9 47

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-4 t^2+6 t-5+6 t^{-1} -4 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{3,t+1\} }[/math]
Determinant and Signature { 27, 2 }
Jones polynomial [math]\displaystyle{ 2 q^5-4 q^4+4 q^3-5 q^2+5 q-3+3 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+4 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^7 a^{-1} +2 z^7 a^{-3} +6 z^6 a^{-2} +3 z^6 a^{-4} +3 z^6+a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} -16 z^4 a^{-2} -7 z^4 a^{-4} -9 z^4-2 a z^3+z^3 a^{-1} +6 z^3 a^{-3} +3 z^3 a^{-5} +11 z^2 a^{-2} +9 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1 }[/math]
The A2 invariant [math]\displaystyle{ -q^6+q^4+q^2+2+2 q^{-2} - q^{-4} + q^{-6} -2 q^{-8} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+4 q^{24}-q^{22}-8 q^{20}+16 q^{18}-16 q^{16}+16 q^{14}-4 q^{12}-11 q^{10}+20 q^8-18 q^6+14 q^4-2 q^2-13+21 q^{-2} -8 q^{-4} +13 q^{-8} -19 q^{-10} +18 q^{-12} -4 q^{-14} -9 q^{-16} +12 q^{-18} -21 q^{-20} +25 q^{-22} -13 q^{-24} +9 q^{-28} -19 q^{-30} +23 q^{-32} -18 q^{-34} +4 q^{-36} +3 q^{-38} -14 q^{-40} +19 q^{-42} -11 q^{-44} -3 q^{-46} +15 q^{-48} -19 q^{-50} +12 q^{-52} + q^{-54} -17 q^{-56} +20 q^{-58} -17 q^{-60} +11 q^{-62} +2 q^{-64} -11 q^{-66} +14 q^{-68} -12 q^{-70} +8 q^{-72} -5 q^{-76} +2 q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_47.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^5+2 q^3+2 q^{-1} - q^{-5} -2 q^{-9} +2 q^{-11} }[/math]
2 [math]\displaystyle{ q^{16}-2 q^{14}-3 q^{12}+5 q^{10}+2 q^8-5 q^6+3 q^4+6 q^2-4- q^{-2} +4 q^{-4} -3 q^{-6} -4 q^{-8} +2 q^{-10} +2 q^{-12} -3 q^{-14} - q^{-16} +7 q^{-18} - q^{-20} -5 q^{-22} +6 q^{-24} -5 q^{-28} + q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ -q^{33}+2 q^{31}+3 q^{29}-2 q^{27}-8 q^{25}-5 q^{23}+11 q^{21}+11 q^{19}-5 q^{17}-17 q^{15}-2 q^{13}+20 q^{11}+15 q^9-14 q^7-20 q^5+6 q^3+24 q- q^{-1} -24 q^{-3} -7 q^{-5} +19 q^{-7} +10 q^{-9} -18 q^{-11} -8 q^{-13} +13 q^{-15} +12 q^{-17} -9 q^{-19} -10 q^{-21} +4 q^{-23} +16 q^{-25} +3 q^{-27} -16 q^{-29} -13 q^{-31} +13 q^{-33} +21 q^{-35} -10 q^{-37} -27 q^{-39} + q^{-41} +27 q^{-43} +5 q^{-45} -17 q^{-47} -12 q^{-49} +11 q^{-51} +10 q^{-53} -3 q^{-55} -4 q^{-57} -2 q^{-59} +2 q^{-61} }[/math]
4 [math]\displaystyle{ q^{56}-2 q^{54}-3 q^{52}+2 q^{50}+5 q^{48}+11 q^{46}-2 q^{44}-16 q^{42}-18 q^{40}-10 q^{38}+29 q^{36}+33 q^{34}+10 q^{32}-26 q^{30}-58 q^{28}-16 q^{26}+35 q^{24}+70 q^{22}+43 q^{20}-51 q^{18}-83 q^{16}-45 q^{14}+56 q^{12}+109 q^{10}+32 q^8-71 q^6-116 q^4-24 q^2+98+97 q^{-2} -5 q^{-4} -109 q^{-6} -72 q^{-8} +49 q^{-10} +97 q^{-12} +32 q^{-14} -69 q^{-16} -68 q^{-18} +17 q^{-20} +70 q^{-22} +30 q^{-24} -42 q^{-26} -53 q^{-28} +54 q^{-32} +32 q^{-34} -20 q^{-36} -52 q^{-38} -34 q^{-40} +32 q^{-42} +60 q^{-44} +38 q^{-46} -47 q^{-48} -95 q^{-50} -23 q^{-52} +75 q^{-54} +116 q^{-56} +9 q^{-58} -120 q^{-60} -102 q^{-62} +18 q^{-64} +135 q^{-66} +84 q^{-68} -58 q^{-70} -108 q^{-72} -55 q^{-74} +60 q^{-76} +84 q^{-78} +16 q^{-80} -38 q^{-82} -49 q^{-84} -6 q^{-86} +22 q^{-88} +17 q^{-90} +4 q^{-92} -8 q^{-94} -5 q^{-96} + q^{-98} + q^{-100} }[/math]
5 [math]\displaystyle{ -q^{85}+2 q^{83}+3 q^{81}-2 q^{79}-5 q^{77}-8 q^{75}-4 q^{73}+7 q^{71}+24 q^{69}+24 q^{67}+3 q^{65}-27 q^{63}-51 q^{61}-48 q^{59}-5 q^{57}+66 q^{55}+95 q^{53}+67 q^{51}-13 q^{49}-112 q^{47}-156 q^{45}-90 q^{43}+58 q^{41}+190 q^{39}+220 q^{37}+93 q^{35}-137 q^{33}-303 q^{31}-264 q^{29}-17 q^{27}+278 q^{25}+409 q^{23}+230 q^{21}-146 q^{19}-452 q^{17}-434 q^{15}-65 q^{13}+380 q^{11}+548 q^9+284 q^7-221 q^5-572 q^3-451 q+40 q^{-1} +495 q^{-3} +539 q^{-5} +130 q^{-7} -371 q^{-9} -539 q^{-11} -236 q^{-13} +254 q^{-15} +480 q^{-17} +282 q^{-19} -146 q^{-21} -402 q^{-23} -276 q^{-25} +86 q^{-27} +317 q^{-29} +229 q^{-31} -48 q^{-33} -254 q^{-35} -192 q^{-37} +48 q^{-39} +206 q^{-41} +151 q^{-43} -34 q^{-45} -183 q^{-47} -159 q^{-49} +13 q^{-51} +173 q^{-53} +191 q^{-55} +55 q^{-57} -150 q^{-59} -255 q^{-61} -160 q^{-63} +92 q^{-65} +319 q^{-67} +312 q^{-69} +19 q^{-71} -345 q^{-73} -469 q^{-75} -191 q^{-77} +296 q^{-79} +590 q^{-81} +398 q^{-83} -168 q^{-85} -624 q^{-87} -572 q^{-89} -30 q^{-91} +534 q^{-93} +663 q^{-95} +242 q^{-97} -364 q^{-99} -619 q^{-101} -370 q^{-103} +129 q^{-105} +472 q^{-107} +404 q^{-109} +42 q^{-111} -276 q^{-113} -319 q^{-115} -135 q^{-117} +99 q^{-119} +199 q^{-121} +136 q^{-123} + q^{-125} -89 q^{-127} -78 q^{-129} -31 q^{-131} +18 q^{-133} +35 q^{-135} +18 q^{-137} -3 q^{-139} -6 q^{-141} -4 q^{-143} -2 q^{-145} +2 q^{-147} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^6+q^4+q^2+2+2 q^{-2} - q^{-4} + q^{-6} -2 q^{-8} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
1,1 [math]\displaystyle{ q^{20}-4 q^{18}+10 q^{16}-22 q^{14}+32 q^{12}-46 q^{10}+56 q^8-52 q^6+48 q^4-22 q^2+4+36 q^{-2} -58 q^{-4} +74 q^{-6} -94 q^{-8} +90 q^{-10} -94 q^{-12} +70 q^{-14} -52 q^{-16} +28 q^{-18} +5 q^{-20} -24 q^{-22} +50 q^{-24} -56 q^{-26} +58 q^{-28} -46 q^{-30} +32 q^{-32} -22 q^{-34} +10 q^{-36} -2 q^{-38} -4 q^{-40} +2 q^{-46} }[/math]
2,0 [math]\displaystyle{ q^{18}-q^{16}-3 q^{14}-q^{12}+2 q^{10}+2 q^8+4 q^4+6 q^2+3-2 q^{-2} + q^{-4} -3 q^{-6} -3 q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} -2 q^{-16} +3 q^{-18} +3 q^{-24} +5 q^{-26} - q^{-28} - q^{-30} + q^{-32} -2 q^{-38} - q^{-48} + q^{-52} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{14}-2 q^{12}+q^8-4 q^6+5 q^4+3 q^2+1+7 q^{-2} +3 q^{-4} -4 q^{-6} -2 q^{-8} -3 q^{-10} -5 q^{-12} -2 q^{-14} +5 q^{-18} + q^{-20} + q^{-22} +5 q^{-24} -4 q^{-26} -3 q^{-28} +3 q^{-30} -3 q^{-32} - q^{-34} +3 q^{-36} }[/math]
1,0,0 [math]\displaystyle{ -q^7+q^5+3 q+ q^{-1} +2 q^{-3} - q^{-11} -2 q^{-15} -2 q^{-19} + q^{-21} + q^{-25} + q^{-27} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+4 q^{24}-q^{22}-8 q^{20}+16 q^{18}-16 q^{16}+16 q^{14}-4 q^{12}-11 q^{10}+20 q^8-18 q^6+14 q^4-2 q^2-13+21 q^{-2} -8 q^{-4} +13 q^{-8} -19 q^{-10} +18 q^{-12} -4 q^{-14} -9 q^{-16} +12 q^{-18} -21 q^{-20} +25 q^{-22} -13 q^{-24} +9 q^{-28} -19 q^{-30} +23 q^{-32} -18 q^{-34} +4 q^{-36} +3 q^{-38} -14 q^{-40} +19 q^{-42} -11 q^{-44} -3 q^{-46} +15 q^{-48} -19 q^{-50} +12 q^{-52} + q^{-54} -17 q^{-56} +20 q^{-58} -17 q^{-60} +11 q^{-62} +2 q^{-64} -11 q^{-66} +14 q^{-68} -12 q^{-70} +8 q^{-72} -5 q^{-76} +2 q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/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["9 47"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-4 t^2+6 t-5+6 t^{-1} -4 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 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{ \{3,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 27, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 2 q^5-4 q^4+4 q^3-5 q^2+5 q-3+3 q^{-1} - q^{-2} }[/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} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+4 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^7 a^{-1} +2 z^7 a^{-3} +6 z^6 a^{-2} +3 z^6 a^{-4} +3 z^6+a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} -16 z^4 a^{-2} -7 z^4 a^{-4} -9 z^4-2 a z^3+z^3 a^{-1} +6 z^3 a^{-3} +3 z^3 a^{-5} +11 z^2 a^{-2} +9 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1 }[/math]
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