8 13

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^2-7 t+11-7 t^{-1} +2 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 29, 0 }
Jones polynomial [math]\displaystyle{ -q^5+2 q^4-3 q^3+5 q^2-5 q+5-4 q^{-1} +3 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^{-2} +z^4-a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +z^2+2 a^{-2} - a^{-4} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +5 z^6 a^{-2} +2 z^6 a^{-4} +3 z^6+4 a z^5+4 z^5 a^{-1} +z^5 a^{-3} +z^5 a^{-5} +3 a^2 z^4-11 z^4 a^{-2} -6 z^4 a^{-4} -2 z^4+a^3 z^3-4 a z^3-9 z^3 a^{-1} -7 z^3 a^{-3} -3 z^3 a^{-5} -2 a^2 z^2+7 z^2 a^{-2} +5 z^2 a^{-4} +a z+3 z a^{-1} +4 z a^{-3} +2 z a^{-5} -2 a^{-2} - a^{-4} }[/math]
The A2 invariant [math]\displaystyle{ -q^{10}+q^8+q^6-q^4+q^2-1+ q^{-2} + q^{-4} + q^{-6} +2 q^{-8} - q^{-10} - q^{-16} }[/math]
The G2 invariant [math]\displaystyle{ q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+9 q^{38}-11 q^{36}+12 q^{34}-8 q^{32}+6 q^{28}-13 q^{26}+18 q^{24}-16 q^{22}+10 q^{20}-8 q^{16}+14 q^{14}-10 q^{12}+4 q^{10}+2 q^8-10 q^6+9 q^4-3 q^2-7+17 q^{-2} -22 q^{-4} +19 q^{-6} -6 q^{-8} -9 q^{-10} +19 q^{-12} -26 q^{-14} +26 q^{-16} -14 q^{-18} +3 q^{-20} +11 q^{-22} -16 q^{-24} +21 q^{-26} -10 q^{-28} + q^{-30} +6 q^{-32} -9 q^{-34} +8 q^{-36} -6 q^{-40} +14 q^{-42} -15 q^{-44} +9 q^{-46} + q^{-48} -13 q^{-50} +18 q^{-52} -19 q^{-54} +11 q^{-56} -4 q^{-58} -6 q^{-60} +11 q^{-62} -13 q^{-64} +10 q^{-66} -4 q^{-68} - q^{-70} +2 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 8_13.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^7+2 q^5-q^3+q+2 q^{-5} - q^{-7} + q^{-9} - q^{-11} }[/math]
2 [math]\displaystyle{ q^{20}-2 q^{18}-q^{16}+5 q^{14}-4 q^{12}-3 q^{10}+7 q^8-3 q^6-3 q^4+6 q^2+1-2 q^{-2} +3 q^{-6} - q^{-8} -5 q^{-10} +4 q^{-12} +3 q^{-14} -6 q^{-16} +3 q^{-18} +4 q^{-20} -5 q^{-22} +3 q^{-26} -2 q^{-28} - q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ -q^{39}+2 q^{37}+q^{35}-3 q^{33}-2 q^{31}+4 q^{29}+6 q^{27}-9 q^{25}-8 q^{23}+10 q^{21}+11 q^{19}-12 q^{17}-15 q^{15}+12 q^{13}+19 q^{11}-8 q^9-18 q^7+5 q^5+16 q^3+q-11 q^{-1} -6 q^{-3} +7 q^{-5} +10 q^{-7} - q^{-9} -12 q^{-11} -3 q^{-13} +14 q^{-15} +8 q^{-17} -15 q^{-19} -11 q^{-21} +13 q^{-23} +13 q^{-25} -10 q^{-27} -17 q^{-29} +8 q^{-31} +17 q^{-33} -2 q^{-35} -17 q^{-37} - q^{-39} +14 q^{-41} +5 q^{-43} -10 q^{-45} -7 q^{-47} +5 q^{-49} +6 q^{-51} -2 q^{-53} -4 q^{-55} +2 q^{-59} + q^{-61} - q^{-63} }[/math]
4 [math]\displaystyle{ q^{64}-2 q^{62}-q^{60}+3 q^{58}+2 q^{54}-7 q^{52}-q^{50}+10 q^{48}+q^{46}+3 q^{44}-21 q^{42}-6 q^{40}+26 q^{38}+13 q^{36}-47 q^{32}-19 q^{30}+43 q^{28}+40 q^{26}+11 q^{24}-67 q^{22}-47 q^{20}+36 q^{18}+60 q^{16}+35 q^{14}-55 q^{12}-60 q^{10}+6 q^8+46 q^6+49 q^4-18 q^2-43-22 q^{-2} +11 q^{-4} +39 q^{-6} +19 q^{-8} -13 q^{-10} -39 q^{-12} -19 q^{-14} +24 q^{-16} +43 q^{-18} +9 q^{-20} -44 q^{-22} -35 q^{-24} +9 q^{-26} +59 q^{-28} +26 q^{-30} -46 q^{-32} -47 q^{-34} -7 q^{-36} +64 q^{-38} +41 q^{-40} -34 q^{-42} -52 q^{-44} -32 q^{-46} +51 q^{-48} +56 q^{-50} -4 q^{-52} -42 q^{-54} -54 q^{-56} +18 q^{-58} +50 q^{-60} +26 q^{-62} -12 q^{-64} -51 q^{-66} -12 q^{-68} +21 q^{-70} +29 q^{-72} +14 q^{-74} -25 q^{-76} -18 q^{-78} -4 q^{-80} +12 q^{-82} +16 q^{-84} -4 q^{-86} -6 q^{-88} -6 q^{-90} +6 q^{-94} + q^{-96} -2 q^{-100} - q^{-102} + q^{-104} }[/math]
5 [math]\displaystyle{ -q^{95}+2 q^{93}+q^{91}-3 q^{89}+q^{83}+2 q^{81}-6 q^{77}-4 q^{75}+7 q^{73}+12 q^{71}+4 q^{69}-13 q^{67}-20 q^{65}-14 q^{63}+20 q^{61}+52 q^{59}+24 q^{57}-36 q^{55}-78 q^{53}-56 q^{51}+38 q^{49}+124 q^{47}+100 q^{45}-36 q^{43}-161 q^{41}-154 q^{39}+5 q^{37}+184 q^{35}+211 q^{33}+41 q^{31}-183 q^{29}-251 q^{27}-94 q^{25}+149 q^{23}+266 q^{21}+146 q^{19}-95 q^{17}-245 q^{15}-175 q^{13}+34 q^{11}+195 q^9+183 q^7+27 q^5-131 q^3-166 q-71 q^{-1} +63 q^{-3} +132 q^{-5} +100 q^{-7} + q^{-9} -98 q^{-11} -115 q^{-13} -48 q^{-15} +65 q^{-17} +128 q^{-19} +82 q^{-21} -39 q^{-23} -136 q^{-25} -111 q^{-27} +27 q^{-29} +148 q^{-31} +130 q^{-33} -18 q^{-35} -157 q^{-37} -154 q^{-39} +6 q^{-41} +171 q^{-43} +177 q^{-45} +11 q^{-47} -176 q^{-49} -200 q^{-51} -41 q^{-53} +166 q^{-55} +225 q^{-57} +77 q^{-59} -142 q^{-61} -231 q^{-63} -119 q^{-65} +90 q^{-67} +226 q^{-69} +164 q^{-71} -37 q^{-73} -191 q^{-75} -185 q^{-77} -36 q^{-79} +139 q^{-81} +190 q^{-83} +86 q^{-85} -73 q^{-87} -162 q^{-89} -121 q^{-91} +8 q^{-93} +117 q^{-95} +127 q^{-97} +42 q^{-99} -62 q^{-101} -107 q^{-103} -66 q^{-105} +13 q^{-107} +70 q^{-109} +69 q^{-111} +18 q^{-113} -35 q^{-115} -53 q^{-117} -29 q^{-119} +7 q^{-121} +30 q^{-123} +28 q^{-125} +6 q^{-127} -14 q^{-129} -17 q^{-131} -7 q^{-133} +2 q^{-135} +8 q^{-137} +8 q^{-139} -4 q^{-143} -3 q^{-145} - q^{-147} +2 q^{-151} + q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{10}+q^8+q^6-q^4+q^2-1+ q^{-2} + q^{-4} + q^{-6} +2 q^{-8} - q^{-10} - q^{-16} }[/math]
1,1 [math]\displaystyle{ q^{28}-4 q^{26}+8 q^{24}-12 q^{22}+20 q^{20}-32 q^{18}+38 q^{16}-44 q^{14}+51 q^{12}-52 q^{10}+44 q^8-30 q^6+13 q^4+12 q^2-36+66 q^{-2} -82 q^{-4} +100 q^{-6} -102 q^{-8} +98 q^{-10} -86 q^{-12} +66 q^{-14} -42 q^{-16} +12 q^{-18} +10 q^{-20} -28 q^{-22} +46 q^{-24} -54 q^{-26} +55 q^{-28} -48 q^{-30} +40 q^{-32} -32 q^{-34} +19 q^{-36} -12 q^{-38} +6 q^{-40} -2 q^{-42} + q^{-44} }[/math]
2,0 [math]\displaystyle{ q^{26}-q^{24}-2 q^{22}+q^{20}+3 q^{18}-5 q^{14}+3 q^{10}-2 q^8-2 q^6+4 q^4+4 q^2+1+ q^{-2} +2 q^{-4} - q^{-6} -2 q^{-8} + q^{-10} - q^{-12} -3 q^{-14} +3 q^{-16} +4 q^{-18} - q^{-20} -2 q^{-22} +2 q^{-24} +2 q^{-26} -2 q^{-28} -3 q^{-30} + q^{-32} + q^{-34} - q^{-36} - q^{-38} + q^{-42} }[/math]

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+9 q^{38}-11 q^{36}+12 q^{34}-8 q^{32}+6 q^{28}-13 q^{26}+18 q^{24}-16 q^{22}+10 q^{20}-8 q^{16}+14 q^{14}-10 q^{12}+4 q^{10}+2 q^8-10 q^6+9 q^4-3 q^2-7+17 q^{-2} -22 q^{-4} +19 q^{-6} -6 q^{-8} -9 q^{-10} +19 q^{-12} -26 q^{-14} +26 q^{-16} -14 q^{-18} +3 q^{-20} +11 q^{-22} -16 q^{-24} +21 q^{-26} -10 q^{-28} + q^{-30} +6 q^{-32} -9 q^{-34} +8 q^{-36} -6 q^{-40} +14 q^{-42} -15 q^{-44} +9 q^{-46} + q^{-48} -13 q^{-50} +18 q^{-52} -19 q^{-54} +11 q^{-56} -4 q^{-58} -6 q^{-60} +11 q^{-62} -13 q^{-64} +10 q^{-66} -4 q^{-68} - q^{-70} +2 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/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["8 13"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^2-7 t+11-7 t^{-1} +2 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 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{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 29, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+2 q^4-3 q^3+5 q^2-5 q+5-4 q^{-1} +3 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^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +z^2+2 a^{-2} - a^{-4} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +5 z^6 a^{-2} +2 z^6 a^{-4} +3 z^6+4 a z^5+4 z^5 a^{-1} +z^5 a^{-3} +z^5 a^{-5} +3 a^2 z^4-11 z^4 a^{-2} -6 z^4 a^{-4} -2 z^4+a^3 z^3-4 a z^3-9 z^3 a^{-1} -7 z^3 a^{-3} -3 z^3 a^{-5} -2 a^2 z^2+7 z^2 a^{-2} +5 z^2 a^{-4} +a z+3 z a^{-1} +4 z a^{-3} +2 z a^{-5} -2 a^{-2} - a^{-4} }[/math]
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