10 16

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -4 t^2+12 t-15+12 t^{-1} -4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -4 z^4-4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 47, 2 }
Jones polynomial	[math]\displaystyle{ q^7-2 q^6+4 q^5-6 q^4+7 q^3-8 q^2+7 q-5+4 q^{-1} -2 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -2 z^4 a^{-2} -z^4 a^{-4} -z^4+a^2 z^2-4 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -z^2+a^2-2 a^{-2} + a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8+2 a z^7-z^7 a^{-1} +3 z^7 a^{-5} +a^2 z^6-13 z^6 a^{-2} -3 z^6 a^{-4} +3 z^6 a^{-6} -6 z^6-7 a z^5-2 z^5 a^{-1} -4 z^5 a^{-3} -7 z^5 a^{-5} +2 z^5 a^{-7} -4 a^2 z^4+17 z^4 a^{-2} +2 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} +4 z^4+5 a z^3+8 z^3 a^{-3} +10 z^3 a^{-5} -3 z^3 a^{-7} +4 a^2 z^2-11 z^2 a^{-2} +2 z^2 a^{-4} +5 z^2 a^{-6} -2 z^2 a^{-8} -2 z^2-4 z a^{-3} -4 z a^{-5} -a^2+2 a^{-2} - a^{-6} +1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{10}+2 q^4+1+ q^{-2} -2 q^{-4} -2 q^{-8} - q^{-14} +2 q^{-16} + q^{-22} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-q^{44}+3 q^{42}-4 q^{40}+4 q^{38}-2 q^{36}-q^{34}+9 q^{32}-13 q^{30}+17 q^{28}-16 q^{26}+7 q^{24}+5 q^{22}-20 q^{20}+31 q^{18}-31 q^{16}+24 q^{14}-5 q^{12}-14 q^{10}+30 q^8-33 q^6+26 q^4-7 q^2-10+22 q^{-2} -20 q^{-4} +11 q^{-6} +8 q^{-8} -19 q^{-10} +23 q^{-12} -18 q^{-14} + q^{-16} +15 q^{-18} -34 q^{-20} +39 q^{-22} -32 q^{-24} +13 q^{-26} +8 q^{-28} -33 q^{-30} +42 q^{-32} -42 q^{-34} +25 q^{-36} -6 q^{-38} -18 q^{-40} +31 q^{-42} -29 q^{-44} +16 q^{-46} + q^{-48} -13 q^{-50} +16 q^{-52} -10 q^{-54} -3 q^{-56} +15 q^{-58} -18 q^{-60} +20 q^{-62} -10 q^{-64} -3 q^{-66} +16 q^{-68} -22 q^{-70} +25 q^{-72} -19 q^{-74} +11 q^{-76} -10 q^{-80} +17 q^{-82} -20 q^{-84} +18 q^{-86} -10 q^{-88} +2 q^{-90} +4 q^{-92} -9 q^{-94} +10 q^{-96} -8 q^{-98} +6 q^{-100} -2 q^{-102} - q^{-104} +2 q^{-106} -3 q^{-108} +2 q^{-110} - q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_16.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^7-q^5+2 q^3-q+2 q^{-1} - q^{-3} - q^{-5} + q^{-7} -2 q^{-9} +2 q^{-11} - q^{-13} + q^{-15} }[/math]
2	[math]\displaystyle{ q^{22}-q^{20}-q^{18}+4 q^{16}-q^{14}-5 q^{12}+6 q^{10}+2 q^8-8 q^6+4 q^4+5 q^2-8+ q^{-2} +6 q^{-4} -5 q^{-6} -2 q^{-8} +5 q^{-10} +2 q^{-12} -4 q^{-14} - q^{-16} +8 q^{-18} -4 q^{-20} -5 q^{-22} +9 q^{-24} -3 q^{-26} -5 q^{-28} +5 q^{-30} -2 q^{-32} -3 q^{-34} +3 q^{-36} - q^{-40} + q^{-42} }[/math]
3	[math]\displaystyle{ q^{45}-q^{43}-q^{41}+q^{39}+3 q^{37}-q^{35}-6 q^{33}+10 q^{29}+3 q^{27}-11 q^{25}-10 q^{23}+12 q^{21}+16 q^{19}-8 q^{17}-21 q^{15}+22 q^{11}+8 q^9-21 q^7-15 q^5+16 q^3+21 q-8 q^{-1} -22 q^{-3} +5 q^{-5} +23 q^{-7} + q^{-9} -23 q^{-11} -2 q^{-13} +19 q^{-15} +8 q^{-17} -16 q^{-19} -11 q^{-21} +9 q^{-23} +15 q^{-25} - q^{-27} -19 q^{-29} -10 q^{-31} +19 q^{-33} +16 q^{-35} -16 q^{-37} -20 q^{-39} +10 q^{-41} +21 q^{-43} -7 q^{-45} -14 q^{-47} +2 q^{-49} +11 q^{-51} -4 q^{-55} + q^{-57} +2 q^{-59} -2 q^{-61} -2 q^{-63} +2 q^{-65} + q^{-67} -2 q^{-69} -2 q^{-71} + q^{-73} + q^{-75} - q^{-79} + q^{-81} }[/math]
4	[math]\displaystyle{ q^{76}-q^{74}-q^{72}+q^{70}+3 q^{66}-3 q^{64}-4 q^{62}+2 q^{60}+2 q^{58}+12 q^{56}-5 q^{54}-16 q^{52}-6 q^{50}+2 q^{48}+32 q^{46}+10 q^{44}-20 q^{42}-29 q^{40}-26 q^{38}+40 q^{36}+43 q^{34}+10 q^{32}-32 q^{30}-69 q^{28}+q^{26}+43 q^{24}+59 q^{22}+20 q^{20}-71 q^{18}-53 q^{16}-12 q^{14}+64 q^{12}+83 q^{10}-14 q^8-65 q^6-81 q^4+16 q^2+103+49 q^{-2} -32 q^{-4} -106 q^{-6} -34 q^{-8} +80 q^{-10} +73 q^{-12} - q^{-14} -89 q^{-16} -47 q^{-18} +51 q^{-20} +66 q^{-22} +9 q^{-24} -64 q^{-26} -46 q^{-28} +24 q^{-30} +61 q^{-32} +21 q^{-34} -35 q^{-36} -52 q^{-38} -21 q^{-40} +48 q^{-42} +55 q^{-44} +23 q^{-46} -55 q^{-48} -86 q^{-50} +5 q^{-52} +71 q^{-54} +91 q^{-56} -14 q^{-58} -114 q^{-60} -55 q^{-62} +30 q^{-64} +114 q^{-66} +44 q^{-68} -75 q^{-70} -69 q^{-72} -30 q^{-74} +68 q^{-76} +60 q^{-78} -14 q^{-80} -33 q^{-82} -49 q^{-84} +13 q^{-86} +34 q^{-88} +13 q^{-90} +6 q^{-92} -30 q^{-94} -7 q^{-96} +7 q^{-98} +7 q^{-100} +15 q^{-102} -10 q^{-104} -3 q^{-106} -3 q^{-108} -2 q^{-110} +8 q^{-112} -3 q^{-114} + q^{-116} - q^{-118} -3 q^{-120} +2 q^{-122} - q^{-124} + q^{-126} - q^{-130} + q^{-132} }[/math]
5	[math]\displaystyle{ q^{115}-q^{113}-q^{111}+q^{109}+q^{103}-q^{101}-3 q^{99}+2 q^{97}+5 q^{95}+2 q^{93}-8 q^{89}-13 q^{87}-5 q^{85}+16 q^{83}+26 q^{81}+16 q^{79}-10 q^{77}-42 q^{75}-43 q^{73}-7 q^{71}+48 q^{69}+76 q^{67}+42 q^{65}-31 q^{63}-96 q^{61}-96 q^{59}-18 q^{57}+94 q^{55}+141 q^{53}+81 q^{51}-39 q^{49}-149 q^{47}-157 q^{45}-46 q^{43}+108 q^{41}+190 q^{39}+144 q^{37}-4 q^{35}-167 q^{33}-222 q^{31}-122 q^{29}+78 q^{27}+241 q^{25}+244 q^{23}+68 q^{21}-189 q^{19}-327 q^{17}-223 q^{15}+76 q^{13}+344 q^{11}+352 q^9+71 q^7-296 q^5-439 q^3-215 q+205 q^{-1} +461 q^{-3} +324 q^{-5} -91 q^{-7} -428 q^{-9} -391 q^{-11} -9 q^{-13} +369 q^{-15} +398 q^{-17} +79 q^{-19} -282 q^{-21} -372 q^{-23} -117 q^{-25} +221 q^{-27} +318 q^{-29} +115 q^{-31} -161 q^{-33} -263 q^{-35} -109 q^{-37} +136 q^{-39} +224 q^{-41} +91 q^{-43} -107 q^{-45} -199 q^{-47} -110 q^{-49} +76 q^{-51} +190 q^{-53} +155 q^{-55} -13 q^{-57} -184 q^{-59} -221 q^{-61} -90 q^{-63} +143 q^{-65} +300 q^{-67} +226 q^{-69} -66 q^{-71} -348 q^{-73} -365 q^{-75} -68 q^{-77} +336 q^{-79} +489 q^{-81} +224 q^{-83} -265 q^{-85} -550 q^{-87} -365 q^{-89} +134 q^{-91} +526 q^{-93} +471 q^{-95} +15 q^{-97} -441 q^{-99} -493 q^{-101} -134 q^{-103} +290 q^{-105} +453 q^{-107} +223 q^{-109} -157 q^{-111} -353 q^{-113} -242 q^{-115} +38 q^{-117} +241 q^{-119} +223 q^{-121} +37 q^{-123} -140 q^{-125} -176 q^{-127} -73 q^{-129} +63 q^{-131} +121 q^{-133} +76 q^{-135} -14 q^{-137} -77 q^{-139} -67 q^{-141} -10 q^{-143} +41 q^{-145} +48 q^{-147} +20 q^{-149} -17 q^{-151} -30 q^{-153} -20 q^{-155} +5 q^{-157} +19 q^{-159} +13 q^{-161} +2 q^{-163} -6 q^{-165} -10 q^{-167} -3 q^{-169} +5 q^{-171} +2 q^{-173} + q^{-175} + q^{-177} -2 q^{-179} -2 q^{-181} + q^{-183} - q^{-187} + q^{-189} - q^{-193} + q^{-195} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{10}+2 q^4+1+ q^{-2} -2 q^{-4} -2 q^{-8} - q^{-14} +2 q^{-16} + q^{-22} }[/math]
1,1	[math]\displaystyle{ q^{28}-2 q^{26}+6 q^{24}-12 q^{22}+23 q^{20}-34 q^{18}+54 q^{16}-72 q^{14}+90 q^{12}-102 q^{10}+110 q^8-104 q^6+81 q^4-52 q^2+12+36 q^{-2} -84 q^{-4} +122 q^{-6} -156 q^{-8} +176 q^{-10} -189 q^{-12} +178 q^{-14} -158 q^{-16} +136 q^{-18} -91 q^{-20} +62 q^{-22} -16 q^{-24} -10 q^{-26} +35 q^{-28} -56 q^{-30} +58 q^{-32} -70 q^{-34} +70 q^{-36} -68 q^{-38} +60 q^{-40} -56 q^{-42} +50 q^{-44} -38 q^{-46} +28 q^{-48} -20 q^{-50} +15 q^{-52} -8 q^{-54} +4 q^{-56} -2 q^{-58} + q^{-60} }[/math]
2,0	[math]\displaystyle{ q^{28}-q^{24}+3 q^{20}+3 q^{18}-2 q^{16}-2 q^{14}+3 q^{12}+3 q^{10}-2 q^8-5 q^6+2 q^4+3 q^2-4-5 q^{-2} +3 q^{-4} + q^{-6} -3 q^{-8} +2 q^{-12} + q^{-14} +5 q^{-18} + q^{-20} -2 q^{-22} +5 q^{-24} +5 q^{-26} -5 q^{-28} -4 q^{-30} +4 q^{-32} +2 q^{-34} -5 q^{-36} -4 q^{-38} +2 q^{-40} -3 q^{-44} +2 q^{-48} + q^{-50} + q^{-56} }[/math]

A3 Invariants.

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

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{46}-q^{44}+3 q^{42}-4 q^{40}+4 q^{38}-2 q^{36}-q^{34}+9 q^{32}-13 q^{30}+17 q^{28}-16 q^{26}+7 q^{24}+5 q^{22}-20 q^{20}+31 q^{18}-31 q^{16}+24 q^{14}-5 q^{12}-14 q^{10}+30 q^8-33 q^6+26 q^4-7 q^2-10+22 q^{-2} -20 q^{-4} +11 q^{-6} +8 q^{-8} -19 q^{-10} +23 q^{-12} -18 q^{-14} + q^{-16} +15 q^{-18} -34 q^{-20} +39 q^{-22} -32 q^{-24} +13 q^{-26} +8 q^{-28} -33 q^{-30} +42 q^{-32} -42 q^{-34} +25 q^{-36} -6 q^{-38} -18 q^{-40} +31 q^{-42} -29 q^{-44} +16 q^{-46} + q^{-48} -13 q^{-50} +16 q^{-52} -10 q^{-54} -3 q^{-56} +15 q^{-58} -18 q^{-60} +20 q^{-62} -10 q^{-64} -3 q^{-66} +16 q^{-68} -22 q^{-70} +25 q^{-72} -19 q^{-74} +11 q^{-76} -10 q^{-80} +17 q^{-82} -20 q^{-84} +18 q^{-86} -10 q^{-88} +2 q^{-90} +4 q^{-92} -9 q^{-94} +10 q^{-96} -8 q^{-98} +6 q^{-100} -2 q^{-102} - q^{-104} +2 q^{-106} -3 q^{-108} +2 q^{-110} - 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 16"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -4 t^2+12 t-15+12 t^{-1} -4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 47, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^7-2 q^6+4 q^5-6 q^4+7 q^3-8 q^2+7 q-5+4 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{ -2 z^4 a^{-2} -z^4 a^{-4} -z^4+a^2 z^2-4 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -z^2+a^2-2 a^{-2} + a^{-6} +1 }[/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} +4 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8+2 a z^7-z^7 a^{-1} +3 z^7 a^{-5} +a^2 z^6-13 z^6 a^{-2} -3 z^6 a^{-4} +3 z^6 a^{-6} -6 z^6-7 a z^5-2 z^5 a^{-1} -4 z^5 a^{-3} -7 z^5 a^{-5} +2 z^5 a^{-7} -4 a^2 z^4+17 z^4 a^{-2} +2 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} +4 z^4+5 a z^3+8 z^3 a^{-3} +10 z^3 a^{-5} -3 z^3 a^{-7} +4 a^2 z^2-11 z^2 a^{-2} +2 z^2 a^{-4} +5 z^2 a^{-6} -2 z^2 a^{-8} -2 z^2-4 z a^{-3} -4 z a^{-5} -a^2+2 a^{-2} - a^{-6} +1 }[/math]