10 81

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+8 t^2-20 t+27-20 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	{ 85, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-7 q^3+11 q^2-13 q+15-13 q^{-1} +11 q^{-2} -7 q^{-3} +3 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+3 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} -z^2-a^4+a^2+ a^{-2} - a^{-4} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^9+z^9 a^{-1} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+5 a^3 z^7+13 a z^7+13 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6+3 z^6 a^{-4} -6 z^6+a^5 z^5-8 a^3 z^5-31 a z^5-31 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4-9 a^2 z^4-9 z^4 a^{-2} -5 z^4 a^{-4} -8 z^4-2 a^5 z^3+5 a^3 z^3+25 a z^3+25 z^3 a^{-1} +5 z^3 a^{-3} -2 z^3 a^{-5} +3 a^4 z^2+6 a^2 z^2+6 z^2 a^{-2} +3 z^2 a^{-4} +6 z^2+a^5 z-2 a^3 z-8 a z-8 z a^{-1} -2 z a^{-3} +z a^{-5} -a^4-a^2- a^{-2} - a^{-4} +1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{12}-3 q^{10}+2 q^8-q^4+4 q^2-1+4 q^{-2} - q^{-4} +2 q^{-8} -3 q^{-10} + q^{-12} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-9 q^{70}+q^{68}+14 q^{66}-35 q^{64}+56 q^{62}-69 q^{60}+55 q^{58}-16 q^{56}-50 q^{54}+129 q^{52}-183 q^{50}+191 q^{48}-130 q^{46}+4 q^{44}+139 q^{42}-255 q^{40}+293 q^{38}-227 q^{36}+79 q^{34}+91 q^{32}-219 q^{30}+247 q^{28}-166 q^{26}+14 q^{24}+137 q^{22}-214 q^{20}+173 q^{18}-34 q^{16}-147 q^{14}+296 q^{12}-335 q^{10}+249 q^8-55 q^6-172 q^4+360 q^2-427+360 q^{-2} -172 q^{-4} -55 q^{-6} +249 q^{-8} -335 q^{-10} +296 q^{-12} -147 q^{-14} -34 q^{-16} +173 q^{-18} -214 q^{-20} +137 q^{-22} +14 q^{-24} -166 q^{-26} +247 q^{-28} -219 q^{-30} +91 q^{-32} +79 q^{-34} -227 q^{-36} +293 q^{-38} -255 q^{-40} +139 q^{-42} +4 q^{-44} -130 q^{-46} +191 q^{-48} -183 q^{-50} +129 q^{-52} -50 q^{-54} -16 q^{-56} +55 q^{-58} -69 q^{-60} +56 q^{-62} -35 q^{-64} +14 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_81.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-4 q^7+4 q^5-2 q^3+2 q+2 q^{-1} -2 q^{-3} +4 q^{-5} -4 q^{-7} +2 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}+8 q^{26}-10 q^{24}-8 q^{22}+26 q^{20}-13 q^{18}-27 q^{16}+38 q^{14}-38 q^{10}+26 q^8+15 q^6-26 q^4+q^2+21+ q^{-2} -26 q^{-4} +15 q^{-6} +26 q^{-8} -38 q^{-10} +38 q^{-14} -27 q^{-16} -13 q^{-18} +26 q^{-20} -8 q^{-22} -10 q^{-24} +8 q^{-26} -2 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}-4 q^{57}-2 q^{55}+12 q^{53}+9 q^{51}-26 q^{49}-25 q^{47}+41 q^{45}+58 q^{43}-47 q^{41}-113 q^{39}+41 q^{37}+173 q^{35}-3 q^{33}-226 q^{31}-68 q^{29}+260 q^{27}+140 q^{25}-250 q^{23}-215 q^{21}+207 q^{19}+260 q^{17}-137 q^{15}-277 q^{13}+64 q^{11}+255 q^9+21 q^7-215 q^5-91 q^3+159 q+159 q^{-1} -91 q^{-3} -215 q^{-5} +21 q^{-7} +255 q^{-9} +64 q^{-11} -277 q^{-13} -137 q^{-15} +260 q^{-17} +207 q^{-19} -215 q^{-21} -250 q^{-23} +140 q^{-25} +260 q^{-27} -68 q^{-29} -226 q^{-31} -3 q^{-33} +173 q^{-35} +41 q^{-37} -113 q^{-39} -47 q^{-41} +58 q^{-43} +41 q^{-45} -25 q^{-47} -26 q^{-49} +9 q^{-51} +12 q^{-53} -2 q^{-55} -4 q^{-57} +2 q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}+4 q^{98}-2 q^{96}-13 q^{92}+q^{90}+31 q^{88}+8 q^{86}-9 q^{84}-80 q^{82}-34 q^{80}+117 q^{78}+121 q^{76}+39 q^{74}-273 q^{72}-277 q^{70}+153 q^{68}+456 q^{66}+436 q^{64}-420 q^{62}-907 q^{60}-292 q^{58}+744 q^{56}+1385 q^{54}+71 q^{52}-1490 q^{50}-1436 q^{48}+256 q^{46}+2270 q^{44}+1356 q^{42}-1174 q^{40}-2491 q^{38}-1058 q^{36}+2127 q^{34}+2496 q^{32}+51 q^{30}-2484 q^{28}-2193 q^{26}+1010 q^{24}+2579 q^{22}+1212 q^{20}-1509 q^{18}-2375 q^{16}-229 q^{14}+1776 q^{12}+1719 q^{10}-323 q^8-1855 q^6-1131 q^4+741 q^2+1799+741 q^{-2} -1131 q^{-4} -1855 q^{-6} -323 q^{-8} +1719 q^{-10} +1776 q^{-12} -229 q^{-14} -2375 q^{-16} -1509 q^{-18} +1212 q^{-20} +2579 q^{-22} +1010 q^{-24} -2193 q^{-26} -2484 q^{-28} +51 q^{-30} +2496 q^{-32} +2127 q^{-34} -1058 q^{-36} -2491 q^{-38} -1174 q^{-40} +1356 q^{-42} +2270 q^{-44} +256 q^{-46} -1436 q^{-48} -1490 q^{-50} +71 q^{-52} +1385 q^{-54} +744 q^{-56} -292 q^{-58} -907 q^{-60} -420 q^{-62} +436 q^{-64} +456 q^{-66} +153 q^{-68} -277 q^{-70} -273 q^{-72} +39 q^{-74} +121 q^{-76} +117 q^{-78} -34 q^{-80} -80 q^{-82} -9 q^{-84} +8 q^{-86} +31 q^{-88} + q^{-90} -13 q^{-92} -2 q^{-96} +4 q^{-98} -2 q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+2 q^{153}-4 q^{149}+2 q^{147}+4 q^{145}+q^{143}+3 q^{141}-6 q^{139}-24 q^{137}-7 q^{135}+34 q^{133}+49 q^{131}+30 q^{129}-49 q^{127}-139 q^{125}-122 q^{123}+73 q^{121}+307 q^{119}+323 q^{117}-3 q^{115}-536 q^{113}-776 q^{111}-304 q^{109}+763 q^{107}+1544 q^{105}+1063 q^{103}-721 q^{101}-2541 q^{99}-2552 q^{97}-33 q^{95}+3502 q^{93}+4831 q^{91}+1897 q^{89}-3746 q^{87}-7535 q^{85}-5268 q^{83}+2547 q^{81}+9987 q^{79}+9878 q^{77}+624 q^{75}-11060 q^{73}-14906 q^{71}-5906 q^{69}+9898 q^{67}+19206 q^{65}+12485 q^{63}-6200 q^{61}-21361 q^{59}-19110 q^{57}+243 q^{55}+20810 q^{53}+24323 q^{51}+6647 q^{49}-17440 q^{47}-26957 q^{45}-13219 q^{43}+12096 q^{41}+26728 q^{39}+18078 q^{37}-5902 q^{35}-23973 q^{33}-20683 q^{31}+113 q^{29}+19538 q^{27}+21025 q^{25}+4588 q^{23}-14499 q^{21}-19760 q^{19}-7842 q^{17}+9640 q^{15}+17570 q^{13}+10084 q^{11}-5387 q^9-15327 q^7-11708 q^5+1734 q^3+13340 q+13340 q^{-1} +1734 q^{-3} -11708 q^{-5} -15327 q^{-7} -5387 q^{-9} +10084 q^{-11} +17570 q^{-13} +9640 q^{-15} -7842 q^{-17} -19760 q^{-19} -14499 q^{-21} +4588 q^{-23} +21025 q^{-25} +19538 q^{-27} +113 q^{-29} -20683 q^{-31} -23973 q^{-33} -5902 q^{-35} +18078 q^{-37} +26728 q^{-39} +12096 q^{-41} -13219 q^{-43} -26957 q^{-45} -17440 q^{-47} +6647 q^{-49} +24323 q^{-51} +20810 q^{-53} +243 q^{-55} -19110 q^{-57} -21361 q^{-59} -6200 q^{-61} +12485 q^{-63} +19206 q^{-65} +9898 q^{-67} -5906 q^{-69} -14906 q^{-71} -11060 q^{-73} +624 q^{-75} +9878 q^{-77} +9987 q^{-79} +2547 q^{-81} -5268 q^{-83} -7535 q^{-85} -3746 q^{-87} +1897 q^{-89} +4831 q^{-91} +3502 q^{-93} -33 q^{-95} -2552 q^{-97} -2541 q^{-99} -721 q^{-101} +1063 q^{-103} +1544 q^{-105} +763 q^{-107} -304 q^{-109} -776 q^{-111} -536 q^{-113} -3 q^{-115} +323 q^{-117} +307 q^{-119} +73 q^{-121} -122 q^{-123} -139 q^{-125} -49 q^{-127} +30 q^{-129} +49 q^{-131} +34 q^{-133} -7 q^{-135} -24 q^{-137} -6 q^{-139} +3 q^{-141} + q^{-143} +4 q^{-145} +2 q^{-147} -4 q^{-149} +2 q^{-153} - q^{-155} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-2 q^{32}+q^{30}+5 q^{28}-10 q^{26}+2 q^{24}+15 q^{22}-23 q^{20}+3 q^{18}+24 q^{16}-31 q^{14}+24 q^{10}-22 q^8-4 q^6+19 q^4+2 q^2-2+2 q^{-2} +19 q^{-4} -4 q^{-6} -22 q^{-8} +24 q^{-10} -31 q^{-14} +24 q^{-16} +3 q^{-18} -23 q^{-20} +15 q^{-22} +2 q^{-24} -10 q^{-26} +5 q^{-28} + q^{-30} -2 q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{17}+q^{15}-3 q^{13}+3 q^{11}-2 q^9+2 q^7-q^5+3 q^3+q+ q^{-1} +3 q^{-3} - q^{-5} +2 q^{-7} -2 q^{-9} +3 q^{-11} -3 q^{-13} + q^{-15} - q^{-17} - q^{-21} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+9 q^{28}-16 q^{26}+22 q^{24}-29 q^{22}+33 q^{20}-35 q^{18}+32 q^{16}-23 q^{14}+12 q^{12}+6 q^{10}-22 q^8+40 q^6-53 q^4+64 q^2-68+64 q^{-2} -53 q^{-4} +40 q^{-6} -22 q^{-8} +6 q^{-10} +12 q^{-12} -23 q^{-14} +32 q^{-16} -35 q^{-18} +33 q^{-20} -29 q^{-22} +22 q^{-24} -16 q^{-26} +9 q^{-28} -5 q^{-30} +2 q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+7 q^{46}-13 q^{42}-9 q^{40}+13 q^{38}+22 q^{36}-4 q^{34}-31 q^{32}-14 q^{30}+28 q^{28}+29 q^{26}-16 q^{24}-38 q^{22}-4 q^{20}+34 q^{18}+15 q^{16}-25 q^{14}-22 q^{12}+16 q^{10}+24 q^8-6 q^6-22 q^4+5 q^2+27+5 q^{-2} -22 q^{-4} -6 q^{-6} +24 q^{-8} +16 q^{-10} -22 q^{-12} -25 q^{-14} +15 q^{-16} +34 q^{-18} -4 q^{-20} -38 q^{-22} -16 q^{-24} +29 q^{-26} +28 q^{-28} -14 q^{-30} -31 q^{-32} -4 q^{-34} +22 q^{-36} +13 q^{-38} -9 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}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-9 q^{70}+q^{68}+14 q^{66}-35 q^{64}+56 q^{62}-69 q^{60}+55 q^{58}-16 q^{56}-50 q^{54}+129 q^{52}-183 q^{50}+191 q^{48}-130 q^{46}+4 q^{44}+139 q^{42}-255 q^{40}+293 q^{38}-227 q^{36}+79 q^{34}+91 q^{32}-219 q^{30}+247 q^{28}-166 q^{26}+14 q^{24}+137 q^{22}-214 q^{20}+173 q^{18}-34 q^{16}-147 q^{14}+296 q^{12}-335 q^{10}+249 q^8-55 q^6-172 q^4+360 q^2-427+360 q^{-2} -172 q^{-4} -55 q^{-6} +249 q^{-8} -335 q^{-10} +296 q^{-12} -147 q^{-14} -34 q^{-16} +173 q^{-18} -214 q^{-20} +137 q^{-22} +14 q^{-24} -166 q^{-26} +247 q^{-28} -219 q^{-30} +91 q^{-32} +79 q^{-34} -227 q^{-36} +293 q^{-38} -255 q^{-40} +139 q^{-42} +4 q^{-44} -130 q^{-46} +191 q^{-48} -183 q^{-50} +129 q^{-52} -50 q^{-54} -16 q^{-56} +55 q^{-58} -69 q^{-60} +56 q^{-62} -35 q^{-64} +14 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 81"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+8 t^2-20 t+27-20 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]=

{ 85, 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+11 q^2-13 q+15-13 q^{-1} +11 q^{-2} -7 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^6+2 a^2 z^4+2 z^4 a^{-2} -2 z^4-a^4 z^2+3 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} -z^2-a^4+a^2+ a^{-2} - a^{-4} +1 }[/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} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+5 a^3 z^7+13 a z^7+13 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6+3 z^6 a^{-4} -6 z^6+a^5 z^5-8 a^3 z^5-31 a z^5-31 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4-9 a^2 z^4-9 z^4 a^{-2} -5 z^4 a^{-4} -8 z^4-2 a^5 z^3+5 a^3 z^3+25 a z^3+25 z^3 a^{-1} +5 z^3 a^{-3} -2 z^3 a^{-5} +3 a^4 z^2+6 a^2 z^2+6 z^2 a^{-2} +3 z^2 a^{-4} +6 z^2+a^5 z-2 a^3 z-8 a z-8 z a^{-1} -2 z a^{-3} +z a^{-5} -a^4-a^2- a^{-2} - a^{-4} +1 }[/math]