10 96

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+7 t^2-22 t+33-22 t^{-1} +7 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 93, 0 }
Jones polynomial	[math]\displaystyle{ q^6-3 q^5+7 q^4-11 q^3+14 q^2-16 q+15-12 q^{-1} +9 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+5 z^2 a^{-2} -3 z^2 a^{-4} -6 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +11 z^8 a^{-2} +4 z^8 a^{-4} +7 z^8+11 a z^7+14 z^7 a^{-1} +6 z^7 a^{-3} +3 z^7 a^{-5} +9 a^2 z^6-17 z^6 a^{-2} -7 z^6 a^{-4} +z^6 a^{-6} +4 a^3 z^5-15 a z^5-34 z^5 a^{-1} -23 z^5 a^{-3} -8 z^5 a^{-5} +a^4 z^4-10 a^2 z^4-4 z^4 a^{-2} -z^4 a^{-4} -3 z^4 a^{-6} -17 z^4-a^3 z^3+7 a z^3+17 z^3 a^{-1} +16 z^3 a^{-3} +7 z^3 a^{-5} +5 a^2 z^2+10 z^2 a^{-2} +6 z^2 a^{-4} +3 z^2 a^{-6} +12 z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}-2 q^{10}+3 q^8+2 q^6-3 q^4+3 q^2-3+ q^{-2} - q^{-6} +3 q^{-8} -3 q^{-10} + q^{-12} + q^{-14} -2 q^{-16} + q^{-18} + q^{-20} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+11 q^{58}-10 q^{56}+4 q^{54}+15 q^{52}-37 q^{50}+63 q^{48}-80 q^{46}+68 q^{44}-36 q^{42}-30 q^{40}+119 q^{38}-191 q^{36}+229 q^{34}-189 q^{32}+78 q^{30}+83 q^{28}-238 q^{26}+334 q^{24}-324 q^{22}+195 q^{20}+4 q^{18}-197 q^{16}+307 q^{14}-272 q^{12}+120 q^{10}+86 q^8-245 q^6+269 q^4-161 q^2-63+294 q^{-2} -425 q^{-4} +394 q^{-6} -202 q^{-8} -84 q^{-10} +357 q^{-12} -513 q^{-14} +493 q^{-16} -322 q^{-18} +52 q^{-20} +219 q^{-22} -388 q^{-24} +414 q^{-26} -276 q^{-28} +57 q^{-30} +160 q^{-32} -281 q^{-34} +251 q^{-36} -98 q^{-38} -109 q^{-40} +278 q^{-42} -326 q^{-44} +228 q^{-46} -21 q^{-48} -206 q^{-50} +357 q^{-52} -373 q^{-54} +255 q^{-56} -70 q^{-58} -122 q^{-60} +239 q^{-62} -260 q^{-64} +201 q^{-66} -91 q^{-68} -11 q^{-70} +76 q^{-72} -97 q^{-74} +81 q^{-76} -47 q^{-78} +18 q^{-80} +4 q^{-82} -13 q^{-84} +13 q^{-86} -10 q^{-88} +6 q^{-90} -2 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_96.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-3 q^7+5 q^5-3 q^3+3 q- q^{-1} -2 q^{-3} +3 q^{-5} -4 q^{-7} +4 q^{-9} -2 q^{-11} + q^{-13} }[/math]
2	[math]\displaystyle{ q^{26}-3 q^{24}+2 q^{22}+9 q^{20}-18 q^{18}+32 q^{14}-31 q^{12}-13 q^{10}+49 q^8-22 q^6-29 q^4+36 q^2+3-28 q^{-2} +3 q^{-4} +26 q^{-6} -10 q^{-8} -29 q^{-10} +34 q^{-12} +13 q^{-14} -47 q^{-16} +21 q^{-18} +30 q^{-20} -38 q^{-22} +26 q^{-26} -14 q^{-28} -7 q^{-30} +9 q^{-32} - q^{-34} -2 q^{-36} + q^{-38} }[/math]
3	[math]\displaystyle{ q^{51}-3 q^{49}+2 q^{47}+6 q^{45}-6 q^{43}-15 q^{41}+9 q^{39}+42 q^{37}-17 q^{35}-83 q^{33}+17 q^{31}+138 q^{29}+14 q^{27}-216 q^{25}-66 q^{23}+273 q^{21}+153 q^{19}-292 q^{17}-247 q^{15}+260 q^{13}+325 q^{11}-180 q^9-361 q^7+72 q^5+341 q^3+46 q-288 q^{-1} -142 q^{-3} +203 q^{-5} +225 q^{-7} -116 q^{-9} -275 q^{-11} +22 q^{-13} +313 q^{-15} +72 q^{-17} -329 q^{-19} -160 q^{-21} +310 q^{-23} +250 q^{-25} -262 q^{-27} -321 q^{-29} +177 q^{-31} +356 q^{-33} -72 q^{-35} -343 q^{-37} -34 q^{-39} +288 q^{-41} +107 q^{-43} -195 q^{-45} -138 q^{-47} +102 q^{-49} +128 q^{-51} -35 q^{-53} -88 q^{-55} -5 q^{-57} +48 q^{-59} +15 q^{-61} -20 q^{-63} -10 q^{-65} +6 q^{-67} +4 q^{-69} - q^{-71} -2 q^{-73} + q^{-75} }[/math]
4	[math]\displaystyle{ q^{84}-3 q^{82}+2 q^{80}+6 q^{78}-9 q^{76}-3 q^{74}-6 q^{72}+25 q^{70}+32 q^{68}-58 q^{66}-49 q^{64}-19 q^{62}+143 q^{60}+177 q^{58}-185 q^{56}-317 q^{54}-183 q^{52}+471 q^{50}+752 q^{48}-180 q^{46}-992 q^{44}-988 q^{42}+660 q^{40}+1977 q^{38}+686 q^{36}-1526 q^{34}-2615 q^{32}-221 q^{30}+2933 q^{28}+2513 q^{26}-723 q^{24}-3792 q^{22}-2158 q^{20}+2197 q^{18}+3763 q^{16}+1268 q^{14}-3046 q^{12}-3458 q^{10}+76 q^8+3107 q^6+2746 q^4-912 q^2-3050-1735 q^{-2} +1271 q^{-4} +2849 q^{-6} +1033 q^{-8} -1732 q^{-10} -2562 q^{-12} -383 q^{-14} +2294 q^{-16} +2268 q^{-18} -527 q^{-20} -2906 q^{-22} -1613 q^{-24} +1659 q^{-26} +3193 q^{-28} +657 q^{-30} -2957 q^{-32} -2812 q^{-34} +590 q^{-36} +3703 q^{-38} +2169 q^{-40} -2095 q^{-42} -3634 q^{-44} -1211 q^{-46} +2985 q^{-48} +3378 q^{-50} -124 q^{-52} -3072 q^{-54} -2775 q^{-56} +941 q^{-58} +3049 q^{-60} +1677 q^{-62} -1121 q^{-64} -2690 q^{-66} -911 q^{-68} +1297 q^{-70} +1830 q^{-72} +567 q^{-74} -1223 q^{-76} -1182 q^{-78} -161 q^{-80} +780 q^{-82} +805 q^{-84} -70 q^{-86} -470 q^{-88} -395 q^{-90} +30 q^{-92} +311 q^{-94} +138 q^{-96} -22 q^{-98} -133 q^{-100} -69 q^{-102} +41 q^{-104} +34 q^{-106} +23 q^{-108} -14 q^{-110} -16 q^{-112} +3 q^{-114} + q^{-116} +4 q^{-118} - q^{-120} -2 q^{-122} + q^{-124} }[/math]
5	[math]\displaystyle{ q^{125}-3 q^{123}+2 q^{121}+6 q^{119}-9 q^{117}-6 q^{115}+6 q^{113}+10 q^{111}+15 q^{109}-3 q^{107}-48 q^{105}-56 q^{103}+42 q^{101}+145 q^{99}+119 q^{97}-95 q^{95}-346 q^{93}-336 q^{91}+136 q^{89}+823 q^{87}+867 q^{85}-140 q^{83}-1585 q^{81}-1997 q^{79}-294 q^{77}+2663 q^{75}+4140 q^{73}+1652 q^{71}-3769 q^{69}-7394 q^{67}-4631 q^{65}+3967 q^{63}+11549 q^{61}+9927 q^{59}-2279 q^{57}-15649 q^{55}-17196 q^{53}-2539 q^{51}+17911 q^{49}+25619 q^{47}+10696 q^{45}-16770 q^{43}-32921 q^{41}-21228 q^{39}+11069 q^{37}+36814 q^{35}+32043 q^{33}-1366 q^{31}-35574 q^{29}-40327 q^{27}-10445 q^{25}+28877 q^{23}+43868 q^{21}+21724 q^{19}-18214 q^{17}-41801 q^{15}-29894 q^{13}+5958 q^{11}+34897 q^9+33650 q^7+5317 q^5-25118 q^3-33074 q-13948 q^{-1} +14857 q^{-3} +29481 q^{-5} +19294 q^{-7} -5762 q^{-9} -24644 q^{-11} -22168 q^{-13} -1138 q^{-15} +20180 q^{-17} +23554 q^{-19} +6143 q^{-21} -16835 q^{-23} -24883 q^{-25} -10068 q^{-27} +14747 q^{-29} +26768 q^{-31} +13965 q^{-33} -13016 q^{-35} -29348 q^{-37} -18765 q^{-39} +10575 q^{-41} +32004 q^{-43} +24653 q^{-45} -6324 q^{-47} -33394 q^{-49} -31233 q^{-51} -408 q^{-53} +32267 q^{-55} +37240 q^{-57} +9286 q^{-59} -27460 q^{-61} -40930 q^{-63} -19220 q^{-65} +18847 q^{-67} +40696 q^{-69} +28154 q^{-71} -7396 q^{-73} -35691 q^{-75} -33819 q^{-77} -4838 q^{-79} +26289 q^{-81} +34643 q^{-83} +15255 q^{-85} -14398 q^{-87} -30268 q^{-89} -21479 q^{-91} +2578 q^{-93} +21920 q^{-95} +22620 q^{-97} +6502 q^{-99} -12116 q^{-101} -19138 q^{-103} -11260 q^{-105} +3337 q^{-107} +13010 q^{-109} +11772 q^{-111} +2499 q^{-113} -6637 q^{-115} -9240 q^{-117} -4935 q^{-119} +1714 q^{-121} +5626 q^{-123} +4792 q^{-125} +934 q^{-127} -2486 q^{-129} -3294 q^{-131} -1704 q^{-133} +514 q^{-135} +1707 q^{-137} +1410 q^{-139} +287 q^{-141} -623 q^{-143} -800 q^{-145} -393 q^{-147} +96 q^{-149} +334 q^{-151} +265 q^{-153} +43 q^{-155} -105 q^{-157} -110 q^{-159} -43 q^{-161} +11 q^{-163} +40 q^{-165} +26 q^{-167} -6 q^{-169} -10 q^{-171} -3 q^{-173} -2 q^{-175} + q^{-177} +4 q^{-179} - q^{-181} -2 q^{-183} + q^{-185} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}-2 q^{10}+3 q^8+2 q^6-3 q^4+3 q^2-3+ q^{-2} - q^{-6} +3 q^{-8} -3 q^{-10} + q^{-12} + q^{-14} -2 q^{-16} + q^{-18} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{32}-2 q^{30}+q^{28}+6 q^{26}-4 q^{24}-9 q^{22}+6 q^{20}+17 q^{18}-9 q^{16}-25 q^{14}+11 q^{12}+26 q^{10}-12 q^8-20 q^6+18 q^4+14 q^2-11-10 q^{-2} +7 q^{-4} -3 q^{-6} -7 q^{-8} +11 q^{-10} -3 q^{-12} -12 q^{-14} +12 q^{-16} +19 q^{-18} -15 q^{-20} -15 q^{-22} +16 q^{-24} +15 q^{-26} -16 q^{-28} -17 q^{-30} +14 q^{-32} +13 q^{-34} -8 q^{-36} -11 q^{-38} +3 q^{-40} +8 q^{-42} -4 q^{-46} - q^{-48} + q^{-50} + q^{-52} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{28}-3 q^{26}+10 q^{22}-11 q^{20}-6 q^{18}+26 q^{16}-18 q^{14}-12 q^{12}+38 q^{10}-18 q^8-18 q^6+31 q^4-12 q^2-17+10 q^{-2} +5 q^{-4} -4 q^{-6} -12 q^{-8} +18 q^{-10} +12 q^{-12} -30 q^{-14} +15 q^{-16} +20 q^{-18} -35 q^{-20} +12 q^{-22} +17 q^{-24} -25 q^{-26} +9 q^{-28} +8 q^{-30} -10 q^{-32} +4 q^{-34} +2 q^{-36} -2 q^{-38} + q^{-40} }[/math]
1,0,0	[math]\displaystyle{ q^{15}-2 q^{13}+4 q^{11}+3 q^7-3 q^5+2 q^3-3 q- q^{-1} +2 q^{-7} - q^{-9} +4 q^{-11} -3 q^{-13} + q^{-15} -2 q^{-17} + q^{-19} -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}-12 q^{22}+21 q^{20}-28 q^{18}+36 q^{16}-40 q^{14}+42 q^{12}-36 q^{10}+26 q^8-10 q^6-9 q^4+30 q^2-51+66 q^{-2} -77 q^{-4} +80 q^{-6} -74 q^{-8} +62 q^{-10} -44 q^{-12} +24 q^{-14} -3 q^{-16} -16 q^{-18} +29 q^{-20} -38 q^{-22} +41 q^{-24} -39 q^{-26} +33 q^{-28} -26 q^{-30} +18 q^{-32} -10 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}+11 q^{36}+4 q^{34}-16 q^{32}-17 q^{30}+9 q^{28}+31 q^{26}+8 q^{24}-33 q^{22}-27 q^{20}+23 q^{18}+42 q^{16}-q^{14}-42 q^{12}-16 q^{10}+33 q^8+27 q^6-23 q^4-32 q^2+10+30 q^{-2} -2 q^{-4} -30 q^{-6} -5 q^{-8} +27 q^{-10} +11 q^{-12} -24 q^{-14} -15 q^{-16} +25 q^{-18} +26 q^{-20} -19 q^{-22} -37 q^{-24} +7 q^{-26} +44 q^{-28} +12 q^{-30} -39 q^{-32} -32 q^{-34} +23 q^{-36} +40 q^{-38} -3 q^{-40} -34 q^{-42} -14 q^{-44} +20 q^{-46} +19 q^{-48} -6 q^{-50} -14 q^{-52} -2 q^{-54} +7 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}+11 q^{58}-10 q^{56}+4 q^{54}+15 q^{52}-37 q^{50}+63 q^{48}-80 q^{46}+68 q^{44}-36 q^{42}-30 q^{40}+119 q^{38}-191 q^{36}+229 q^{34}-189 q^{32}+78 q^{30}+83 q^{28}-238 q^{26}+334 q^{24}-324 q^{22}+195 q^{20}+4 q^{18}-197 q^{16}+307 q^{14}-272 q^{12}+120 q^{10}+86 q^8-245 q^6+269 q^4-161 q^2-63+294 q^{-2} -425 q^{-4} +394 q^{-6} -202 q^{-8} -84 q^{-10} +357 q^{-12} -513 q^{-14} +493 q^{-16} -322 q^{-18} +52 q^{-20} +219 q^{-22} -388 q^{-24} +414 q^{-26} -276 q^{-28} +57 q^{-30} +160 q^{-32} -281 q^{-34} +251 q^{-36} -98 q^{-38} -109 q^{-40} +278 q^{-42} -326 q^{-44} +228 q^{-46} -21 q^{-48} -206 q^{-50} +357 q^{-52} -373 q^{-54} +255 q^{-56} -70 q^{-58} -122 q^{-60} +239 q^{-62} -260 q^{-64} +201 q^{-66} -91 q^{-68} -11 q^{-70} +76 q^{-72} -97 q^{-74} +81 q^{-76} -47 q^{-78} +18 q^{-80} +4 q^{-82} -13 q^{-84} +13 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 96"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+7 t^2-22 t+33-22 t^{-1} +7 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6+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]=

{ 93, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-3 q^5+7 q^4-11 q^3+14 q^2-16 q+15-12 q^{-1} +9 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+5 z^2 a^{-2} -3 z^2 a^{-4} -6 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +11 z^8 a^{-2} +4 z^8 a^{-4} +7 z^8+11 a z^7+14 z^7 a^{-1} +6 z^7 a^{-3} +3 z^7 a^{-5} +9 a^2 z^6-17 z^6 a^{-2} -7 z^6 a^{-4} +z^6 a^{-6} +4 a^3 z^5-15 a z^5-34 z^5 a^{-1} -23 z^5 a^{-3} -8 z^5 a^{-5} +a^4 z^4-10 a^2 z^4-4 z^4 a^{-2} -z^4 a^{-4} -3 z^4 a^{-6} -17 z^4-a^3 z^3+7 a z^3+17 z^3 a^{-1} +16 z^3 a^{-3} +7 z^3 a^{-5} +5 a^2 z^2+10 z^2 a^{-2} +6 z^2 a^{-4} +3 z^2 a^{-6} +12 z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3 }[/math]