10 75

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+7 t^2-19 t+27-19 t^{-1} +7 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{3,t+1\} }[/math]
Determinant and Signature	{ 81, 0 }
Jones polynomial	[math]\displaystyle{ q^6-3 q^5+6 q^4-10 q^3+12 q^2-13 q+14-10 q^{-1} +7 q^{-2} -4 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+a^2 z^4+3 z^4 a^{-2} -3 z^4+a^2 z^2+6 z^2 a^{-2} -3 z^2 a^{-4} -4 z^2+3 a^{-2} -3 a^{-4} + a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +7 z^8 a^{-2} +3 z^8 a^{-4} +4 z^8+7 a z^7+13 z^7 a^{-1} +9 z^7 a^{-3} +3 z^7 a^{-5} +7 a^2 z^6-4 z^6 a^{-2} -3 z^6 a^{-4} +z^6 a^{-6} +7 z^6+4 a^3 z^5-5 a z^5-29 z^5 a^{-1} -29 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-8 a^2 z^4-21 z^4 a^{-2} -9 z^4 a^{-4} -3 z^4 a^{-6} -24 z^4-3 a^3 z^3-a z^3+17 z^3 a^{-1} +24 z^3 a^{-3} +9 z^3 a^{-5} +4 a^2 z^2+20 z^2 a^{-2} +12 z^2 a^{-4} +3 z^2 a^{-6} +15 z^2-a z-5 z a^{-1} -7 z a^{-3} -3 z a^{-5} -3 a^{-2} -3 a^{-4} - a^{-6} }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}-2 q^{10}+q^8-3 q^4+4 q^2+3 q^{-2} + q^{-4} - q^{-6} +2 q^{-8} -3 q^{-10} -2 q^{-16} + q^{-18} + q^{-20} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+9 q^{58}-6 q^{56}-2 q^{54}+19 q^{52}-32 q^{50}+50 q^{48}-55 q^{46}+39 q^{44}-9 q^{42}-40 q^{40}+87 q^{38}-126 q^{36}+134 q^{34}-107 q^{32}+37 q^{30}+62 q^{28}-144 q^{26}+193 q^{24}-183 q^{22}+114 q^{20}-13 q^{18}-96 q^{16}+154 q^{14}-150 q^{12}+86 q^{10}+31 q^8-114 q^6+133 q^4-79 q^2-29+154 q^{-2} -237 q^{-4} +223 q^{-6} -125 q^{-8} -26 q^{-10} +202 q^{-12} -306 q^{-14} +309 q^{-16} -212 q^{-18} +55 q^{-20} +106 q^{-22} -224 q^{-24} +248 q^{-26} -183 q^{-28} +72 q^{-30} +62 q^{-32} -144 q^{-34} +145 q^{-36} -70 q^{-38} -41 q^{-40} +132 q^{-42} -177 q^{-44} +133 q^{-46} -32 q^{-48} -92 q^{-50} +197 q^{-52} -228 q^{-54} +181 q^{-56} -76 q^{-58} -50 q^{-60} +136 q^{-62} -175 q^{-64} +153 q^{-66} -90 q^{-68} +16 q^{-70} +43 q^{-72} -71 q^{-74} +70 q^{-76} -46 q^{-78} +22 q^{-80} -11 q^{-84} +12 q^{-86} -10 q^{-88} +6 q^{-90} -2 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_75.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-3 q^7+3 q^5-3 q^3+4 q+ q^{-1} - q^{-3} +2 q^{-5} -4 q^{-7} +3 q^{-9} -2 q^{-11} + q^{-13} }[/math]
2	[math]\displaystyle{ q^{26}-3 q^{24}+9 q^{20}-11 q^{18}-2 q^{16}+22 q^{14}-22 q^{12}-10 q^{10}+33 q^8-18 q^6-18 q^4+29 q^2+2-16 q^{-2} +4 q^{-4} +16 q^{-6} -8 q^{-8} -21 q^{-10} +22 q^{-12} +7 q^{-14} -33 q^{-16} +17 q^{-18} +20 q^{-20} -28 q^{-22} +4 q^{-24} +19 q^{-26} -13 q^{-28} -4 q^{-30} +8 q^{-32} -2 q^{-34} -2 q^{-36} + q^{-38} }[/math]
3	[math]\displaystyle{ q^{51}-3 q^{49}+6 q^{45}+q^{43}-11 q^{41}-5 q^{39}+26 q^{37}+2 q^{35}-43 q^{33}-9 q^{31}+72 q^{29}+23 q^{27}-105 q^{25}-42 q^{23}+133 q^{21}+78 q^{19}-156 q^{17}-116 q^{15}+147 q^{13}+156 q^{11}-126 q^9-173 q^7+70 q^5+184 q^3-10 q-157 q^{-1} -44 q^{-3} +120 q^{-5} +102 q^{-7} -79 q^{-9} -137 q^{-11} +22 q^{-13} +166 q^{-15} +18 q^{-17} -174 q^{-19} -74 q^{-21} +174 q^{-23} +117 q^{-25} -154 q^{-27} -154 q^{-29} +122 q^{-31} +179 q^{-33} -72 q^{-35} -183 q^{-37} +20 q^{-39} +170 q^{-41} +20 q^{-43} -131 q^{-45} -53 q^{-47} +88 q^{-49} +62 q^{-51} -48 q^{-53} -54 q^{-55} +16 q^{-57} +37 q^{-59} + q^{-61} -20 q^{-63} -4 q^{-65} +8 q^{-67} +3 q^{-69} -2 q^{-71} -2 q^{-73} + q^{-75} }[/math]
4	[math]\displaystyle{ q^{84}-3 q^{82}+6 q^{78}-2 q^{76}+q^{74}-14 q^{72}+5 q^{70}+26 q^{68}-16 q^{66}-9 q^{64}-42 q^{62}+37 q^{60}+102 q^{58}-42 q^{56}-89 q^{54}-143 q^{52}+115 q^{50}+316 q^{48}-16 q^{46}-283 q^{44}-423 q^{42}+169 q^{40}+724 q^{38}+224 q^{36}-493 q^{34}-943 q^{32}-14 q^{30}+1135 q^{28}+760 q^{26}-408 q^{24}-1459 q^{22}-553 q^{20}+1126 q^{18}+1297 q^{16}+139 q^{14}-1448 q^{12}-1119 q^{10}+498 q^8+1325 q^6+808 q^4-776 q^2-1241-341 q^{-2} +787 q^{-4} +1111 q^{-6} +110 q^{-8} -892 q^{-10} -941 q^{-12} +83 q^{-14} +1040 q^{-16} +800 q^{-18} -407 q^{-20} -1235 q^{-22} -485 q^{-24} +811 q^{-26} +1258 q^{-28} +74 q^{-30} -1310 q^{-32} -950 q^{-34} +436 q^{-36} +1509 q^{-38} +612 q^{-40} -1086 q^{-42} -1303 q^{-44} -172 q^{-46} +1402 q^{-48} +1119 q^{-50} -483 q^{-52} -1295 q^{-54} -819 q^{-56} +816 q^{-58} +1244 q^{-60} +256 q^{-62} -785 q^{-64} -1077 q^{-66} +56 q^{-68} +830 q^{-70} +627 q^{-72} -99 q^{-74} -759 q^{-76} -350 q^{-78} +221 q^{-80} +464 q^{-82} +254 q^{-84} -257 q^{-86} -278 q^{-88} -93 q^{-90} +137 q^{-92} +197 q^{-94} -2 q^{-96} -75 q^{-98} -84 q^{-100} -7 q^{-102} +58 q^{-104} +19 q^{-106} + q^{-108} -20 q^{-110} -11 q^{-112} +8 q^{-114} +3 q^{-116} +3 q^{-118} -2 q^{-120} -2 q^{-122} + q^{-124} }[/math]
5	[math]\displaystyle{ q^{125}-3 q^{123}+6 q^{119}-2 q^{117}-2 q^{115}-2 q^{113}-4 q^{111}+5 q^{109}+14 q^{107}-6 q^{105}-27 q^{103}-11 q^{101}+30 q^{99}+60 q^{97}+21 q^{95}-65 q^{93}-149 q^{91}-80 q^{89}+171 q^{87}+321 q^{85}+147 q^{83}-276 q^{81}-612 q^{79}-391 q^{77}+452 q^{75}+1136 q^{73}+772 q^{71}-598 q^{69}-1843 q^{67}-1513 q^{65}+611 q^{63}+2804 q^{61}+2676 q^{59}-373 q^{57}-3862 q^{55}-4272 q^{53}-408 q^{51}+4782 q^{49}+6318 q^{47}+1824 q^{45}-5292 q^{43}-8403 q^{41}-3932 q^{39}+4942 q^{37}+10239 q^{35}+6485 q^{33}-3625 q^{31}-11185 q^{29}-9089 q^{27}+1333 q^{25}+10980 q^{23}+11083 q^{21}+1516 q^{19}-9338 q^{17}-12117 q^{15}-4443 q^{13}+6725 q^{11}+11765 q^9+6815 q^7-3402 q^5-10244 q^3-8354 q+140 q^{-1} +7907 q^{-3} +8861 q^{-5} +2721 q^{-7} -5196 q^{-9} -8601 q^{-11} -4950 q^{-13} +2662 q^{-15} +7894 q^{-17} +6446 q^{-19} -426 q^{-21} -7047 q^{-23} -7609 q^{-25} -1330 q^{-27} +6335 q^{-29} +8415 q^{-31} +2851 q^{-33} -5657 q^{-35} -9279 q^{-37} -4256 q^{-39} +5029 q^{-41} +9984 q^{-43} +5791 q^{-45} -4091 q^{-47} -10585 q^{-49} -7436 q^{-51} +2722 q^{-53} +10720 q^{-55} +9145 q^{-57} -773 q^{-59} -10198 q^{-61} -10594 q^{-63} -1627 q^{-65} +8759 q^{-67} +11476 q^{-69} +4246 q^{-71} -6478 q^{-73} -11431 q^{-75} -6592 q^{-77} +3532 q^{-79} +10242 q^{-81} +8235 q^{-83} -365 q^{-85} -8093 q^{-87} -8765 q^{-89} -2409 q^{-91} +5236 q^{-93} +8065 q^{-95} +4393 q^{-97} -2297 q^{-99} -6419 q^{-101} -5148 q^{-103} -187 q^{-105} +4199 q^{-107} +4831 q^{-109} +1816 q^{-111} -2045 q^{-113} -3733 q^{-115} -2428 q^{-117} +365 q^{-119} +2334 q^{-121} +2244 q^{-123} +615 q^{-125} -1094 q^{-127} -1633 q^{-129} -913 q^{-131} +249 q^{-133} +926 q^{-135} +794 q^{-137} +169 q^{-139} -397 q^{-141} -517 q^{-143} -250 q^{-145} +96 q^{-147} +252 q^{-149} +192 q^{-151} +29 q^{-153} -100 q^{-155} -108 q^{-157} -37 q^{-159} +26 q^{-161} +40 q^{-163} +28 q^{-165} + q^{-167} -20 q^{-169} -11 q^{-171} + q^{-173} +3 q^{-175} +3 q^{-177} +3 q^{-179} -2 q^{-181} -2 q^{-183} + q^{-185} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}-2 q^{10}+q^8-3 q^4+4 q^2+3 q^{-2} + q^{-4} - q^{-6} +2 q^{-8} -3 q^{-10} -2 q^{-16} + q^{-18} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+4 q^{26}-q^{24}-5 q^{22}+3 q^{20}+11 q^{18}-6 q^{16}-15 q^{14}+8 q^{12}+16 q^{10}-15 q^8-17 q^6+15 q^4+11 q^2-6-3 q^{-2} +14 q^{-4} + q^{-6} -3 q^{-8} +8 q^{-10} - q^{-12} -12 q^{-14} +3 q^{-16} +8 q^{-18} -16 q^{-20} -11 q^{-22} +11 q^{-24} +11 q^{-26} -11 q^{-28} -7 q^{-30} +14 q^{-32} +9 q^{-34} -8 q^{-36} -7 q^{-38} +5 q^{-40} +6 q^{-42} -2 q^{-44} -5 q^{-46} - q^{-48} + q^{-50} + q^{-52} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{28}-3 q^{26}+8 q^{22}-9 q^{20}-2 q^{18}+19 q^{16}-15 q^{14}-12 q^{12}+24 q^{10}-16 q^8-16 q^6+27 q^4-2 q^2-6+14 q^{-2} +8 q^{-4} -6 q^{-6} -13 q^{-8} +9 q^{-10} +4 q^{-12} -24 q^{-14} +12 q^{-16} +17 q^{-18} -24 q^{-20} +9 q^{-22} +14 q^{-24} -19 q^{-26} +5 q^{-28} +7 q^{-30} -9 q^{-32} +3 q^{-34} +2 q^{-36} -2 q^{-38} + q^{-40} }[/math]
1,0,0	[math]\displaystyle{ q^{15}-2 q^{13}+2 q^{11}-2 q^9+q^7-3 q^5+3 q^3+2 q^{-1} +2 q^{-3} + q^{-5} +2 q^{-7} - q^{-9} +3 q^{-11} -3 q^{-13} -3 q^{-17} -2 q^{-21} + q^{-23} + q^{-25} + q^{-27} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{28}-3 q^{26}+6 q^{24}-10 q^{22}+15 q^{20}-20 q^{18}+25 q^{16}-29 q^{14}+28 q^{12}-26 q^{10}+18 q^8-8 q^6-5 q^4+22 q^2-34+48 q^{-2} -52 q^{-4} +58 q^{-6} -53 q^{-8} +47 q^{-10} -34 q^{-12} +20 q^{-14} -6 q^{-16} -9 q^{-18} +18 q^{-20} -27 q^{-22} +28 q^{-24} -29 q^{-26} +25 q^{-28} -21 q^{-30} +15 q^{-32} -9 q^{-34} +6 q^{-36} -2 q^{-38} + q^{-40} }[/math]
1,0	[math]\displaystyle{ q^{46}-3 q^{42}-3 q^{40}+3 q^{38}+9 q^{36}+2 q^{34}-12 q^{32}-11 q^{30}+9 q^{28}+22 q^{26}+4 q^{24}-25 q^{22}-20 q^{20}+13 q^{18}+28 q^{16}-2 q^{14}-31 q^{12}-13 q^{10}+23 q^8+23 q^6-12 q^4-19 q^2+9+26 q^{-2} +2 q^{-4} -18 q^{-6} -5 q^{-8} +17 q^{-10} +6 q^{-12} -18 q^{-14} -14 q^{-16} +14 q^{-18} +16 q^{-20} -14 q^{-22} -26 q^{-24} +4 q^{-26} +32 q^{-28} +11 q^{-30} -26 q^{-32} -24 q^{-34} +16 q^{-36} +30 q^{-38} + q^{-40} -25 q^{-42} -13 q^{-44} +14 q^{-46} +16 q^{-48} -4 q^{-50} -12 q^{-52} -3 q^{-54} +6 q^{-56} +4 q^{-58} -2 q^{-60} -2 q^{-62} + q^{-66} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+9 q^{58}-6 q^{56}-2 q^{54}+19 q^{52}-32 q^{50}+50 q^{48}-55 q^{46}+39 q^{44}-9 q^{42}-40 q^{40}+87 q^{38}-126 q^{36}+134 q^{34}-107 q^{32}+37 q^{30}+62 q^{28}-144 q^{26}+193 q^{24}-183 q^{22}+114 q^{20}-13 q^{18}-96 q^{16}+154 q^{14}-150 q^{12}+86 q^{10}+31 q^8-114 q^6+133 q^4-79 q^2-29+154 q^{-2} -237 q^{-4} +223 q^{-6} -125 q^{-8} -26 q^{-10} +202 q^{-12} -306 q^{-14} +309 q^{-16} -212 q^{-18} +55 q^{-20} +106 q^{-22} -224 q^{-24} +248 q^{-26} -183 q^{-28} +72 q^{-30} +62 q^{-32} -144 q^{-34} +145 q^{-36} -70 q^{-38} -41 q^{-40} +132 q^{-42} -177 q^{-44} +133 q^{-46} -32 q^{-48} -92 q^{-50} +197 q^{-52} -228 q^{-54} +181 q^{-56} -76 q^{-58} -50 q^{-60} +136 q^{-62} -175 q^{-64} +153 q^{-66} -90 q^{-68} +16 q^{-70} +43 q^{-72} -71 q^{-74} +70 q^{-76} -46 q^{-78} +22 q^{-80} -11 q^{-84} +12 q^{-86} -10 q^{-88} +6 q^{-90} -2 q^{-92} + q^{-94} }[/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 75"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+7 t^2-19 t+27-19 t^{-1} +7 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6+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{ \{3,t+1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-3 q^5+6 q^4-10 q^3+12 q^2-13 q+14-10 q^{-1} +7 q^{-2} -4 q^{-3} + q^{-4} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^6+a^2 z^4+3 z^4 a^{-2} -3 z^4+a^2 z^2+6 z^2 a^{-2} -3 z^2 a^{-4} -4 z^2+3 a^{-2} -3 a^{-4} + a^{-6} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +7 z^8 a^{-2} +3 z^8 a^{-4} +4 z^8+7 a z^7+13 z^7 a^{-1} +9 z^7 a^{-3} +3 z^7 a^{-5} +7 a^2 z^6-4 z^6 a^{-2} -3 z^6 a^{-4} +z^6 a^{-6} +7 z^6+4 a^3 z^5-5 a z^5-29 z^5 a^{-1} -29 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-8 a^2 z^4-21 z^4 a^{-2} -9 z^4 a^{-4} -3 z^4 a^{-6} -24 z^4-3 a^3 z^3-a z^3+17 z^3 a^{-1} +24 z^3 a^{-3} +9 z^3 a^{-5} +4 a^2 z^2+20 z^2 a^{-2} +12 z^2 a^{-4} +3 z^2 a^{-6} +15 z^2-a z-5 z a^{-1} -7 z a^{-3} -3 z a^{-5} -3 a^{-2} -3 a^{-4} - a^{-6} }[/math]