10 107

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+8 t^2-22 t+31-22 t^{-1} +8 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{ \{1\} }[/math]
Determinant and Signature	{ 93, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-7 q^3+12 q^2-14 q+16-15 q^{-1} +12 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -2 z^4-a^4 z^2+2 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} -2 z^2+2 a^{-2} - a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+5 z^8 a^{-2} +11 z^8+7 a^3 z^7+11 a z^7+9 z^7 a^{-1} +5 z^7 a^{-3} +4 a^4 z^6-5 a^2 z^6-4 z^6 a^{-2} +3 z^6 a^{-4} -16 z^6+a^5 z^5-12 a^3 z^5-27 a z^5-22 z^5 a^{-1} -7 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} -5 z^4 a^{-4} +5 z^4-a^5 z^3+6 a^3 z^3+17 a z^3+15 z^3 a^{-1} +3 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2+2 a^2 z^2+3 z^2 a^{-2} +3 z^2 a^{-4} -a^3 z-3 a z-3 z a^{-1} +z a^{-5} -2 a^{-2} - a^{-4} }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-3 q^{10}+2 q^8-q^6-2 q^4+3 q^2-2+4 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} + q^{-12} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-14 q^{70}+3 q^{68}+22 q^{66}-52 q^{64}+86 q^{62}-103 q^{60}+81 q^{58}-21 q^{56}-81 q^{54}+193 q^{52}-265 q^{50}+263 q^{48}-160 q^{46}-20 q^{44}+222 q^{42}-364 q^{40}+386 q^{38}-262 q^{36}+37 q^{34}+184 q^{32}-320 q^{30}+303 q^{28}-144 q^{26}-75 q^{24}+258 q^{22}-309 q^{20}+196 q^{18}+30 q^{16}-286 q^{14}+447 q^{12}-447 q^{10}+279 q^8-290 q^4+500 q^2-540+409 q^{-2} -151 q^{-4} -144 q^{-6} +361 q^{-8} -422 q^{-10} +318 q^{-12} -95 q^{-14} -130 q^{-16} +276 q^{-18} -268 q^{-20} +115 q^{-22} +106 q^{-24} -293 q^{-26} +355 q^{-28} -262 q^{-30} +54 q^{-32} +177 q^{-34} -333 q^{-36} +372 q^{-38} -279 q^{-40} +115 q^{-42} +54 q^{-44} -183 q^{-46} +223 q^{-48} -191 q^{-50} +117 q^{-52} -32 q^{-54} -29 q^{-56} +60 q^{-58} -67 q^{-60} +53 q^{-62} -33 q^{-64} +13 q^{-66} + q^{-68} -9 q^{-70} +9 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_107.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+3 q^9-4 q^7+4 q^5-3 q^3+q+2 q^{-1} -2 q^{-3} +5 q^{-5} -4 q^{-7} +2 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-3 q^{30}+11 q^{26}-13 q^{24}-11 q^{22}+34 q^{20}-13 q^{18}-36 q^{16}+44 q^{14}+6 q^{12}-46 q^{10}+26 q^8+22 q^6-29 q^4-6 q^2+24+4 q^{-2} -34 q^{-4} +15 q^{-6} +36 q^{-8} -43 q^{-10} -5 q^{-12} +47 q^{-14} -27 q^{-16} -19 q^{-18} +28 q^{-20} -5 q^{-22} -11 q^{-24} +7 q^{-26} -2 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+3 q^{61}-7 q^{57}-2 q^{55}+17 q^{53}+14 q^{51}-40 q^{49}-38 q^{47}+59 q^{45}+92 q^{43}-62 q^{41}-174 q^{39}+34 q^{37}+257 q^{35}+44 q^{33}-315 q^{31}-159 q^{29}+327 q^{27}+272 q^{25}-281 q^{23}-358 q^{21}+188 q^{19}+399 q^{17}-82 q^{15}-384 q^{13}-18 q^{11}+328 q^9+115 q^7-253 q^5-188 q^3+159 q+257 q^{-1} -64 q^{-3} -308 q^{-5} -47 q^{-7} +342 q^{-9} +162 q^{-11} -341 q^{-13} -267 q^{-15} +293 q^{-17} +351 q^{-19} -201 q^{-21} -384 q^{-23} +85 q^{-25} +365 q^{-27} +11 q^{-29} -282 q^{-31} -86 q^{-33} +188 q^{-35} +104 q^{-37} -100 q^{-39} -83 q^{-41} +37 q^{-43} +52 q^{-45} -10 q^{-47} -26 q^{-49} +3 q^{-51} +10 q^{-53} - q^{-55} -3 q^{-57} +2 q^{-61} - q^{-63} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-3 q^{32}+q^{30}+7 q^{28}-13 q^{26}+4 q^{24}+19 q^{22}-28 q^{20}+6 q^{18}+28 q^{16}-36 q^{14}+4 q^{12}+28 q^{10}-24 q^8-4 q^6+16 q^4-q^2-9-4 q^{-2} +21 q^{-4} -4 q^{-6} -23 q^{-8} +34 q^{-10} +2 q^{-12} -33 q^{-14} +30 q^{-16} -25 q^{-20} +16 q^{-22} + q^{-24} -10 q^{-26} +5 q^{-28} + q^{-30} -2 q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}+q^{19}+2 q^{15}-3 q^{13}+3 q^{11}-3 q^9+q^7-2 q^5+2 q^3+3 q^{-3} - q^{-5} +3 q^{-7} - q^{-9} +4 q^{-11} -3 q^{-13} + q^{-15} - q^{-17} - q^{-21} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+3 q^{32}-7 q^{30}+13 q^{28}-21 q^{26}+30 q^{24}-37 q^{22}+42 q^{20}-42 q^{18}+36 q^{16}-24 q^{14}+8 q^{12}+12 q^{10}-34 q^8+54 q^6-70 q^4+79 q^2-81+74 q^{-2} -59 q^{-4} +42 q^{-6} -19 q^{-8} +20 q^{-12} -31 q^{-14} +40 q^{-16} -42 q^{-18} +39 q^{-20} -32 q^{-22} +23 q^{-24} -16 q^{-26} +9 q^{-28} -5 q^{-30} +2 q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+10 q^{46}-17 q^{42}-12 q^{40}+18 q^{38}+28 q^{36}-6 q^{34}-39 q^{32}-15 q^{30}+37 q^{28}+34 q^{26}-22 q^{24}-44 q^{22}+q^{20}+42 q^{18}+15 q^{16}-32 q^{14}-22 q^{12}+22 q^{10}+25 q^8-14 q^6-27 q^4+7 q^2+29- q^{-2} -30 q^{-4} -5 q^{-6} +31 q^{-8} +17 q^{-10} -29 q^{-12} -27 q^{-14} +24 q^{-16} +43 q^{-18} -7 q^{-20} -45 q^{-22} -14 q^{-24} +37 q^{-26} +30 q^{-28} -19 q^{-30} -35 q^{-32} -2 q^{-34} +24 q^{-36} +12 q^{-38} -10 q^{-40} -13 q^{-42} +7 q^{-46} +3 q^{-48} -2 q^{-50} -2 q^{-52} + q^{-56} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-14 q^{70}+3 q^{68}+22 q^{66}-52 q^{64}+86 q^{62}-103 q^{60}+81 q^{58}-21 q^{56}-81 q^{54}+193 q^{52}-265 q^{50}+263 q^{48}-160 q^{46}-20 q^{44}+222 q^{42}-364 q^{40}+386 q^{38}-262 q^{36}+37 q^{34}+184 q^{32}-320 q^{30}+303 q^{28}-144 q^{26}-75 q^{24}+258 q^{22}-309 q^{20}+196 q^{18}+30 q^{16}-286 q^{14}+447 q^{12}-447 q^{10}+279 q^8-290 q^4+500 q^2-540+409 q^{-2} -151 q^{-4} -144 q^{-6} +361 q^{-8} -422 q^{-10} +318 q^{-12} -95 q^{-14} -130 q^{-16} +276 q^{-18} -268 q^{-20} +115 q^{-22} +106 q^{-24} -293 q^{-26} +355 q^{-28} -262 q^{-30} +54 q^{-32} +177 q^{-34} -333 q^{-36} +372 q^{-38} -279 q^{-40} +115 q^{-42} +54 q^{-44} -183 q^{-46} +223 q^{-48} -191 q^{-50} +117 q^{-52} -32 q^{-54} -29 q^{-56} +60 q^{-58} -67 q^{-60} +53 q^{-62} -33 q^{-64} +13 q^{-66} + q^{-68} -9 q^{-70} +9 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} }[/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["10 107"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+8 t^2-22 t+31-22 t^{-1} +8 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{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 93, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+3 q^4-7 q^3+12 q^2-14 q+16-15 q^{-1} +12 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -2 z^4-a^4 z^2+2 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} -2 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{ 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+5 z^8 a^{-2} +11 z^8+7 a^3 z^7+11 a z^7+9 z^7 a^{-1} +5 z^7 a^{-3} +4 a^4 z^6-5 a^2 z^6-4 z^6 a^{-2} +3 z^6 a^{-4} -16 z^6+a^5 z^5-12 a^3 z^5-27 a z^5-22 z^5 a^{-1} -7 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} -5 z^4 a^{-4} +5 z^4-a^5 z^3+6 a^3 z^3+17 a z^3+15 z^3 a^{-1} +3 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2+2 a^2 z^2+3 z^2 a^{-2} +3 z^2 a^{-4} -a^3 z-3 a z-3 z a^{-1} +z a^{-5} -2 a^{-2} - a^{-4} }[/math]