10 33

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t^2-16 t+25-16 t^{-1} +4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 65, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-5 q^3+8 q^2-10 q+11-10 q^{-1} +8 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+2 z^4+2 z^2+1+z^4 a^{-2} -z^2 a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^9+z^9 a^{-1} +3 a^2 z^8+3 z^8 a^{-2} +6 z^8+4 a^3 z^7+5 a z^7+5 z^7 a^{-1} +4 z^7 a^{-3} +3 a^4 z^6-4 a^2 z^6-4 z^6 a^{-2} +3 z^6 a^{-4} -14 z^6+a^5 z^5-9 a^3 z^5-16 a z^5-16 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4+a^2 z^4+z^4 a^{-2} -7 z^4 a^{-4} +16 z^4-2 a^5 z^3+6 a^3 z^3+18 a z^3+18 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} +3 a^4 z^2+3 z^2 a^{-4} -6 z^2-2 a^3 z-6 a z-6 z a^{-1} -2 z a^{-3} +1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-2 q^{10}+2 q^8-q^4+2 q^2-1+2 q^{-2} - q^{-4} +2 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+6 q^{72}-5 q^{70}-2 q^{68}+14 q^{66}-23 q^{64}+32 q^{62}-32 q^{60}+19 q^{58}+3 q^{56}-33 q^{54}+60 q^{52}-73 q^{50}+67 q^{48}-37 q^{46}-9 q^{44}+57 q^{42}-88 q^{40}+96 q^{38}-70 q^{36}+20 q^{34}+31 q^{32}-69 q^{30}+73 q^{28}-46 q^{26}+45 q^{22}-65 q^{20}+47 q^{18}-3 q^{16}-55 q^{14}+102 q^{12}-111 q^{10}+80 q^8-14 q^6-61 q^4+124 q^2-145+124 q^{-2} -61 q^{-4} -14 q^{-6} +80 q^{-8} -111 q^{-10} +102 q^{-12} -55 q^{-14} -3 q^{-16} +47 q^{-18} -65 q^{-20} +45 q^{-22} -46 q^{-26} +73 q^{-28} -69 q^{-30} +31 q^{-32} +20 q^{-34} -70 q^{-36} +96 q^{-38} -88 q^{-40} +57 q^{-42} -9 q^{-44} -37 q^{-46} +67 q^{-48} -73 q^{-50} +60 q^{-52} -33 q^{-54} +3 q^{-56} +19 q^{-58} -32 q^{-60} +32 q^{-62} -23 q^{-64} +14 q^{-66} -2 q^{-68} -5 q^{-70} +6 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_33.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-2 q^7+3 q^5-2 q^3+q+ q^{-1} -2 q^{-3} +3 q^{-5} -2 q^{-7} +2 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+6 q^{26}-5 q^{24}-6 q^{22}+13 q^{20}-4 q^{18}-13 q^{16}+18 q^{14}+q^{12}-18 q^{10}+13 q^8+6 q^6-13 q^4+q^2+9+ q^{-2} -13 q^{-4} +6 q^{-6} +13 q^{-8} -18 q^{-10} + q^{-12} +18 q^{-14} -13 q^{-16} -4 q^{-18} +13 q^{-20} -6 q^{-22} -5 q^{-24} +6 q^{-26} - q^{-28} -2 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+q^{59}-3 q^{57}-3 q^{55}+5 q^{53}+8 q^{51}-9 q^{49}-14 q^{47}+10 q^{45}+23 q^{43}-10 q^{41}-35 q^{39}+4 q^{37}+51 q^{35}+6 q^{33}-61 q^{31}-25 q^{29}+69 q^{27}+45 q^{25}-66 q^{23}-63 q^{21}+56 q^{19}+76 q^{17}-42 q^{15}-78 q^{13}+20 q^{11}+73 q^9+q^7-61 q^5-21 q^3+44 q+44 q^{-1} -21 q^{-3} -61 q^{-5} + q^{-7} +73 q^{-9} +20 q^{-11} -78 q^{-13} -42 q^{-15} +76 q^{-17} +56 q^{-19} -63 q^{-21} -66 q^{-23} +45 q^{-25} +69 q^{-27} -25 q^{-29} -61 q^{-31} +6 q^{-33} +51 q^{-35} +4 q^{-37} -35 q^{-39} -10 q^{-41} +23 q^{-43} +10 q^{-45} -14 q^{-47} -9 q^{-49} +8 q^{-51} +5 q^{-53} -3 q^{-55} -3 q^{-57} + q^{-59} +2 q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}-q^{100}+3 q^{98}+3 q^{94}-8 q^{92}-3 q^{90}+11 q^{88}+q^{86}+9 q^{84}-24 q^{82}-15 q^{80}+28 q^{78}+16 q^{76}+24 q^{74}-58 q^{72}-54 q^{70}+38 q^{68}+61 q^{66}+91 q^{64}-82 q^{62}-151 q^{60}-31 q^{58}+98 q^{56}+245 q^{54}-q^{52}-237 q^{50}-218 q^{48}+6 q^{46}+393 q^{44}+216 q^{42}-179 q^{40}-398 q^{38}-220 q^{36}+375 q^{34}+411 q^{32}+21 q^{30}-408 q^{28}-406 q^{26}+200 q^{24}+430 q^{22}+202 q^{20}-256 q^{18}-418 q^{16}-3 q^{14}+297 q^{12}+276 q^{10}-65 q^8-310 q^6-165 q^4+118 q^2+283+118 q^{-2} -165 q^{-4} -310 q^{-6} -65 q^{-8} +276 q^{-10} +297 q^{-12} -3 q^{-14} -418 q^{-16} -256 q^{-18} +202 q^{-20} +430 q^{-22} +200 q^{-24} -406 q^{-26} -408 q^{-28} +21 q^{-30} +411 q^{-32} +375 q^{-34} -220 q^{-36} -398 q^{-38} -179 q^{-40} +216 q^{-42} +393 q^{-44} +6 q^{-46} -218 q^{-48} -237 q^{-50} - q^{-52} +245 q^{-54} +98 q^{-56} -31 q^{-58} -151 q^{-60} -82 q^{-62} +91 q^{-64} +61 q^{-66} +38 q^{-68} -54 q^{-70} -58 q^{-72} +24 q^{-74} +16 q^{-76} +28 q^{-78} -15 q^{-80} -24 q^{-82} +9 q^{-84} + q^{-86} +11 q^{-88} -3 q^{-90} -8 q^{-92} +3 q^{-94} +3 q^{-98} - q^{-100} -2 q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+2 q^{153}+q^{151}-3 q^{149}+3 q^{141}+2 q^{139}-7 q^{137}-5 q^{135}+7 q^{133}+9 q^{131}+5 q^{129}-7 q^{127}-22 q^{125}-20 q^{123}+17 q^{121}+55 q^{119}+38 q^{117}-21 q^{115}-89 q^{113}-98 q^{111}-4 q^{109}+142 q^{107}+197 q^{105}+68 q^{103}-161 q^{101}-321 q^{99}-234 q^{97}+111 q^{95}+454 q^{93}+475 q^{91}+74 q^{89}-504 q^{87}-780 q^{85}-421 q^{83}+393 q^{81}+1046 q^{79}+927 q^{77}-58 q^{75}-1179 q^{73}-1467 q^{71}-522 q^{69}+1052 q^{67}+1955 q^{65}+1250 q^{63}-658 q^{61}-2208 q^{59}-1983 q^{57}+6 q^{55}+2170 q^{53}+2578 q^{51}+734 q^{49}-1836 q^{47}-2882 q^{45}-1428 q^{43}+1275 q^{41}+2871 q^{39}+1948 q^{37}-653 q^{35}-2575 q^{33}-2178 q^{31}+55 q^{29}+2088 q^{27}+2182 q^{25}+404 q^{23}-1548 q^{21}-1986 q^{19}-711 q^{17}+1020 q^{15}+1708 q^{13}+910 q^{11}-563 q^9-1444 q^7-1048 q^5+181 q^3+1215 q+1215 q^{-1} +181 q^{-3} -1048 q^{-5} -1444 q^{-7} -563 q^{-9} +910 q^{-11} +1708 q^{-13} +1020 q^{-15} -711 q^{-17} -1986 q^{-19} -1548 q^{-21} +404 q^{-23} +2182 q^{-25} +2088 q^{-27} +55 q^{-29} -2178 q^{-31} -2575 q^{-33} -653 q^{-35} +1948 q^{-37} +2871 q^{-39} +1275 q^{-41} -1428 q^{-43} -2882 q^{-45} -1836 q^{-47} +734 q^{-49} +2578 q^{-51} +2170 q^{-53} +6 q^{-55} -1983 q^{-57} -2208 q^{-59} -658 q^{-61} +1250 q^{-63} +1955 q^{-65} +1052 q^{-67} -522 q^{-69} -1467 q^{-71} -1179 q^{-73} -58 q^{-75} +927 q^{-77} +1046 q^{-79} +393 q^{-81} -421 q^{-83} -780 q^{-85} -504 q^{-87} +74 q^{-89} +475 q^{-91} +454 q^{-93} +111 q^{-95} -234 q^{-97} -321 q^{-99} -161 q^{-101} +68 q^{-103} +197 q^{-105} +142 q^{-107} -4 q^{-109} -98 q^{-111} -89 q^{-113} -21 q^{-115} +38 q^{-117} +55 q^{-119} +17 q^{-121} -20 q^{-123} -22 q^{-125} -7 q^{-127} +5 q^{-129} +9 q^{-131} +7 q^{-133} -5 q^{-135} -7 q^{-137} +2 q^{-139} +3 q^{-141} -3 q^{-149} + q^{-151} +2 q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-2 q^{10}+2 q^8-q^4+2 q^2-1+2 q^{-2} - q^{-4} +2 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+10 q^{40}-22 q^{38}+44 q^{36}-74 q^{34}+112 q^{32}-166 q^{30}+222 q^{28}-278 q^{26}+330 q^{24}-362 q^{22}+371 q^{20}-340 q^{18}+270 q^{16}-158 q^{14}+8 q^{12}+164 q^{10}-340 q^8+510 q^6-645 q^4+732 q^2-762+732 q^{-2} -645 q^{-4} +510 q^{-6} -340 q^{-8} +164 q^{-10} +8 q^{-12} -158 q^{-14} +270 q^{-16} -340 q^{-18} +371 q^{-20} -362 q^{-22} +330 q^{-24} -278 q^{-26} +222 q^{-28} -166 q^{-30} +112 q^{-32} -74 q^{-34} +44 q^{-36} -22 q^{-38} +10 q^{-40} -4 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{40}-2 q^{38}+q^{36}+4 q^{34}-8 q^{30}-q^{28}+9 q^{26}+q^{24}-11 q^{22}-q^{20}+14 q^{18}+4 q^{16}-13 q^{14}+12 q^{10}-2 q^8-7 q^6+2 q^4+2 q^2-2+2 q^{-2} +2 q^{-4} -7 q^{-6} -2 q^{-8} +12 q^{-10} -13 q^{-14} +4 q^{-16} +14 q^{-18} - q^{-20} -11 q^{-22} + q^{-24} +9 q^{-26} - q^{-28} -8 q^{-30} +4 q^{-34} + q^{-36} -2 q^{-38} - q^{-40} + q^{-42} }[/math]

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-4 q^{30}+7 q^{28}-10 q^{26}+13 q^{24}-16 q^{22}+17 q^{20}-17 q^{18}+16 q^{16}-10 q^{14}+5 q^{12}+5 q^{10}-13 q^8+21 q^6-28 q^4+33 q^2-36+33 q^{-2} -28 q^{-4} +21 q^{-6} -13 q^{-8} +5 q^{-10} +5 q^{-12} -10 q^{-14} +16 q^{-16} -17 q^{-18} +17 q^{-20} -16 q^{-22} +13 q^{-24} -10 q^{-26} +7 q^{-28} -4 q^{-30} +2 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}-8 q^{42}-4 q^{40}+9 q^{38}+12 q^{36}-4 q^{34}-16 q^{32}-5 q^{30}+15 q^{28}+15 q^{26}-9 q^{24}-19 q^{22}-2 q^{20}+17 q^{18}+8 q^{16}-12 q^{14}-11 q^{12}+7 q^{10}+12 q^8-3 q^6-11 q^4+2 q^2+13+2 q^{-2} -11 q^{-4} -3 q^{-6} +12 q^{-8} +7 q^{-10} -11 q^{-12} -12 q^{-14} +8 q^{-16} +17 q^{-18} -2 q^{-20} -19 q^{-22} -9 q^{-24} +15 q^{-26} +15 q^{-28} -5 q^{-30} -16 q^{-32} -4 q^{-34} +12 q^{-36} +9 q^{-38} -4 q^{-40} -8 q^{-42} - q^{-44} +5 q^{-46} +2 q^{-48} -2 q^{-50} -2 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}-5 q^{70}-2 q^{68}+14 q^{66}-23 q^{64}+32 q^{62}-32 q^{60}+19 q^{58}+3 q^{56}-33 q^{54}+60 q^{52}-73 q^{50}+67 q^{48}-37 q^{46}-9 q^{44}+57 q^{42}-88 q^{40}+96 q^{38}-70 q^{36}+20 q^{34}+31 q^{32}-69 q^{30}+73 q^{28}-46 q^{26}+45 q^{22}-65 q^{20}+47 q^{18}-3 q^{16}-55 q^{14}+102 q^{12}-111 q^{10}+80 q^8-14 q^6-61 q^4+124 q^2-145+124 q^{-2} -61 q^{-4} -14 q^{-6} +80 q^{-8} -111 q^{-10} +102 q^{-12} -55 q^{-14} -3 q^{-16} +47 q^{-18} -65 q^{-20} +45 q^{-22} -46 q^{-26} +73 q^{-28} -69 q^{-30} +31 q^{-32} +20 q^{-34} -70 q^{-36} +96 q^{-38} -88 q^{-40} +57 q^{-42} -9 q^{-44} -37 q^{-46} +67 q^{-48} -73 q^{-50} +60 q^{-52} -33 q^{-54} +3 q^{-56} +19 q^{-58} -32 q^{-60} +32 q^{-62} -23 q^{-64} +14 q^{-66} -2 q^{-68} -5 q^{-70} +6 q^{-72} -7 q^{-74} +4 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 33"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 4 t^2-16 t+25-16 t^{-1} +4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 65, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+3 q^4-5 q^3+8 q^2-10 q+11-10 q^{-1} +8 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+2 z^4+2 z^2+1+z^4 a^{-2} -z^2 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+3 z^8 a^{-2} +6 z^8+4 a^3 z^7+5 a z^7+5 z^7 a^{-1} +4 z^7 a^{-3} +3 a^4 z^6-4 a^2 z^6-4 z^6 a^{-2} +3 z^6 a^{-4} -14 z^6+a^5 z^5-9 a^3 z^5-16 a z^5-16 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4+a^2 z^4+z^4 a^{-2} -7 z^4 a^{-4} +16 z^4-2 a^5 z^3+6 a^3 z^3+18 a z^3+18 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} +3 a^4 z^2+3 z^2 a^{-4} -6 z^2-2 a^3 z-6 a z-6 z a^{-1} -2 z a^{-3} +1 }[/math]