10 110

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-8 t^2+20 t-25+20 t^{-1} -8 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6-2 z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 83, -2 }
Jones polynomial [math]\displaystyle{ q^3-3 q^2+7 q-10+13 q^{-1} -14 q^{-2} +13 q^{-3} -11 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^6+a^6-2 z^4 a^4-3 z^2 a^4-a^4+z^6 a^2+2 z^4 a^2+z^2 a^2-2 z^4-3 z^2+z^2 a^{-2} + a^{-2} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 a^3 z^9+2 a z^9+6 a^4 z^8+10 a^2 z^8+4 z^8+8 a^5 z^7+9 a^3 z^7+4 a z^7+3 z^7 a^{-1} +6 a^6 z^6-5 a^4 z^6-20 a^2 z^6+z^6 a^{-2} -8 z^6+3 a^7 z^5-13 a^5 z^5-27 a^3 z^5-19 a z^5-8 z^5 a^{-1} +a^8 z^4-7 a^6 z^4-4 a^4 z^4+8 a^2 z^4-3 z^4 a^{-2} +z^4-2 a^7 z^3+12 a^5 z^3+21 a^3 z^3+13 a z^3+6 z^3 a^{-1} -a^8 z^2+5 a^6 z^2+6 a^4 z^2-a^2 z^2+3 z^2 a^{-2} +2 z^2-4 a^5 z-6 a^3 z-3 a z-z a^{-1} -a^6-a^4- a^{-2} }[/math]
The A2 invariant [math]\displaystyle{ q^{22}-q^{18}+3 q^{16}-2 q^{14}+q^{10}-3 q^8+2 q^6-3 q^4+2 q^2+1- q^{-2} +3 q^{-4} - q^{-6} + q^{-10} }[/math]
The G2 invariant [math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+6 q^{106}-5 q^{104}+10 q^{100}-20 q^{98}+31 q^{96}-39 q^{94}+35 q^{92}-20 q^{90}-10 q^{88}+55 q^{86}-94 q^{84}+121 q^{82}-118 q^{80}+71 q^{78}+10 q^{76}-112 q^{74}+200 q^{72}-226 q^{70}+178 q^{68}-61 q^{66}-84 q^{64}+200 q^{62}-231 q^{60}+167 q^{58}-31 q^{56}-116 q^{54}+198 q^{52}-176 q^{50}+53 q^{48}+118 q^{46}-250 q^{44}+281 q^{42}-194 q^{40}+14 q^{38}+182 q^{36}-325 q^{34}+358 q^{32}-275 q^{30}+101 q^{28}+99 q^{26}-256 q^{24}+317 q^{22}-265 q^{20}+124 q^{18}+44 q^{16}-179 q^{14}+221 q^{12}-157 q^{10}+20 q^8+134 q^6-223 q^4+206 q^2-87-82 q^{-2} +226 q^{-4} -279 q^{-6} +228 q^{-8} -96 q^{-10} -60 q^{-12} +176 q^{-14} -213 q^{-16} +180 q^{-18} -94 q^{-20} +3 q^{-22} +60 q^{-24} -87 q^{-26} +76 q^{-28} -46 q^{-30} +19 q^{-32} +3 q^{-34} -12 q^{-36} +13 q^{-38} -10 q^{-40} +6 q^{-42} -2 q^{-44} + q^{-46} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_110.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{15}-2 q^{13}+4 q^{11}-4 q^9+2 q^7-q^5-q^3+3 q-3 q^{-1} +4 q^{-3} -2 q^{-5} + q^{-7} }[/math]
2 [math]\displaystyle{ q^{42}-2 q^{40}+q^{38}+5 q^{36}-11 q^{34}+3 q^{32}+19 q^{30}-26 q^{28}-5 q^{26}+37 q^{24}-23 q^{22}-19 q^{20}+32 q^{18}-2 q^{16}-22 q^{14}+8 q^{12}+19 q^{10}-13 q^8-18 q^6+28 q^4+3 q^2-35+22 q^{-2} +20 q^{-4} -33 q^{-6} +4 q^{-8} +23 q^{-10} -14 q^{-12} -6 q^{-14} +9 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} }[/math]
3 [math]\displaystyle{ q^{81}-2 q^{79}+q^{77}+2 q^{75}-2 q^{73}-5 q^{71}+6 q^{69}+13 q^{67}-18 q^{65}-30 q^{63}+34 q^{61}+63 q^{59}-40 q^{57}-122 q^{55}+31 q^{53}+189 q^{51}+8 q^{49}-235 q^{47}-84 q^{45}+253 q^{43}+158 q^{41}-218 q^{39}-215 q^{37}+144 q^{35}+239 q^{33}-52 q^{31}-227 q^{29}-37 q^{27}+190 q^{25}+109 q^{23}-140 q^{21}-169 q^{19}+90 q^{17}+211 q^{15}-36 q^{13}-244 q^{11}-16 q^9+257 q^7+86 q^5-250 q^3-154 q+213 q^{-1} +210 q^{-3} -142 q^{-5} -241 q^{-7} +54 q^{-9} +233 q^{-11} +27 q^{-13} -182 q^{-15} -81 q^{-17} +110 q^{-19} +99 q^{-21} -48 q^{-23} -77 q^{-25} +3 q^{-27} +46 q^{-29} +12 q^{-31} -20 q^{-33} -9 q^{-35} +6 q^{-37} +4 q^{-39} - q^{-41} -2 q^{-43} + q^{-45} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{22}-q^{18}+3 q^{16}-2 q^{14}+q^{10}-3 q^8+2 q^6-3 q^4+2 q^2+1- q^{-2} +3 q^{-4} - q^{-6} + q^{-10} }[/math]
2,0 [math]\displaystyle{ q^{56}-q^{52}+q^{50}+2 q^{48}-q^{46}-6 q^{44}+3 q^{42}+10 q^{40}-9 q^{38}-11 q^{36}+7 q^{34}+15 q^{32}-10 q^{30}-16 q^{28}+14 q^{26}+10 q^{24}-10 q^{22}-4 q^{20}+12 q^{18}-2 q^{16}-3 q^{14}+6 q^{12}-q^{10}-9 q^8+2 q^6+12 q^4-12 q^2-10+15 q^{-2} +9 q^{-4} -13 q^{-6} -7 q^{-8} +11 q^{-10} +7 q^{-12} -9 q^{-14} -6 q^{-16} +6 q^{-18} +5 q^{-20} - q^{-22} -2 q^{-24} + q^{-28} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{48}-2 q^{46}+6 q^{42}-7 q^{40}-4 q^{38}+18 q^{36}-12 q^{34}-13 q^{32}+27 q^{30}-13 q^{28}-16 q^{26}+27 q^{24}-6 q^{22}-11 q^{20}+12 q^{18}+3 q^{16}-5 q^{14}-10 q^{12}+9 q^{10}+6 q^8-23 q^6+9 q^4+18 q^2-25+8 q^{-2} +17 q^{-4} -20 q^{-6} +8 q^{-8} +8 q^{-10} -9 q^{-12} +4 q^{-14} +2 q^{-16} -2 q^{-18} + q^{-20} }[/math]
1,0,0 [math]\displaystyle{ q^{29}+q^{25}-q^{23}+3 q^{21}-3 q^{19}+2 q^{17}-2 q^{15}+q^{13}-2 q^{11}-2 q^5+2 q^3-q+3 q^{-1} -2 q^{-3} +3 q^{-5} - q^{-7} + q^{-9} + q^{-13} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{48}-2 q^{46}+4 q^{44}-8 q^{42}+13 q^{40}-18 q^{38}+26 q^{36}-30 q^{34}+33 q^{32}-31 q^{30}+25 q^{28}-14 q^{26}-q^{24}+16 q^{22}-33 q^{20}+46 q^{18}-59 q^{16}+63 q^{14}-62 q^{12}+55 q^{10}-42 q^8+27 q^6-9 q^4-6 q^2+19-28 q^{-2} +33 q^{-4} -32 q^{-6} +30 q^{-8} -24 q^{-10} +17 q^{-12} -10 q^{-14} +6 q^{-16} -2 q^{-18} + q^{-20} }[/math]
1,0 [math]\displaystyle{ q^{78}-2 q^{74}-2 q^{72}+2 q^{70}+7 q^{68}+2 q^{66}-10 q^{64}-11 q^{62}+5 q^{60}+22 q^{58}+7 q^{56}-23 q^{54}-23 q^{52}+12 q^{50}+33 q^{48}+3 q^{46}-33 q^{44}-17 q^{42}+25 q^{40}+26 q^{38}-13 q^{36}-27 q^{34}+5 q^{32}+27 q^{30}+4 q^{28}-24 q^{26}-8 q^{24}+19 q^{22}+12 q^{20}-17 q^{18}-17 q^{16}+14 q^{14}+21 q^{12}-11 q^{10}-30 q^8+q^6+33 q^4+15 q^2-28-29 q^{-2} +16 q^{-4} +35 q^{-6} + q^{-8} -28 q^{-10} -14 q^{-12} +18 q^{-14} +18 q^{-16} -5 q^{-18} -13 q^{-20} -2 q^{-22} +7 q^{-24} +4 q^{-26} -2 q^{-28} -2 q^{-30} + q^{-34} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+6 q^{106}-5 q^{104}+10 q^{100}-20 q^{98}+31 q^{96}-39 q^{94}+35 q^{92}-20 q^{90}-10 q^{88}+55 q^{86}-94 q^{84}+121 q^{82}-118 q^{80}+71 q^{78}+10 q^{76}-112 q^{74}+200 q^{72}-226 q^{70}+178 q^{68}-61 q^{66}-84 q^{64}+200 q^{62}-231 q^{60}+167 q^{58}-31 q^{56}-116 q^{54}+198 q^{52}-176 q^{50}+53 q^{48}+118 q^{46}-250 q^{44}+281 q^{42}-194 q^{40}+14 q^{38}+182 q^{36}-325 q^{34}+358 q^{32}-275 q^{30}+101 q^{28}+99 q^{26}-256 q^{24}+317 q^{22}-265 q^{20}+124 q^{18}+44 q^{16}-179 q^{14}+221 q^{12}-157 q^{10}+20 q^8+134 q^6-223 q^4+206 q^2-87-82 q^{-2} +226 q^{-4} -279 q^{-6} +228 q^{-8} -96 q^{-10} -60 q^{-12} +176 q^{-14} -213 q^{-16} +180 q^{-18} -94 q^{-20} +3 q^{-22} +60 q^{-24} -87 q^{-26} +76 q^{-28} -46 q^{-30} +19 q^{-32} +3 q^{-34} -12 q^{-36} +13 q^{-38} -10 q^{-40} +6 q^{-42} -2 q^{-44} + q^{-46} }[/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["10 110"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-8 t^2+20 t-25+20 t^{-1} -8 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6-2 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]= { 83, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^3-3 q^2+7 q-10+13 q^{-1} -14 q^{-2} +13 q^{-3} -11 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^2 a^6+a^6-2 z^4 a^4-3 z^2 a^4-a^4+z^6 a^2+2 z^4 a^2+z^2 a^2-2 z^4-3 z^2+z^2 a^{-2} + a^{-2} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 a^3 z^9+2 a z^9+6 a^4 z^8+10 a^2 z^8+4 z^8+8 a^5 z^7+9 a^3 z^7+4 a z^7+3 z^7 a^{-1} +6 a^6 z^6-5 a^4 z^6-20 a^2 z^6+z^6 a^{-2} -8 z^6+3 a^7 z^5-13 a^5 z^5-27 a^3 z^5-19 a z^5-8 z^5 a^{-1} +a^8 z^4-7 a^6 z^4-4 a^4 z^4+8 a^2 z^4-3 z^4 a^{-2} +z^4-2 a^7 z^3+12 a^5 z^3+21 a^3 z^3+13 a z^3+6 z^3 a^{-1} -a^8 z^2+5 a^6 z^2+6 a^4 z^2-a^2 z^2+3 z^2 a^{-2} +2 z^2-4 a^5 z-6 a^3 z-3 a z-z a^{-1} -a^6-a^4- a^{-2} }[/math]
Retrieved from "https://katlas.org/index.php?title=10_110&oldid=4377"