10 22
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ -2 t^3+6 t^2-10 t+13-10 t^{-1} +6 t^{-2} -2 t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ -2 z^6-6 z^4-4 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 49, 0 } |
| Jones polynomial | [math]\displaystyle{ q^6-2 q^5+4 q^4-6 q^3+7 q^2-8 q+8-6 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} -4 z^4+3 a^2 z^2-5 z^2 a^{-2} +3 z^2 a^{-4} -5 z^2+2 a^2-2 a^{-2} +2 a^{-4} -1 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8+3 a z^7-z^7 a^{-3} +2 z^7 a^{-5} +3 a^2 z^6-12 z^6 a^{-2} -6 z^6 a^{-4} +z^6 a^{-6} -2 z^6+2 a^3 z^5-6 a z^5-z^5 a^{-1} -7 z^5 a^{-5} +a^4 z^4-6 a^2 z^4+16 z^4 a^{-2} +6 z^4 a^{-4} -4 z^4 a^{-6} -z^4-3 a^3 z^3+7 a z^3-4 z^3 a^{-3} +6 z^3 a^{-5} -2 a^4 z^2+6 a^2 z^2-12 z^2 a^{-2} -6 z^2 a^{-4} +4 z^2 a^{-6} +6 z^2-a z+z a^{-1} +z a^{-3} -z a^{-5} -2 a^2+2 a^{-2} +2 a^{-4} -1 }[/math] |
| The A2 invariant | [math]\displaystyle{ q^{12}+q^8+q^6-q^4+2 q^2-1- q^{-4} -2 q^{-6} + q^{-8} - q^{-10} + q^{-12} + q^{-14} + q^{-18} }[/math] |
| The G2 invariant | [math]\displaystyle{ q^{66}-q^{64}+2 q^{62}-3 q^{60}+2 q^{58}-q^{56}-2 q^{54}+6 q^{52}-7 q^{50}+10 q^{48}-10 q^{46}+6 q^{44}+q^{42}-10 q^{40}+18 q^{38}-22 q^{36}+21 q^{34}-15 q^{32}+4 q^{30}+12 q^{28}-22 q^{26}+33 q^{24}-29 q^{22}+21 q^{20}-6 q^{18}-10 q^{16}+24 q^{14}-27 q^{12}+23 q^{10}-3 q^8-11 q^6+19 q^4-18 q^2+3+17 q^{-2} -36 q^{-4} +35 q^{-6} -27 q^{-8} +29 q^{-12} -52 q^{-14} +54 q^{-16} -43 q^{-18} +14 q^{-20} +13 q^{-22} -39 q^{-24} +48 q^{-26} -42 q^{-28} +24 q^{-30} + q^{-32} -21 q^{-34} +31 q^{-36} -23 q^{-38} +7 q^{-40} +12 q^{-42} -25 q^{-44} +26 q^{-46} -14 q^{-48} -7 q^{-50} +31 q^{-52} -40 q^{-54} +39 q^{-56} -20 q^{-58} -5 q^{-60} +26 q^{-62} -36 q^{-64} +37 q^{-66} -25 q^{-68} +8 q^{-70} +8 q^{-72} -18 q^{-74} +20 q^{-76} -15 q^{-78} +9 q^{-80} -2 q^{-82} -3 q^{-84} +4 q^{-86} -5 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 10_22.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | [math]\displaystyle{ q^9-q^7+2 q^5-2 q^3+2 q- q^{-3} + q^{-5} -2 q^{-7} +2 q^{-9} - q^{-11} + q^{-13} }[/math] |
| 2 | [math]\displaystyle{ q^{26}-q^{24}+3 q^{20}-3 q^{18}-q^{16}+6 q^{14}-6 q^{12}-3 q^{10}+10 q^8-7 q^6-5 q^4+10 q^2-1-4 q^{-2} +2 q^{-4} +5 q^{-6} -3 q^{-8} -5 q^{-10} +8 q^{-12} + q^{-14} -9 q^{-16} +6 q^{-18} +5 q^{-20} -10 q^{-22} +2 q^{-24} +6 q^{-26} -6 q^{-28} - q^{-30} +4 q^{-32} - q^{-34} - q^{-36} + q^{-38} }[/math] |
| 3 | [math]\displaystyle{ q^{51}-q^{49}+q^{45}+q^{43}-2 q^{41}-q^{39}+2 q^{37}+q^{35}-4 q^{33}-q^{31}+6 q^{29}+q^{27}-10 q^{25}-q^{23}+16 q^{21}+3 q^{19}-22 q^{17}-9 q^{15}+29 q^{13}+15 q^{11}-29 q^9-20 q^7+22 q^5+26 q^3-13 q-24 q^{-1} +4 q^{-3} +21 q^{-5} +11 q^{-7} -17 q^{-9} -19 q^{-11} +9 q^{-13} +26 q^{-15} -7 q^{-17} -30 q^{-19} +31 q^{-23} +7 q^{-25} -31 q^{-27} -12 q^{-29} +27 q^{-31} +20 q^{-33} -21 q^{-35} -25 q^{-37} +12 q^{-39} +30 q^{-41} -3 q^{-43} -26 q^{-45} -6 q^{-47} +21 q^{-49} +11 q^{-51} -14 q^{-53} -12 q^{-55} +5 q^{-57} +10 q^{-59} - q^{-61} -6 q^{-63} - q^{-65} +3 q^{-67} + q^{-69} - q^{-71} - q^{-73} + q^{-75} }[/math] |
| 4 | [math]\displaystyle{ q^{84}-q^{82}+q^{78}-q^{76}+2 q^{74}-3 q^{72}+2 q^{68}-4 q^{66}+6 q^{64}-3 q^{62}+2 q^{58}-10 q^{56}+9 q^{54}-q^{52}+7 q^{50}+4 q^{48}-25 q^{46}-q^{42}+32 q^{40}+29 q^{38}-42 q^{36}-40 q^{34}-30 q^{32}+64 q^{30}+94 q^{28}-22 q^{26}-85 q^{24}-105 q^{22}+47 q^{20}+154 q^{18}+47 q^{16}-74 q^{14}-160 q^{12}-23 q^{10}+130 q^8+101 q^6+3 q^4-127 q^2-81+40 q^{-2} +90 q^{-4} +67 q^{-6} -39 q^{-8} -85 q^{-10} -47 q^{-12} +41 q^{-14} +91 q^{-16} +32 q^{-18} -69 q^{-20} -95 q^{-22} +11 q^{-24} +98 q^{-26} +74 q^{-28} -59 q^{-30} -127 q^{-32} -9 q^{-34} +100 q^{-36} +111 q^{-38} -38 q^{-40} -143 q^{-42} -49 q^{-44} +67 q^{-46} +143 q^{-48} +21 q^{-50} -119 q^{-52} -95 q^{-54} -11 q^{-56} +126 q^{-58} +87 q^{-60} -36 q^{-62} -89 q^{-64} -90 q^{-66} +49 q^{-68} +94 q^{-70} +44 q^{-72} -24 q^{-74} -96 q^{-76} -24 q^{-78} +36 q^{-80} +56 q^{-82} +32 q^{-84} -44 q^{-86} -34 q^{-88} -12 q^{-90} +19 q^{-92} +33 q^{-94} -4 q^{-96} -9 q^{-98} -15 q^{-100} -3 q^{-102} +12 q^{-104} + q^{-106} +2 q^{-108} -4 q^{-110} -3 q^{-112} +3 q^{-114} + q^{-118} - q^{-120} - q^{-122} + q^{-124} }[/math] |
| 5 | [math]\displaystyle{ q^{125}-q^{123}+q^{119}-q^{117}+q^{113}-2 q^{111}-q^{109}+2 q^{107}+4 q^{101}-2 q^{99}-6 q^{97}-2 q^{95}+q^{93}+6 q^{91}+11 q^{89}+q^{87}-14 q^{85}-17 q^{83}-7 q^{81}+16 q^{79}+31 q^{77}+23 q^{75}-13 q^{73}-51 q^{71}-53 q^{69}-3 q^{67}+69 q^{65}+102 q^{63}+51 q^{61}-71 q^{59}-172 q^{57}-136 q^{55}+44 q^{53}+241 q^{51}+261 q^{49}+38 q^{47}-286 q^{45}-417 q^{43}-178 q^{41}+281 q^{39}+560 q^{37}+357 q^{35}-192 q^{33}-644 q^{31}-563 q^{29}+41 q^{27}+643 q^{25}+710 q^{23}+156 q^{21}-535 q^{19}-766 q^{17}-355 q^{15}+352 q^{13}+724 q^{11}+477 q^9-132 q^7-572 q^5-525 q^3-74 q+381 q^{-1} +490 q^{-3} +219 q^{-5} -177 q^{-7} -388 q^{-9} -301 q^{-11} +2 q^{-13} +293 q^{-15} +334 q^{-17} +105 q^{-19} -199 q^{-21} -345 q^{-23} -188 q^{-25} +157 q^{-27} +367 q^{-29} +222 q^{-31} -146 q^{-33} -395 q^{-35} -273 q^{-37} +148 q^{-39} +458 q^{-41} +326 q^{-43} -142 q^{-45} -516 q^{-47} -409 q^{-49} +103 q^{-51} +553 q^{-53} +514 q^{-55} -17 q^{-57} -552 q^{-59} -602 q^{-61} -119 q^{-63} +479 q^{-65} +663 q^{-67} +278 q^{-69} -336 q^{-71} -653 q^{-73} -430 q^{-75} +140 q^{-77} +567 q^{-79} +522 q^{-81} +80 q^{-83} -395 q^{-85} -539 q^{-87} -265 q^{-89} +186 q^{-91} +452 q^{-93} +370 q^{-95} +37 q^{-97} -303 q^{-99} -390 q^{-101} -189 q^{-103} +119 q^{-105} +309 q^{-107} +266 q^{-109} +44 q^{-111} -187 q^{-113} -255 q^{-115} -137 q^{-117} +56 q^{-119} +177 q^{-121} +165 q^{-123} +41 q^{-125} -91 q^{-127} -132 q^{-129} -77 q^{-131} +15 q^{-133} +76 q^{-135} +75 q^{-137} +25 q^{-139} -29 q^{-141} -48 q^{-143} -32 q^{-145} + q^{-147} +20 q^{-149} +22 q^{-151} +11 q^{-153} -6 q^{-155} -12 q^{-157} -6 q^{-159} + q^{-163} +5 q^{-165} +2 q^{-167} -3 q^{-169} - q^{-171} + q^{-173} + q^{-179} - q^{-181} - q^{-183} + q^{-185} }[/math] |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{12}+q^8+q^6-q^4+2 q^2-1- q^{-4} -2 q^{-6} + q^{-8} - q^{-10} + q^{-12} + q^{-14} + q^{-18} }[/math] |
| 1,1 | [math]\displaystyle{ q^{36}-2 q^{34}+4 q^{32}-8 q^{30}+15 q^{28}-18 q^{26}+28 q^{24}-40 q^{22}+54 q^{20}-62 q^{18}+72 q^{16}-90 q^{14}+96 q^{12}-92 q^{10}+88 q^8-82 q^6+56 q^4-18 q^2-24+70 q^{-2} -124 q^{-4} +172 q^{-6} -206 q^{-8} +234 q^{-10} -233 q^{-12} +224 q^{-14} -186 q^{-16} +148 q^{-18} -95 q^{-20} +28 q^{-22} +24 q^{-24} -70 q^{-26} +102 q^{-28} -130 q^{-30} +134 q^{-32} -120 q^{-34} +104 q^{-36} -86 q^{-38} +62 q^{-40} -38 q^{-42} +25 q^{-44} -14 q^{-46} +6 q^{-48} -2 q^{-50} + q^{-52} }[/math] |
| 2,0 | [math]\displaystyle{ q^{32}+2 q^{26}+2 q^{24}-q^{22}+2 q^{18}-q^{16}-5 q^{14}+3 q^{10}-5 q^8-4 q^6+4 q^4+2 q^2-2+ q^{-2} +5 q^{-4} - q^{-6} -2 q^{-8} +4 q^{-10} +2 q^{-12} -2 q^{-14} +3 q^{-16} +5 q^{-18} -2 q^{-20} -2 q^{-22} + q^{-24} -5 q^{-28} -3 q^{-30} +2 q^{-32} -2 q^{-36} +2 q^{-40} + q^{-42} + q^{-48} }[/math] |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | [math]\displaystyle{ q^{28}-q^{26}+2 q^{22}-3 q^{20}+7 q^{16}-4 q^{14}-2 q^{12}+10 q^{10}-3 q^8-5 q^6+8 q^4-2 q^2-5+2 q^{-2} -3 q^{-6} -4 q^{-8} +4 q^{-10} +2 q^{-12} -7 q^{-14} +4 q^{-16} +6 q^{-18} -7 q^{-20} +3 q^{-22} +6 q^{-24} -5 q^{-26} +2 q^{-28} +2 q^{-30} -4 q^{-32} + q^{-34} + q^{-36} - q^{-38} + q^{-40} }[/math] |
| 1,0,0 | [math]\displaystyle{ q^{15}+2 q^{11}+2 q^7-q^5+2 q^3-q- q^{-3} -2 q^{-5} - q^{-7} -2 q^{-9} + q^{-11} - q^{-13} +2 q^{-15} +2 q^{-19} + q^{-23} }[/math] |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | [math]\displaystyle{ q^{34}+q^{28}-q^{24}+q^{22}+4 q^{20}+2 q^{18}-q^{16}+3 q^{14}+6 q^{12}-2 q^{10}-6 q^8+3 q^6-q^4-7 q^2-3+3 q^{-2} -2 q^{-4} -4 q^{-6} +5 q^{-8} +4 q^{-10} -3 q^{-12} + q^{-14} +8 q^{-16} -4 q^{-18} -5 q^{-20} +5 q^{-22} +2 q^{-24} -5 q^{-26} - q^{-28} +5 q^{-30} -3 q^{-34} + q^{-36} +2 q^{-38} -2 q^{-40} +2 q^{-44} + q^{-50} }[/math] |
| 1,0,0,0 | [math]\displaystyle{ q^{18}+2 q^{14}+q^{12}+q^{10}+2 q^8-q^6+2 q^4-q^2- q^{-2} - q^{-4} -2 q^{-6} -2 q^{-8} - q^{-10} -2 q^{-12} + q^{-14} - q^{-16} +2 q^{-18} + q^{-20} + q^{-22} +2 q^{-24} + q^{-28} }[/math] |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | [math]\displaystyle{ q^{28}-q^{26}+2 q^{24}-4 q^{22}+5 q^{20}-6 q^{18}+9 q^{16}-8 q^{14}+10 q^{12}-8 q^{10}+7 q^8-3 q^6+6 q^2-11+14 q^{-2} -18 q^{-4} +17 q^{-6} -20 q^{-8} +16 q^{-10} -14 q^{-12} +9 q^{-14} -4 q^{-16} +5 q^{-20} -7 q^{-22} +10 q^{-24} -9 q^{-26} +10 q^{-28} -8 q^{-30} +6 q^{-32} -5 q^{-34} +3 q^{-36} - q^{-38} + q^{-40} }[/math] |
| 1,0 | [math]\displaystyle{ q^{46}-q^{42}-q^{40}+q^{38}+3 q^{36}-4 q^{32}-3 q^{30}+3 q^{28}+8 q^{26}+q^{24}-7 q^{22}-6 q^{20}+5 q^{18}+10 q^{16}+q^{14}-9 q^{12}-5 q^{10}+6 q^8+8 q^6-4 q^4-8 q^2+1+7 q^{-2} + q^{-4} -7 q^{-6} -3 q^{-8} +4 q^{-10} +4 q^{-12} -5 q^{-14} -4 q^{-16} +4 q^{-18} +6 q^{-20} -3 q^{-22} -8 q^{-24} + q^{-26} +10 q^{-28} +5 q^{-30} -8 q^{-32} -9 q^{-34} +4 q^{-36} +11 q^{-38} +2 q^{-40} -8 q^{-42} -6 q^{-44} +5 q^{-46} +6 q^{-48} - q^{-50} -5 q^{-52} -2 q^{-54} +3 q^{-56} +2 q^{-58} - q^{-60} - q^{-62} + q^{-66} }[/math] |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | [math]\displaystyle{ q^{38}-q^{36}+q^{34}-2 q^{32}+3 q^{30}-4 q^{28}+4 q^{26}-4 q^{24}+8 q^{22}-6 q^{20}+7 q^{18}-5 q^{16}+10 q^{14}-4 q^{12}+4 q^{10}-3 q^8+2 q^6+3 q^4-6 q^2+6-12 q^{-2} +11 q^{-4} -14 q^{-6} +12 q^{-8} -17 q^{-10} +13 q^{-12} -13 q^{-14} +11 q^{-16} -10 q^{-18} +6 q^{-20} -3 q^{-22} +2 q^{-24} +2 q^{-26} -2 q^{-28} +8 q^{-30} -5 q^{-32} +8 q^{-34} -7 q^{-36} +9 q^{-38} -7 q^{-40} +5 q^{-42} -6 q^{-44} +4 q^{-46} -3 q^{-48} +2 q^{-50} - q^{-52} + q^{-54} }[/math] |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{66}-q^{64}+2 q^{62}-3 q^{60}+2 q^{58}-q^{56}-2 q^{54}+6 q^{52}-7 q^{50}+10 q^{48}-10 q^{46}+6 q^{44}+q^{42}-10 q^{40}+18 q^{38}-22 q^{36}+21 q^{34}-15 q^{32}+4 q^{30}+12 q^{28}-22 q^{26}+33 q^{24}-29 q^{22}+21 q^{20}-6 q^{18}-10 q^{16}+24 q^{14}-27 q^{12}+23 q^{10}-3 q^8-11 q^6+19 q^4-18 q^2+3+17 q^{-2} -36 q^{-4} +35 q^{-6} -27 q^{-8} +29 q^{-12} -52 q^{-14} +54 q^{-16} -43 q^{-18} +14 q^{-20} +13 q^{-22} -39 q^{-24} +48 q^{-26} -42 q^{-28} +24 q^{-30} + q^{-32} -21 q^{-34} +31 q^{-36} -23 q^{-38} +7 q^{-40} +12 q^{-42} -25 q^{-44} +26 q^{-46} -14 q^{-48} -7 q^{-50} +31 q^{-52} -40 q^{-54} +39 q^{-56} -20 q^{-58} -5 q^{-60} +26 q^{-62} -36 q^{-64} +37 q^{-66} -25 q^{-68} +8 q^{-70} +8 q^{-72} -18 q^{-74} +20 q^{-76} -15 q^{-78} +9 q^{-80} -2 q^{-82} -3 q^{-84} +4 q^{-86} -5 q^{-88} +3 q^{-90} - 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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 22"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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[math]\displaystyle{ -2 t^3+6 t^2-10 t+13-10 t^{-1} +6 t^{-2} -2 t^{-3} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ -2 z^6-6 z^4-4 z^2+1 }[/math] |
In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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[math]\displaystyle{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 49, 0 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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[math]\displaystyle{ q^6-2 q^5+4 q^4-6 q^3+7 q^2-8 q+8-6 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4} }[/math] |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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[math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} -4 z^4+3 a^2 z^2-5 z^2 a^{-2} +3 z^2 a^{-4} -5 z^2+2 a^2-2 a^{-2} +2 a^{-4} -1 }[/math] |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8+3 a z^7-z^7 a^{-3} +2 z^7 a^{-5} +3 a^2 z^6-12 z^6 a^{-2} -6 z^6 a^{-4} +z^6 a^{-6} -2 z^6+2 a^3 z^5-6 a z^5-z^5 a^{-1} -7 z^5 a^{-5} +a^4 z^4-6 a^2 z^4+16 z^4 a^{-2} +6 z^4 a^{-4} -4 z^4 a^{-6} -z^4-3 a^3 z^3+7 a z^3-4 z^3 a^{-3} +6 z^3 a^{-5} -2 a^4 z^2+6 a^2 z^2-12 z^2 a^{-2} -6 z^2 a^{-4} +4 z^2 a^{-6} +6 z^2-a z+z a^{-1} +z a^{-3} -z a^{-5} -2 a^2+2 a^{-2} +2 a^{-4} -1 }[/math] |
