10 34

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t^2-9 t+13-9 t^{-1} +3 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 3 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 37, 0 }
Jones polynomial [math]\displaystyle{ -q^7+2 q^6-3 q^5+4 q^4-5 q^3+6 q^2-5 q+5-3 q^{-1} +2 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^{-2} +z^4 a^{-4} +z^4-a^2 z^2+z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} +2 z^2-a^2+ a^{-4} - a^{-6} +2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +4 z^8 a^{-4} +2 z^8 a^{-6} +2 z^7 a^{-1} -z^7 a^{-3} -2 z^7 a^{-5} +z^7 a^{-7} -5 z^6 a^{-2} -17 z^6 a^{-4} -10 z^6 a^{-6} +2 z^6+2 a z^5-2 z^5 a^{-1} -5 z^5 a^{-3} -6 z^5 a^{-5} -5 z^5 a^{-7} +2 a^2 z^4+4 z^4 a^{-2} +20 z^4 a^{-4} +14 z^4 a^{-6} +a^3 z^3-z^3 a^{-1} +5 z^3 a^{-3} +12 z^3 a^{-5} +7 z^3 a^{-7} -2 a^2 z^2-3 z^2 a^{-2} -8 z^2 a^{-4} -6 z^2 a^{-6} -3 z^2-a^3 z-a z-z a^{-3} -4 z a^{-5} -3 z a^{-7} +a^2+ a^{-4} + a^{-6} +2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{10}-q^4+2 q^2+1+ q^{-2} + q^{-4} + q^{-8} + q^{-14} - q^{-16} - q^{-22} }[/math]
The G2 invariant [math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-3 q^{40}+4 q^{38}-5 q^{36}+4 q^{34}-4 q^{32}+3 q^{28}-6 q^{26}+7 q^{24}-7 q^{22}+4 q^{20}-2 q^{18}-2 q^{16}+5 q^{14}-5 q^{12}+6 q^{10}-2 q^8+2 q^6+q^2+2- q^{-2} +5 q^{-4} -4 q^{-6} +4 q^{-8} - q^{-10} -2 q^{-12} +4 q^{-14} -3 q^{-16} +2 q^{-18} +2 q^{-20} -4 q^{-22} +3 q^{-24} + q^{-26} -6 q^{-28} +11 q^{-30} -11 q^{-32} +7 q^{-34} + q^{-36} -7 q^{-38} +15 q^{-40} -15 q^{-42} +12 q^{-44} -6 q^{-46} -2 q^{-48} +8 q^{-50} -10 q^{-52} +11 q^{-54} -6 q^{-56} +2 q^{-58} +3 q^{-60} -6 q^{-62} +5 q^{-64} - q^{-66} -6 q^{-68} +9 q^{-70} -9 q^{-72} +4 q^{-74} +4 q^{-76} -12 q^{-78} +16 q^{-80} -15 q^{-82} +7 q^{-84} -10 q^{-88} +12 q^{-90} -12 q^{-92} +8 q^{-94} -2 q^{-96} -2 q^{-98} +3 q^{-100} -4 q^{-102} +3 q^{-104} - q^{-106} + q^{-108} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_34.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^7+q^5-q^3+2 q+ q^{-3} + q^{-5} - q^{-7} + q^{-9} - q^{-11} + q^{-13} - q^{-15} }[/math]
2 [math]\displaystyle{ q^{20}-q^{18}-q^{16}+2 q^{14}-2 q^{12}-q^{10}+2 q^8-2 q^6+q^4+2 q^2-2+2 q^{-2} +3 q^{-4} - q^{-6} - q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} +2 q^{-18} -2 q^{-20} -2 q^{-22} +3 q^{-24} - q^{-26} -4 q^{-28} +3 q^{-30} +2 q^{-32} -4 q^{-34} + q^{-36} +3 q^{-38} -2 q^{-40} - q^{-42} + q^{-44} }[/math]
3 [math]\displaystyle{ -q^{39}+q^{37}+q^{35}-2 q^{31}+q^{29}+3 q^{27}-q^{25}-4 q^{23}-2 q^{21}+5 q^{19}+2 q^{17}-4 q^{15}-6 q^{13}+3 q^{11}+8 q^9+q^7-11 q^5-2 q^3+8 q+8 q^{-1} -5 q^{-3} -7 q^{-5} +2 q^{-7} +7 q^{-9} +6 q^{-11} -2 q^{-13} -6 q^{-15} - q^{-17} +9 q^{-19} +3 q^{-21} -8 q^{-23} -6 q^{-25} +7 q^{-27} +3 q^{-29} -7 q^{-31} -5 q^{-33} +6 q^{-35} +5 q^{-37} -3 q^{-39} -6 q^{-41} + q^{-43} +5 q^{-45} +2 q^{-47} -4 q^{-49} -6 q^{-51} + q^{-53} +8 q^{-55} +3 q^{-57} -6 q^{-59} -7 q^{-61} +4 q^{-63} +9 q^{-65} - q^{-67} -8 q^{-69} -3 q^{-71} +6 q^{-73} +5 q^{-75} -3 q^{-77} -4 q^{-79} +2 q^{-83} + q^{-85} - q^{-87} }[/math]
4 [math]\displaystyle{ q^{64}-q^{62}-q^{60}+3 q^{54}-3 q^{52}-q^{50}+q^{48}+2 q^{46}+7 q^{44}-7 q^{42}-6 q^{40}-q^{38}+7 q^{36}+15 q^{34}-10 q^{32}-16 q^{30}-8 q^{28}+14 q^{26}+29 q^{24}-9 q^{22}-29 q^{20}-21 q^{18}+19 q^{16}+46 q^{14}-39 q^{10}-39 q^8+15 q^6+57 q^4+19 q^2-30-48 q^{-2} -7 q^{-4} +43 q^{-6} +36 q^{-8} -33 q^{-12} -28 q^{-14} +6 q^{-16} +23 q^{-18} +25 q^{-20} +3 q^{-22} -25 q^{-24} -22 q^{-26} + q^{-28} +25 q^{-30} +21 q^{-32} -14 q^{-34} -27 q^{-36} -7 q^{-38} +21 q^{-40} +24 q^{-42} -13 q^{-44} -32 q^{-46} -6 q^{-48} +25 q^{-50} +29 q^{-52} -10 q^{-54} -38 q^{-56} -13 q^{-58} +22 q^{-60} +35 q^{-62} +3 q^{-64} -34 q^{-66} -22 q^{-68} +6 q^{-70} +29 q^{-72} +19 q^{-74} -14 q^{-76} -18 q^{-78} -12 q^{-80} +7 q^{-82} +17 q^{-84} +4 q^{-86} +3 q^{-88} -11 q^{-90} -13 q^{-92} - q^{-94} +2 q^{-96} +19 q^{-98} +8 q^{-100} -7 q^{-102} -13 q^{-104} -18 q^{-106} +10 q^{-108} +17 q^{-110} +10 q^{-112} -2 q^{-114} -21 q^{-116} -7 q^{-118} +4 q^{-120} +12 q^{-122} +11 q^{-124} -7 q^{-126} -7 q^{-128} -5 q^{-130} + q^{-132} +6 q^{-134} + q^{-136} -2 q^{-140} - q^{-142} + q^{-144} }[/math]
5 [math]\displaystyle{ -q^{95}+q^{93}+q^{91}-q^{85}-q^{83}+q^{81}+q^{79}-2 q^{77}-q^{75}-2 q^{73}+2 q^{71}+6 q^{69}+3 q^{67}-5 q^{65}-9 q^{63}-5 q^{61}+6 q^{59}+16 q^{57}+12 q^{55}-9 q^{53}-25 q^{51}-16 q^{49}+13 q^{47}+30 q^{45}+22 q^{43}-12 q^{41}-44 q^{39}-26 q^{37}+24 q^{35}+52 q^{33}+23 q^{31}-34 q^{29}-68 q^{27}-28 q^{25}+56 q^{23}+86 q^{21}+27 q^{19}-73 q^{17}-115 q^{15}-36 q^{13}+87 q^{11}+139 q^9+59 q^7-81 q^5-162 q^3-91 q+67 q^{-1} +160 q^{-3} +122 q^{-5} -14 q^{-7} -138 q^{-9} -140 q^{-11} -30 q^{-13} +93 q^{-15} +132 q^{-17} +75 q^{-19} -28 q^{-21} -102 q^{-23} -99 q^{-25} -25 q^{-27} +60 q^{-29} +93 q^{-31} +66 q^{-33} -15 q^{-35} -85 q^{-37} -80 q^{-39} -9 q^{-41} +65 q^{-43} +81 q^{-45} +18 q^{-47} -60 q^{-49} -76 q^{-51} -14 q^{-53} +61 q^{-55} +75 q^{-57} +8 q^{-59} -72 q^{-61} -83 q^{-63} -10 q^{-65} +78 q^{-67} +100 q^{-69} +19 q^{-71} -84 q^{-73} -115 q^{-75} -42 q^{-77} +76 q^{-79} +134 q^{-81} +71 q^{-83} -60 q^{-85} -138 q^{-87} -98 q^{-89} +26 q^{-91} +134 q^{-93} +128 q^{-95} +9 q^{-97} -114 q^{-99} -138 q^{-101} -50 q^{-103} +77 q^{-105} +135 q^{-107} +81 q^{-109} -33 q^{-111} -114 q^{-113} -95 q^{-115} -9 q^{-117} +73 q^{-119} +91 q^{-121} +42 q^{-123} -30 q^{-125} -67 q^{-127} -49 q^{-129} -6 q^{-131} +29 q^{-133} +37 q^{-135} +25 q^{-137} +3 q^{-139} -14 q^{-141} -18 q^{-143} -21 q^{-145} -15 q^{-147} - q^{-149} +19 q^{-151} +29 q^{-153} +22 q^{-155} +3 q^{-157} -25 q^{-159} -37 q^{-161} -23 q^{-163} +9 q^{-165} +32 q^{-167} +33 q^{-169} +14 q^{-171} -17 q^{-173} -33 q^{-175} -25 q^{-177} +19 q^{-181} +24 q^{-183} +15 q^{-185} -6 q^{-187} -17 q^{-189} -14 q^{-191} -3 q^{-193} +5 q^{-195} +9 q^{-197} +7 q^{-199} - q^{-201} -4 q^{-203} -3 q^{-205} - q^{-207} +2 q^{-211} + q^{-213} - q^{-215} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{10}-q^4+2 q^2+1+ q^{-2} + q^{-4} + q^{-8} + q^{-14} - q^{-16} - q^{-22} }[/math]
1,1 [math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-6 q^{22}+9 q^{20}-12 q^{18}+14 q^{16}-18 q^{14}+17 q^{12}-20 q^{10}+16 q^8-16 q^6+12 q^4-8 q^2+8-2 q^{-2} +10 q^{-4} -4 q^{-6} +10 q^{-8} -2 q^{-10} + q^{-12} +12 q^{-14} -24 q^{-16} +38 q^{-18} -53 q^{-20} +62 q^{-22} -70 q^{-24} +68 q^{-26} -65 q^{-28} +52 q^{-30} -38 q^{-32} +18 q^{-34} + q^{-36} -20 q^{-38} +36 q^{-40} -46 q^{-42} +49 q^{-44} -46 q^{-46} +40 q^{-48} -30 q^{-50} +19 q^{-52} -12 q^{-54} +6 q^{-56} -2 q^{-58} + q^{-60} }[/math]
2,0 [math]\displaystyle{ q^{26}-q^{22}+q^{18}-q^{16}-2 q^{14}+q^{12}+q^{10}-4 q^8-3 q^6+3 q^4-1+2 q^{-2} +5 q^{-4} +3 q^{-6} + q^{-8} +3 q^{-10} +2 q^{-12} +2 q^{-16} -2 q^{-20} - q^{-22} + q^{-24} - q^{-26} -2 q^{-28} - q^{-30} + q^{-32} -3 q^{-36} - q^{-38} + q^{-40} + q^{-42} - q^{-44} - q^{-46} +2 q^{-48} + q^{-50} - q^{-52} - q^{-54} + q^{-58} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{22}-q^{20}+2 q^{16}-3 q^{14}-3 q^{12}+2 q^{10}-2 q^8-3 q^6+4 q^4+2 q^2+3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} +2 q^{-10} +2 q^{-12} - q^{-14} + q^{-16} +3 q^{-18} -2 q^{-20} +2 q^{-24} -2 q^{-26} - q^{-28} -3 q^{-32} - q^{-34} + q^{-36} -2 q^{-38} + q^{-40} + q^{-42} - q^{-44} + q^{-46} }[/math]
1,0,0 [math]\displaystyle{ -q^{13}-q^9-q^5+2 q^3+q+2 q^{-1} + q^{-3} + q^{-5} + q^{-11} + q^{-15} + q^{-19} - q^{-21} - q^{-25} - q^{-29} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{22}+q^{20}-2 q^{18}+2 q^{16}-3 q^{14}+3 q^{12}-4 q^{10}+4 q^8-3 q^6+4 q^4-2 q^2+2+ q^{-2} - q^{-4} +5 q^{-6} -5 q^{-8} +8 q^{-10} -8 q^{-12} +9 q^{-14} -9 q^{-16} +7 q^{-18} -6 q^{-20} +4 q^{-22} -2 q^{-24} +3 q^{-28} -4 q^{-30} +5 q^{-32} -5 q^{-34} +5 q^{-36} -4 q^{-38} +3 q^{-40} -3 q^{-42} + q^{-44} - q^{-46} }[/math]
1,0 [math]\displaystyle{ q^{36}-q^{32}-q^{30}+q^{28}+2 q^{26}-3 q^{22}-3 q^{20}+3 q^{16}+q^{14}-3 q^{12}-3 q^{10}+4 q^6+2 q^4-q^2-2+2 q^{-2} +4 q^{-4} +2 q^{-6} -3 q^{-8} - q^{-10} +3 q^{-12} +4 q^{-14} - q^{-16} -2 q^{-18} + q^{-20} +3 q^{-22} -2 q^{-26} + q^{-28} +3 q^{-30} -4 q^{-34} -2 q^{-36} +3 q^{-38} +4 q^{-40} - q^{-42} -6 q^{-44} -2 q^{-46} +4 q^{-48} +3 q^{-50} -3 q^{-52} -5 q^{-54} +4 q^{-58} + q^{-60} -3 q^{-62} -2 q^{-64} +2 q^{-66} +2 q^{-68} - q^{-70} - q^{-72} + q^{-76} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-3 q^{40}+4 q^{38}-5 q^{36}+4 q^{34}-4 q^{32}+3 q^{28}-6 q^{26}+7 q^{24}-7 q^{22}+4 q^{20}-2 q^{18}-2 q^{16}+5 q^{14}-5 q^{12}+6 q^{10}-2 q^8+2 q^6+q^2+2- q^{-2} +5 q^{-4} -4 q^{-6} +4 q^{-8} - q^{-10} -2 q^{-12} +4 q^{-14} -3 q^{-16} +2 q^{-18} +2 q^{-20} -4 q^{-22} +3 q^{-24} + q^{-26} -6 q^{-28} +11 q^{-30} -11 q^{-32} +7 q^{-34} + q^{-36} -7 q^{-38} +15 q^{-40} -15 q^{-42} +12 q^{-44} -6 q^{-46} -2 q^{-48} +8 q^{-50} -10 q^{-52} +11 q^{-54} -6 q^{-56} +2 q^{-58} +3 q^{-60} -6 q^{-62} +5 q^{-64} - q^{-66} -6 q^{-68} +9 q^{-70} -9 q^{-72} +4 q^{-74} +4 q^{-76} -12 q^{-78} +16 q^{-80} -15 q^{-82} +7 q^{-84} -10 q^{-88} +12 q^{-90} -12 q^{-92} +8 q^{-94} -2 q^{-96} -2 q^{-98} +3 q^{-100} -4 q^{-102} +3 q^{-104} - q^{-106} + q^{-108} }[/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 34"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t^2-9 t+13-9 t^{-1} +3 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 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]= { 37, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^7+2 q^6-3 q^5+4 q^4-5 q^3+6 q^2-5 q+5-3 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^4 a^{-2} +z^4 a^{-4} +z^4-a^2 z^2+z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} +2 z^2-a^2+ a^{-4} - a^{-6} +2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +4 z^8 a^{-4} +2 z^8 a^{-6} +2 z^7 a^{-1} -z^7 a^{-3} -2 z^7 a^{-5} +z^7 a^{-7} -5 z^6 a^{-2} -17 z^6 a^{-4} -10 z^6 a^{-6} +2 z^6+2 a z^5-2 z^5 a^{-1} -5 z^5 a^{-3} -6 z^5 a^{-5} -5 z^5 a^{-7} +2 a^2 z^4+4 z^4 a^{-2} +20 z^4 a^{-4} +14 z^4 a^{-6} +a^3 z^3-z^3 a^{-1} +5 z^3 a^{-3} +12 z^3 a^{-5} +7 z^3 a^{-7} -2 a^2 z^2-3 z^2 a^{-2} -8 z^2 a^{-4} -6 z^2 a^{-6} -3 z^2-a^3 z-a z-z a^{-3} -4 z a^{-5} -3 z a^{-7} +a^2+ a^{-4} + a^{-6} +2 }[/math]
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