}-139 q^{73}-26 q^{71}+167 q^{69}+283 q^{67}+174 q^{65}-149 q^{63}-450 q^{61}-449 q^{59}-31 q^{57}+571 q^{55}+826 q^{53}+417 q^{51}-472 q^{49}-1195 q^{47}-1057 q^{45}+59 q^{43}+1388 q^{41}+1786 q^{39}+737 q^{37}-1158 q^{35}-2424 q^{33}-1840 q^{31}+430 q^{29}+2697 q^{27}+2975 q^{25}+785 q^{23}-2366 q^{21}-3903 q^{19}-2282 q^{17}+1468 q^{15}+4314 q^{13}+3714 q^{11}-81 q^9-4092 q^7-4841 q^5-1459 q^3+3323 q+5406 q^{-1} +2871 q^{-3} -2184 q^{-5} -5405 q^{-7} -3922 q^{-9} +968 q^{-11} +4944 q^{-13} +4501 q^{-15} +95 q^{-17} -4211 q^{-19} -4618 q^{-21} -908 q^{-23} +3416 q^{-25} +4430 q^{-27} +1417 q^{-29} -2678 q^{-31} -4082 q^{-33} -1718 q^{-35} +2030 q^{-37} +3713 q^{-39} +1961 q^{-41} -1447 q^{-43} -3398 q^{-45} -2251 q^{-47} +820 q^{-49} +3094 q^{-51} +2688 q^{-53} -13 q^{-55} -2745 q^{-57} -3233 q^{-59} -1017 q^{-61} +2183 q^{-63} +3765 q^{-65} +2282 q^{-67} -1304 q^{-69} -4123 q^{-71} -3622 q^{-73} +86 q^{-75} +4064 q^{-77} +4820 q^{-79} +1406 q^{-81} -3501 q^{-83} -5627 q^{-85} -2893 q^{-87} +2444 q^{-89} +5778 q^{-91} +4138 q^{-93} -1068 q^{-95} -5269 q^{-97} -4847 q^{-99} -306 q^{-101} +4206 q^{-103} +4863 q^{-105} +1425 q^{-107} -2857 q^{-109} -4306 q^{-111} -2051 q^{-113} +1572 q^{-115} +3346 q^{-117} +2142 q^{-119} -550 q^{-121} -2285 q^{-123} -1857 q^{-125} -66 q^{-127} +1372 q^{-129} +1367 q^{-131} +306 q^{-133} -693 q^{-135} -874 q^{-137} -330 q^{-139} +297 q^{-141} +498 q^{-143} +237 q^{-145} -113 q^{-147} -238 q^{-149} -135 q^{-151} +29 q^{-153} +106 q^{-155} +70 q^{-157} -16 q^{-159} -46 q^{-161} -20 q^{-163} +8 q^{-165} +15 q^{-167} +6 q^{-169} -2 q^{-171} -11 q^{-173} -2 q^{-175} +9 q^{-177} + q^{-179} -3 q^{-181} + q^{-183} - q^{-185} -2 q^{-187} +3 q^{-189} -2 q^{-193} + q^{-195} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{20}-q^{18}+2 q^{16}+3 q^{14}-4 q^{12}+5 q^{10}+5 q^8-13 q^6+8 q^4+5 q^2-19+10 q^{-2} +8 q^{-4} -16 q^{-6} +6 q^{-8} +11 q^{-10} -6 q^{-12} -2 q^{-14} +4 q^{-16} +5 q^{-18} -9 q^{-20} -6 q^{-22} +16 q^{-24} -10 q^{-26} -10 q^{-28} +20 q^{-30} -6 q^{-32} -10 q^{-34} +14 q^{-36} -2 q^{-38} -7 q^{-40} +5 q^{-42} -2 q^{-46} + q^{-48} }[/math]
1,0,0	[math]\displaystyle{ q^{13}+q^{11}+2 q^9+2 q^5-3 q^3-3 q^{-1} + q^{-3} - q^{-5} +2 q^{-7} +2 q^{-9} + q^{-13} -2 q^{-15} + q^{-17} -3 q^{-19} +2 q^{-21} - q^{-23} + q^{-25} + q^{-29} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{20}-q^{18}+4 q^{16}-5 q^{14}+10 q^{12}-13 q^{10}+17 q^8-19 q^6+20 q^4-19 q^2+13-8 q^{-2} -2 q^{-4} +12 q^{-6} -22 q^{-8} +31 q^{-10} -36 q^{-12} +40 q^{-14} -38 q^{-16} +33 q^{-18} -25 q^{-20} +16 q^{-22} -6 q^{-24} -4 q^{-26} +12 q^{-28} -18 q^{-30} +20 q^{-32} -20 q^{-34} +18 q^{-36} -14 q^{-38} +11 q^{-40} -7 q^{-42} +4 q^{-44} -2 q^{-46} + q^{-48} }[/math]
1,0	[math]\displaystyle{ q^{34}-q^{30}-q^{28}+3 q^{26}+4 q^{24}-q^{22}-7 q^{20}-2 q^{18}+10 q^{16}+10 q^{14}-8 q^{12}-17 q^{10}-q^8+20 q^6+11 q^4-17 q^2-19+6 q^{-2} +22 q^{-4} +4 q^{-6} -18 q^{-8} -10 q^{-10} +12 q^{-12} +12 q^{-14} -7 q^{-16} -11 q^{-18} +5 q^{-20} +13 q^{-22} -2 q^{-24} -14 q^{-26} - q^{-28} +15 q^{-30} +5 q^{-32} -15 q^{-34} -11 q^{-36} +13 q^{-38} +16 q^{-40} -8 q^{-42} -21 q^{-44} - q^{-46} +21 q^{-48} +11 q^{-50} -13 q^{-52} -17 q^{-54} +3 q^{-56} +16 q^{-58} +6 q^{-60} -8 q^{-62} -9 q^{-64} + q^{-66} +6 q^{-68} +2 q^{-70} -2 q^{-72} -2 q^{-74} + q^{-78} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{46}-q^{44}+4 q^{42}-5 q^{40}+6 q^{38}-5 q^{36}+12 q^{32}-23 q^{30}+36 q^{28}-37 q^{26}+25 q^{24}+5 q^{22}-42 q^{20}+80 q^{18}-97 q^{16}+85 q^{14}-40 q^{12}-32 q^{10}+96 q^8-132 q^6+124 q^4-70 q^2-6+70 q^{-2} -108 q^{-4} +90 q^{-6} -38 q^{-8} -31 q^{-10} +81 q^{-12} -86 q^{-14} +46 q^{-16} +27 q^{-18} -97 q^{-20} +140 q^{-22} -132 q^{-24} +74 q^{-26} +17 q^{-28} -112 q^{-30} +178 q^{-32} -181 q^{-34} +132 q^{-36} -36 q^{-38} -62 q^{-40} +129 q^{-42} -147 q^{-44} +107 q^{-46} -35 q^{-48} -39 q^{-50} +81 q^{-52} -76 q^{-54} +28 q^{-56} +37 q^{-58} -86 q^{-60} +94 q^{-62} -64 q^{-64} + q^{-66} +61 q^{-68} -107 q^{-70} +119 q^{-72} -89 q^{-74} +41 q^{-76} +16 q^{-78} -61 q^{-80} +80 q^{-82} -76 q^{-84} +57 q^{-86} -25 q^{-88} -3 q^{-90} +23 q^{-92} -32 q^{-94} +30 q^{-96} -21 q^{-98} +12 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }[/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 70"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-7 t^2+16 t-19+16 t^{-1} -7 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^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]=

{ 67, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^7-3 q^6+6 q^5-9 q^4+11 q^3-11 q^2+10 q-8+5 q^{-1} -2 q^{-2} + q^{-3} }[/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} +3 z^4 a^{-2} -2 z^4 a^{-4} -2 z^4+a^2 z^2+4 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -5 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+2 a z^7+3 z^7 a^{-1} +6 z^7 a^{-3} +5 z^7 a^{-5} +a^2 z^6-5 z^6 a^{-2} +3 z^6 a^{-4} +5 z^6 a^{-6} -2 z^6-6 a z^5-10 z^5 a^{-1} -11 z^5 a^{-3} -4 z^5 a^{-5} +3 z^5 a^{-7} -4 a^2 z^4-8 z^4 a^{-2} -12 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} -7 z^4+5 a z^3+4 z^3 a^{-1} +2 z^3 a^{-3} -3 z^3 a^{-7} +5 a^2 z^2+9 z^2 a^{-2} +9 z^2 a^{-4} +4 z^2 a^{-6} -z^2 a^{-8} +10 z^2-a z+z a^{-3} +z a^{-5} +z a^{-7} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3 }[/math]