10 88

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+8 t^2-24 t+35-24 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	{ 101, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-8 q^3+13 q^2-16 q+17-16 q^{-1} +13 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+2 z^2 a^{-2} -z^2 a^{-4} -3 z^2+a^2+ a^{-2} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+6 z^8 a^{-2} +12 z^8+7 a^3 z^7+14 a z^7+14 z^7 a^{-1} +7 z^7 a^{-3} +4 a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} +4 z^6 a^{-4} -12 z^6+a^5 z^5-11 a^3 z^5-32 a z^5-32 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-10 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -8 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+19 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +3 a^4 z^2+7 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} +8 z^2-a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^2- a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-3 q^{10}+3 q^8-2 q^4+3 q^2-3+3 q^{-2} -2 q^{-4} +3 q^{-8} -3 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-15 q^{70}+4 q^{68}+21 q^{66}-53 q^{64}+91 q^{62}-111 q^{60}+94 q^{58}-33 q^{56}-75 q^{54}+204 q^{52}-299 q^{50}+314 q^{48}-218 q^{46}+18 q^{44}+223 q^{42}-417 q^{40}+487 q^{38}-379 q^{36}+134 q^{34}+154 q^{32}-373 q^{30}+418 q^{28}-277 q^{26}+21 q^{24}+235 q^{22}-365 q^{20}+303 q^{18}-66 q^{16}-243 q^{14}+490 q^{12}-562 q^{10}+415 q^8-96 q^6-286 q^4+588 q^2-699+588 q^{-2} -286 q^{-4} -96 q^{-6} +415 q^{-8} -562 q^{-10} +490 q^{-12} -243 q^{-14} -66 q^{-16} +303 q^{-18} -365 q^{-20} +235 q^{-22} +21 q^{-24} -277 q^{-26} +418 q^{-28} -373 q^{-30} +154 q^{-32} +134 q^{-34} -379 q^{-36} +487 q^{-38} -417 q^{-40} +223 q^{-42} +18 q^{-44} -218 q^{-46} +314 q^{-48} -299 q^{-50} +204 q^{-52} -75 q^{-54} -33 q^{-56} +94 q^{-58} -111 q^{-60} +91 q^{-62} -53 q^{-64} +21 q^{-66} +4 q^{-68} -15 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_88.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+3 q^9-4 q^7+5 q^5-3 q^3+q+ q^{-1} -3 q^{-3} +5 q^{-5} -4 q^{-7} +3 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-3 q^{30}+11 q^{26}-14 q^{24}-10 q^{22}+37 q^{20}-18 q^{18}-37 q^{16}+54 q^{14}-53 q^{10}+36 q^8+21 q^6-37 q^4-q^2+29- q^{-2} -37 q^{-4} +21 q^{-6} +36 q^{-8} -53 q^{-10} +54 q^{-14} -37 q^{-16} -18 q^{-18} +37 q^{-20} -10 q^{-22} -14 q^{-24} +11 q^{-26} -3 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+3 q^{61}-7 q^{57}-2 q^{55}+18 q^{53}+13 q^{51}-44 q^{49}-36 q^{47}+70 q^{45}+93 q^{43}-88 q^{41}-184 q^{39}+77 q^{37}+291 q^{35}-15 q^{33}-384 q^{31}-101 q^{29}+441 q^{27}+232 q^{25}-426 q^{23}-355 q^{21}+345 q^{19}+441 q^{17}-229 q^{15}-461 q^{13}+93 q^{11}+432 q^9+41 q^7-362 q^5-164 q^3+271 q+271 q^{-1} -164 q^{-3} -362 q^{-5} +41 q^{-7} +432 q^{-9} +93 q^{-11} -461 q^{-13} -229 q^{-15} +441 q^{-17} +345 q^{-19} -355 q^{-21} -426 q^{-23} +232 q^{-25} +441 q^{-27} -101 q^{-29} -384 q^{-31} -15 q^{-33} +291 q^{-35} +77 q^{-37} -184 q^{-39} -88 q^{-41} +93 q^{-43} +70 q^{-45} -36 q^{-47} -44 q^{-49} +13 q^{-51} +18 q^{-53} -2 q^{-55} -7 q^{-57} +3 q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-3 q^{102}+7 q^{98}-2 q^{96}-2 q^{94}-21 q^{92}+4 q^{90}+54 q^{88}+13 q^{86}-29 q^{84}-144 q^{82}-46 q^{80}+234 q^{78}+234 q^{76}+24 q^{74}-554 q^{72}-518 q^{70}+359 q^{68}+966 q^{66}+786 q^{64}-922 q^{62}-1844 q^{60}-517 q^{58}+1658 q^{56}+2747 q^{54}+35 q^{52}-3124 q^{50}-2894 q^{48}+724 q^{46}+4612 q^{44}+2698 q^{42}-2489 q^{40}-5095 q^{38}-2009 q^{36}+4299 q^{34}+5074 q^{32}+92 q^{30}-5038 q^{28}-4396 q^{26}+1926 q^{24}+5220 q^{22}+2525 q^{20}-2979 q^{18}-4789 q^{16}-623 q^{14}+3557 q^{12}+3551 q^{10}-574 q^8-3759 q^6-2380 q^4+1504 q^2+3699+1504 q^{-2} -2380 q^{-4} -3759 q^{-6} -574 q^{-8} +3551 q^{-10} +3557 q^{-12} -623 q^{-14} -4789 q^{-16} -2979 q^{-18} +2525 q^{-20} +5220 q^{-22} +1926 q^{-24} -4396 q^{-26} -5038 q^{-28} +92 q^{-30} +5074 q^{-32} +4299 q^{-34} -2009 q^{-36} -5095 q^{-38} -2489 q^{-40} +2698 q^{-42} +4612 q^{-44} +724 q^{-46} -2894 q^{-48} -3124 q^{-50} +35 q^{-52} +2747 q^{-54} +1658 q^{-56} -517 q^{-58} -1844 q^{-60} -922 q^{-62} +786 q^{-64} +966 q^{-66} +359 q^{-68} -518 q^{-70} -554 q^{-72} +24 q^{-74} +234 q^{-76} +234 q^{-78} -46 q^{-80} -144 q^{-82} -29 q^{-84} +13 q^{-86} +54 q^{-88} +4 q^{-90} -21 q^{-92} -2 q^{-94} -2 q^{-96} +7 q^{-98} -3 q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+3 q^{153}-7 q^{149}+2 q^{147}+6 q^{145}+5 q^{143}+4 q^{141}-14 q^{139}-41 q^{137}-10 q^{135}+70 q^{133}+105 q^{131}+43 q^{129}-135 q^{127}-298 q^{125}-225 q^{123}+222 q^{121}+732 q^{119}+682 q^{117}-159 q^{115}-1354 q^{113}-1787 q^{111}-496 q^{109}+2095 q^{107}+3755 q^{105}+2268 q^{103}-2201 q^{101}-6450 q^{99}-5989 q^{97}+635 q^{95}+9175 q^{93}+11773 q^{91}+3865 q^{89}-10226 q^{87}-18885 q^{85}-12188 q^{83}+7671 q^{81}+25415 q^{79}+23813 q^{77}-4 q^{75}-28588 q^{73}-36643 q^{71}-13038 q^{69}+26035 q^{67}+47593 q^{65}+29531 q^{63}-16857 q^{61}-53210 q^{59}-46175 q^{57}+1965 q^{55}+51702 q^{53}+59244 q^{51}+15546 q^{49}-43098 q^{47}-65778 q^{45}-32065 q^{43}+29307 q^{41}+65010 q^{39}+44420 q^{37}-13624 q^{35}-57872 q^{33}-50838 q^{31}-1066 q^{29}+46663 q^{27}+51692 q^{25}+12623 q^{23}-34147 q^{21}-48336 q^{19}-20550 q^{17}+22334 q^{15}+42967 q^{13}+25616 q^{11}-12308 q^9-37451 q^7-29220 q^5+3904 q^3+32846 q+32846 q^{-1} +3904 q^{-3} -29220 q^{-5} -37451 q^{-7} -12308 q^{-9} +25616 q^{-11} +42967 q^{-13} +22334 q^{-15} -20550 q^{-17} -48336 q^{-19} -34147 q^{-21} +12623 q^{-23} +51692 q^{-25} +46663 q^{-27} -1066 q^{-29} -50838 q^{-31} -57872 q^{-33} -13624 q^{-35} +44420 q^{-37} +65010 q^{-39} +29307 q^{-41} -32065 q^{-43} -65778 q^{-45} -43098 q^{-47} +15546 q^{-49} +59244 q^{-51} +51702 q^{-53} +1965 q^{-55} -46175 q^{-57} -53210 q^{-59} -16857 q^{-61} +29531 q^{-63} +47593 q^{-65} +26035 q^{-67} -13038 q^{-69} -36643 q^{-71} -28588 q^{-73} -4 q^{-75} +23813 q^{-77} +25415 q^{-79} +7671 q^{-81} -12188 q^{-83} -18885 q^{-85} -10226 q^{-87} +3865 q^{-89} +11773 q^{-91} +9175 q^{-93} +635 q^{-95} -5989 q^{-97} -6450 q^{-99} -2201 q^{-101} +2268 q^{-103} +3755 q^{-105} +2095 q^{-107} -496 q^{-109} -1787 q^{-111} -1354 q^{-113} -159 q^{-115} +682 q^{-117} +732 q^{-119} +222 q^{-121} -225 q^{-123} -298 q^{-125} -135 q^{-127} +43 q^{-129} +105 q^{-131} +70 q^{-133} -10 q^{-135} -41 q^{-137} -14 q^{-139} +4 q^{-141} +5 q^{-143} +6 q^{-145} +2 q^{-147} -7 q^{-149} +3 q^{-153} - q^{-155} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-3 q^{32}+q^{30}+7 q^{28}-14 q^{26}+4 q^{24}+22 q^{22}-30 q^{20}+5 q^{18}+35 q^{16}-41 q^{14}+2 q^{12}+35 q^{10}-30 q^8-6 q^6+22 q^4-2 q^2-10-2 q^{-2} +22 q^{-4} -6 q^{-6} -30 q^{-8} +35 q^{-10} +2 q^{-12} -41 q^{-14} +35 q^{-16} +5 q^{-18} -30 q^{-20} +22 q^{-22} +4 q^{-24} -14 q^{-26} +7 q^{-28} + q^{-30} -3 q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}+q^{19}+2 q^{15}-3 q^{13}+4 q^{11}-2 q^9+2 q^7-2 q^5+2 q^3-q- q^{-1} +2 q^{-3} -2 q^{-5} +2 q^{-7} -2 q^{-9} +4 q^{-11} -3 q^{-13} +2 q^{-15} + q^{-19} - q^{-21} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+3 q^{32}-7 q^{30}+13 q^{28}-22 q^{26}+32 q^{24}-40 q^{22}+48 q^{20}-49 q^{18}+45 q^{16}-33 q^{14}+16 q^{12}+7 q^{10}-32 q^8+56 q^6-76 q^4+90 q^2-96+90 q^{-2} -76 q^{-4} +56 q^{-6} -32 q^{-8} +7 q^{-10} +16 q^{-12} -33 q^{-14} +45 q^{-16} -49 q^{-18} +48 q^{-20} -40 q^{-22} +32 q^{-24} -22 q^{-26} +13 q^{-28} -7 q^{-30} +3 q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+10 q^{46}-18 q^{42}-13 q^{40}+19 q^{38}+31 q^{36}-4 q^{34}-43 q^{32}-19 q^{30}+40 q^{28}+42 q^{26}-21 q^{24}-53 q^{22}-4 q^{20}+49 q^{18}+22 q^{16}-36 q^{14}-31 q^{12}+22 q^{10}+33 q^8-11 q^6-33 q^4+4 q^2+35+4 q^{-2} -33 q^{-4} -11 q^{-6} +33 q^{-8} +22 q^{-10} -31 q^{-12} -36 q^{-14} +22 q^{-16} +49 q^{-18} -4 q^{-20} -53 q^{-22} -21 q^{-24} +42 q^{-26} +40 q^{-28} -19 q^{-30} -43 q^{-32} -4 q^{-34} +31 q^{-36} +19 q^{-38} -13 q^{-40} -18 q^{-42} +10 q^{-46} +4 q^{-48} -3 q^{-50} -3 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}-15 q^{70}+4 q^{68}+21 q^{66}-53 q^{64}+91 q^{62}-111 q^{60}+94 q^{58}-33 q^{56}-75 q^{54}+204 q^{52}-299 q^{50}+314 q^{48}-218 q^{46}+18 q^{44}+223 q^{42}-417 q^{40}+487 q^{38}-379 q^{36}+134 q^{34}+154 q^{32}-373 q^{30}+418 q^{28}-277 q^{26}+21 q^{24}+235 q^{22}-365 q^{20}+303 q^{18}-66 q^{16}-243 q^{14}+490 q^{12}-562 q^{10}+415 q^8-96 q^6-286 q^4+588 q^2-699+588 q^{-2} -286 q^{-4} -96 q^{-6} +415 q^{-8} -562 q^{-10} +490 q^{-12} -243 q^{-14} -66 q^{-16} +303 q^{-18} -365 q^{-20} +235 q^{-22} +21 q^{-24} -277 q^{-26} +418 q^{-28} -373 q^{-30} +154 q^{-32} +134 q^{-34} -379 q^{-36} +487 q^{-38} -417 q^{-40} +223 q^{-42} +18 q^{-44} -218 q^{-46} +314 q^{-48} -299 q^{-50} +204 q^{-52} -75 q^{-54} -33 q^{-56} +94 q^{-58} -111 q^{-60} +91 q^{-62} -53 q^{-64} +21 q^{-66} +4 q^{-68} -15 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 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["10 88"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+8 t^2-24 t+35-24 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]=

{ 101, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+4 q^4-8 q^3+13 q^2-16 q+17-16 q^{-1} +13 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+2 z^2 a^{-2} -z^2 a^{-4} -3 z^2+a^2+ a^{-2} -1 }[/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+6 z^8 a^{-2} +12 z^8+7 a^3 z^7+14 a z^7+14 z^7 a^{-1} +7 z^7 a^{-3} +4 a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} +4 z^6 a^{-4} -12 z^6+a^5 z^5-11 a^3 z^5-32 a z^5-32 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-10 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -8 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+19 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +3 a^4 z^2+7 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} +8 z^2-a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^2- a^{-2} -1 }[/math]