10 31

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t^2-14 t+21-14 t^{-1} +4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 57, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+2 q^4-4 q^3+7 q^2-8 q+10-9 q^{-1} +7 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4+z^4 a^2-a^2+2 z^4+3 z^2+2+z^4 a^{-2} +z^2 a^{-2} + a^{-2} -z^2 a^{-4} - a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^9+z^9 a^{-1} +3 a^2 z^8+2 z^8 a^{-2} +5 z^8+4 a^3 z^7+3 a z^7+z^7 a^{-1} +2 z^7 a^{-3} +3 a^4 z^6-6 a^2 z^6-3 z^6 a^{-2} +2 z^6 a^{-4} -14 z^6+a^5 z^5-10 a^3 z^5-12 a z^5-4 z^5 a^{-1} -2 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4+5 a^2 z^4+3 z^4 a^{-2} -5 z^4 a^{-4} +20 z^4-2 a^5 z^3+7 a^3 z^3+15 a z^3+6 z^3 a^{-1} -3 z^3 a^{-3} -3 z^3 a^{-5} +2 a^4 z^2-3 a^2 z^2-2 z^2 a^{-2} +3 z^2 a^{-4} -10 z^2-2 a^3 z-4 a z-2 z a^{-1} +2 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-2 q^{10}+q^8-q^6-q^4+2 q^2+3 q^{-2} + q^{-6} +2 q^{-8} -2 q^{-10} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+6 q^{72}-4 q^{70}-3 q^{68}+14 q^{66}-21 q^{64}+28 q^{62}-28 q^{60}+14 q^{58}+6 q^{56}-32 q^{54}+53 q^{52}-58 q^{50}+48 q^{48}-19 q^{46}-17 q^{44}+51 q^{42}-68 q^{40}+61 q^{38}-34 q^{36}-8 q^{34}+37 q^{32}-49 q^{30}+35 q^{28}-4 q^{26}-27 q^{24}+49 q^{22}-47 q^{20}+15 q^{18}+25 q^{16}-67 q^{14}+87 q^{12}-75 q^{10}+39 q^8+14 q^6-60 q^4+93 q^2-92+66 q^{-2} -20 q^{-4} -28 q^{-6} +61 q^{-8} -62 q^{-10} +44 q^{-12} -6 q^{-14} -22 q^{-16} +39 q^{-18} -32 q^{-20} +6 q^{-22} +27 q^{-24} -50 q^{-26} +56 q^{-28} -36 q^{-30} + q^{-32} +34 q^{-34} -54 q^{-36} +61 q^{-38} -47 q^{-40} +22 q^{-42} +3 q^{-44} -27 q^{-46} +36 q^{-48} -37 q^{-50} +28 q^{-52} -15 q^{-54} +2 q^{-56} +7 q^{-58} -15 q^{-60} +15 q^{-62} -13 q^{-64} +8 q^{-66} -3 q^{-68} -2 q^{-70} +3 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_31.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-2 q^7+2 q^5-2 q^3+q+2 q^{-1} - q^{-3} +3 q^{-5} -2 q^{-7} + q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+6 q^{26}-4 q^{24}-6 q^{22}+11 q^{20}-2 q^{18}-13 q^{16}+12 q^{14}+4 q^{12}-14 q^{10}+8 q^8+7 q^6-8 q^4-2 q^2+6+4 q^{-2} -11 q^{-4} +3 q^{-6} +13 q^{-8} -12 q^{-10} -2 q^{-12} +14 q^{-14} -8 q^{-16} -5 q^{-18} +8 q^{-20} -3 q^{-22} -3 q^{-24} +3 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+q^{59}-3 q^{57}-3 q^{55}+4 q^{53}+8 q^{51}-7 q^{49}-13 q^{47}+7 q^{45}+21 q^{43}-2 q^{41}-31 q^{39}-8 q^{37}+39 q^{35}+22 q^{33}-40 q^{31}-40 q^{29}+34 q^{27}+54 q^{25}-25 q^{23}-62 q^{21}+12 q^{19}+63 q^{17}+2 q^{15}-56 q^{13}-12 q^{11}+43 q^9+25 q^7-30 q^5-28 q^3+10 q+39 q^{-1} +7 q^{-3} -43 q^{-5} -24 q^{-7} +43 q^{-9} +40 q^{-11} -40 q^{-13} -51 q^{-15} +30 q^{-17} +59 q^{-19} -17 q^{-21} -54 q^{-23} + q^{-25} +49 q^{-27} +7 q^{-29} -33 q^{-31} -15 q^{-33} +21 q^{-35} +12 q^{-37} -11 q^{-39} -9 q^{-41} +5 q^{-43} +5 q^{-45} -3 q^{-47} -2 q^{-49} +2 q^{-51} + q^{-53} -2 q^{-55} + q^{-59} + q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}-q^{100}+3 q^{98}+3 q^{94}-7 q^{92}-3 q^{90}+9 q^{88}+9 q^{84}-20 q^{82}-15 q^{80}+21 q^{78}+15 q^{76}+30 q^{74}-39 q^{72}-54 q^{70}+4 q^{68}+34 q^{66}+102 q^{64}-7 q^{62}-104 q^{60}-91 q^{58}-16 q^{56}+184 q^{54}+121 q^{52}-58 q^{50}-199 q^{48}-175 q^{46}+161 q^{44}+257 q^{42}+100 q^{40}-198 q^{38}-326 q^{36}+22 q^{34}+272 q^{32}+244 q^{30}-85 q^{28}-343 q^{26}-108 q^{24}+172 q^{22}+267 q^{20}+37 q^{18}-236 q^{16}-163 q^{14}+46 q^{12}+202 q^{10}+108 q^8-95 q^6-170 q^4-62 q^2+117+163 q^{-2} +47 q^{-4} -179 q^{-6} -175 q^{-8} +31 q^{-10} +219 q^{-12} +194 q^{-14} -153 q^{-16} -275 q^{-18} -96 q^{-20} +212 q^{-22} +323 q^{-24} -43 q^{-26} -279 q^{-28} -225 q^{-30} +89 q^{-32} +334 q^{-34} +99 q^{-36} -144 q^{-38} -246 q^{-40} -68 q^{-42} +201 q^{-44} +141 q^{-46} +14 q^{-48} -141 q^{-50} -122 q^{-52} +50 q^{-54} +77 q^{-56} +71 q^{-58} -31 q^{-60} -74 q^{-62} -9 q^{-64} +8 q^{-66} +44 q^{-68} +9 q^{-70} -22 q^{-72} -6 q^{-74} -12 q^{-76} +13 q^{-78} +6 q^{-80} -3 q^{-82} +4 q^{-84} -7 q^{-86} + q^{-88} - q^{-92} +4 q^{-94} - q^{-96} - q^{-100} - q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+2 q^{153}+q^{151}-3 q^{149}+2 q^{141}+2 q^{139}-5 q^{137}-4 q^{135}+7 q^{133}+8 q^{131}+3 q^{129}-7 q^{127}-18 q^{125}-22 q^{123}+7 q^{121}+45 q^{119}+43 q^{117}+7 q^{115}-53 q^{113}-90 q^{111}-59 q^{109}+46 q^{107}+145 q^{105}+136 q^{103}+20 q^{101}-151 q^{99}-254 q^{97}-168 q^{95}+91 q^{93}+343 q^{91}+371 q^{89}+100 q^{87}-338 q^{85}-597 q^{83}-409 q^{81}+186 q^{79}+757 q^{77}+783 q^{75}+141 q^{73}-754 q^{71}-1150 q^{69}-598 q^{67}+565 q^{65}+1389 q^{63}+1088 q^{61}-199 q^{59}-1434 q^{57}-1513 q^{55}-266 q^{53}+1281 q^{51}+1770 q^{49}+718 q^{47}-970 q^{45}-1814 q^{43}-1072 q^{41}+581 q^{39}+1676 q^{37}+1267 q^{35}-211 q^{33}-1399 q^{31}-1290 q^{29}-101 q^{27}+1059 q^{25}+1210 q^{23}+310 q^{21}-744 q^{19}-1033 q^{17}-441 q^{15}+436 q^{13}+885 q^{11}+544 q^9-225 q^7-733 q^5-635 q^3-4 q+659 q^{-1} +780 q^{-3} +216 q^{-5} -604 q^{-7} -961 q^{-9} -477 q^{-11} +527 q^{-13} +1177 q^{-15} +799 q^{-17} -390 q^{-19} -1363 q^{-21} -1155 q^{-23} +145 q^{-25} +1439 q^{-27} +1513 q^{-29} +204 q^{-31} -1359 q^{-33} -1769 q^{-35} -621 q^{-37} +1090 q^{-39} +1859 q^{-41} +1011 q^{-43} -664 q^{-45} -1720 q^{-47} -1306 q^{-49} +178 q^{-51} +1403 q^{-53} +1375 q^{-55} +277 q^{-57} -927 q^{-59} -1281 q^{-61} -585 q^{-63} +470 q^{-65} +994 q^{-67} +703 q^{-69} -59 q^{-71} -662 q^{-73} -670 q^{-75} -179 q^{-77} +339 q^{-79} +514 q^{-81} +287 q^{-83} -104 q^{-85} -335 q^{-87} -278 q^{-89} -33 q^{-91} +181 q^{-93} +213 q^{-95} +85 q^{-97} -71 q^{-99} -138 q^{-101} -89 q^{-103} +13 q^{-105} +78 q^{-107} +68 q^{-109} +11 q^{-111} -37 q^{-113} -41 q^{-115} -20 q^{-117} +12 q^{-119} +27 q^{-121} +14 q^{-123} -3 q^{-125} -8 q^{-127} -10 q^{-129} -6 q^{-131} +5 q^{-133} +6 q^{-135} + q^{-137} +2 q^{-139} - q^{-141} -4 q^{-143} - q^{-145} + q^{-147} + q^{-151} + q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-2 q^{10}+q^8-q^6-q^4+2 q^2+3 q^{-2} + q^{-6} +2 q^{-8} -2 q^{-10} - q^{-16} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+10 q^{40}-22 q^{38}+42 q^{36}-66 q^{34}+98 q^{32}-142 q^{30}+182 q^{28}-214 q^{26}+240 q^{24}-248 q^{22}+228 q^{20}-176 q^{18}+94 q^{16}+2 q^{14}-118 q^{12}+238 q^{10}-342 q^8+424 q^6-469 q^4+480 q^2-448+388 q^{-2} -294 q^{-4} +186 q^{-6} -70 q^{-8} -26 q^{-10} +111 q^{-12} -174 q^{-14} +210 q^{-16} -214 q^{-18} +203 q^{-20} -184 q^{-22} +154 q^{-24} -124 q^{-26} +94 q^{-28} -72 q^{-30} +50 q^{-32} -34 q^{-34} +21 q^{-36} -12 q^{-38} +6 q^{-40} -2 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{40}-2 q^{38}+q^{36}+4 q^{34}+q^{32}-7 q^{30}-2 q^{28}+7 q^{26}+2 q^{24}-8 q^{22}-3 q^{20}+9 q^{18}+3 q^{16}-9 q^{14}-q^{12}+8 q^{10}-2 q^8-4 q^6+2 q^4+q^2-1+3 q^{-2} +4 q^{-4} -5 q^{-6} +11 q^{-10} +3 q^{-12} -10 q^{-14} +2 q^{-16} +9 q^{-18} - q^{-20} -9 q^{-22} -2 q^{-24} +5 q^{-26} - q^{-28} -4 q^{-30} +2 q^{-34} - q^{-38} + q^{-42} }[/math]

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-4 q^{30}+7 q^{28}-9 q^{26}+11 q^{24}-14 q^{22}+14 q^{20}-13 q^{18}+10 q^{16}-5 q^{14}+7 q^{10}-14 q^8+20 q^6-24 q^4+27 q^2-26+24 q^{-2} -18 q^{-4} +13 q^{-6} -5 q^{-8} +7 q^{-12} -9 q^{-14} +13 q^{-16} -13 q^{-18} +12 q^{-20} -11 q^{-22} +8 q^{-24} -6 q^{-26} +4 q^{-28} -3 q^{-30} + q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+2 q^{48}+5 q^{46}-q^{44}-7 q^{42}-3 q^{40}+8 q^{38}+10 q^{36}-5 q^{34}-14 q^{32}-3 q^{30}+13 q^{28}+10 q^{26}-9 q^{24}-14 q^{22}+q^{20}+13 q^{18}+4 q^{16}-10 q^{14}-6 q^{12}+7 q^{10}+7 q^8-5 q^6-8 q^4+4 q^2+9- q^{-2} -9 q^{-4} + q^{-6} +11 q^{-8} +5 q^{-10} -9 q^{-12} -6 q^{-14} +10 q^{-16} +14 q^{-18} -3 q^{-20} -14 q^{-22} -5 q^{-24} +12 q^{-26} +9 q^{-28} -6 q^{-30} -12 q^{-32} -3 q^{-34} +7 q^{-36} +4 q^{-38} -3 q^{-40} -5 q^{-42} - q^{-44} +3 q^{-46} +2 q^{-48} - q^{-50} - q^{-52} + q^{-56} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+6 q^{72}-4 q^{70}-3 q^{68}+14 q^{66}-21 q^{64}+28 q^{62}-28 q^{60}+14 q^{58}+6 q^{56}-32 q^{54}+53 q^{52}-58 q^{50}+48 q^{48}-19 q^{46}-17 q^{44}+51 q^{42}-68 q^{40}+61 q^{38}-34 q^{36}-8 q^{34}+37 q^{32}-49 q^{30}+35 q^{28}-4 q^{26}-27 q^{24}+49 q^{22}-47 q^{20}+15 q^{18}+25 q^{16}-67 q^{14}+87 q^{12}-75 q^{10}+39 q^8+14 q^6-60 q^4+93 q^2-92+66 q^{-2} -20 q^{-4} -28 q^{-6} +61 q^{-8} -62 q^{-10} +44 q^{-12} -6 q^{-14} -22 q^{-16} +39 q^{-18} -32 q^{-20} +6 q^{-22} +27 q^{-24} -50 q^{-26} +56 q^{-28} -36 q^{-30} + q^{-32} +34 q^{-34} -54 q^{-36} +61 q^{-38} -47 q^{-40} +22 q^{-42} +3 q^{-44} -27 q^{-46} +36 q^{-48} -37 q^{-50} +28 q^{-52} -15 q^{-54} +2 q^{-56} +7 q^{-58} -15 q^{-60} +15 q^{-62} -13 q^{-64} +8 q^{-66} -3 q^{-68} -2 q^{-70} +3 q^{-72} -4 q^{-74} +3 q^{-76} - 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 31"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 4 t^2-14 t+21-14 t^{-1} +4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 4 z^4+2 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]=

{ 57, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+2 q^4-4 q^3+7 q^2-8 q+10-9 q^{-1} +7 q^{-2} -5 q^{-3} +3 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^2 a^4+z^4 a^2-a^2+2 z^4+3 z^2+2+z^4 a^{-2} +z^2 a^{-2} + a^{-2} -z^2 a^{-4} - a^{-4} }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ a z^9+z^9 a^{-1} +3 a^2 z^8+2 z^8 a^{-2} +5 z^8+4 a^3 z^7+3 a z^7+z^7 a^{-1} +2 z^7 a^{-3} +3 a^4 z^6-6 a^2 z^6-3 z^6 a^{-2} +2 z^6 a^{-4} -14 z^6+a^5 z^5-10 a^3 z^5-12 a z^5-4 z^5 a^{-1} -2 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4+5 a^2 z^4+3 z^4 a^{-2} -5 z^4 a^{-4} +20 z^4-2 a^5 z^3+7 a^3 z^3+15 a z^3+6 z^3 a^{-1} -3 z^3 a^{-3} -3 z^3 a^{-5} +2 a^4 z^2-3 a^2 z^2-2 z^2 a^{-2} +3 z^2 a^{-4} -10 z^2-2 a^3 z-4 a z-2 z a^{-1} +2 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]