10 59

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-7 t^2+18 t-23+18 t^{-1} -7 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6-z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 75, 2 }
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+6 q^5-10 q^4+12 q^3-12 q^2+12 q-9+6 q^{-1} -3 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +3 z^4 a^{-2} -2 z^4 a^{-4} -2 z^4+a^2 z^2+5 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -4 z^2+a^2+4 a^{-2} -3 a^{-4} + a^{-6} -2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +7 z^8 a^{-2} +4 z^8 a^{-4} +3 z^8+3 a z^7+8 z^7 a^{-1} +11 z^7 a^{-3} +6 z^7 a^{-5} +a^2 z^6-9 z^6 a^{-2} +z^6 a^{-4} +5 z^6 a^{-6} -4 z^6-9 a z^5-27 z^5 a^{-1} -28 z^5 a^{-3} -7 z^5 a^{-5} +3 z^5 a^{-7} -3 a^2 z^4-8 z^4 a^{-2} -11 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} -6 z^4+8 a z^3+21 z^3 a^{-1} +20 z^3 a^{-3} +4 z^3 a^{-5} -3 z^3 a^{-7} +3 a^2 z^2+11 z^2 a^{-2} +10 z^2 a^{-4} +3 z^2 a^{-6} -z^2 a^{-8} +8 z^2-2 a z-5 z a^{-1} -4 z a^{-3} +z a^{-7} -a^2-4 a^{-2} -3 a^{-4} - a^{-6} -2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{10}-q^6+2 q^4-2 q^2+2 q^{-2} - q^{-4} +4 q^{-6} - q^{-8} + q^{-10} - q^{-12} -3 q^{-14} +2 q^{-16} - q^{-18} + q^{-22} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-2 q^{44}+6 q^{42}-10 q^{40}+12 q^{38}-10 q^{36}-q^{34}+23 q^{32}-44 q^{30}+64 q^{28}-63 q^{26}+33 q^{24}+18 q^{22}-83 q^{20}+135 q^{18}-149 q^{16}+112 q^{14}-28 q^{12}-76 q^{10}+158 q^8-183 q^6+145 q^4-55 q^2-52+123 q^{-2} -140 q^{-4} +86 q^{-6} +9 q^{-8} -96 q^{-10} +143 q^{-12} -111 q^{-14} +24 q^{-16} +88 q^{-18} -182 q^{-20} +220 q^{-22} -177 q^{-24} +68 q^{-26} +74 q^{-28} -192 q^{-30} +256 q^{-32} -225 q^{-34} +126 q^{-36} +6 q^{-38} -123 q^{-40} +179 q^{-42} -167 q^{-44} +85 q^{-46} +20 q^{-48} -97 q^{-50} +117 q^{-52} -74 q^{-54} -17 q^{-56} +101 q^{-58} -146 q^{-60} +127 q^{-62} -63 q^{-64} -31 q^{-66} +113 q^{-68} -154 q^{-70} +149 q^{-72} -93 q^{-74} +23 q^{-76} +41 q^{-78} -84 q^{-80} +92 q^{-82} -79 q^{-84} +52 q^{-86} -16 q^{-88} -10 q^{-90} +27 q^{-92} -32 q^{-94} +28 q^{-96} -19 q^{-98} +11 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_59.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^7-2 q^5+3 q^3-3 q+3 q^{-1} +2 q^{-7} -4 q^{-9} +3 q^{-11} -2 q^{-13} + q^{-15} }[/math]
2	[math]\displaystyle{ q^{22}-2 q^{20}-2 q^{18}+8 q^{16}-3 q^{14}-12 q^{12}+17 q^{10}+4 q^8-26 q^6+15 q^4+17 q^2-27+4 q^{-2} +22 q^{-4} -14 q^{-6} -9 q^{-8} +14 q^{-10} +6 q^{-12} -16 q^{-14} -3 q^{-16} +24 q^{-18} -15 q^{-20} -18 q^{-22} +29 q^{-24} -5 q^{-26} -19 q^{-28} +17 q^{-30} + q^{-32} -9 q^{-34} +5 q^{-36} -2 q^{-40} + q^{-42} }[/math]
3	[math]\displaystyle{ q^{45}-2 q^{43}-2 q^{41}+3 q^{39}+8 q^{37}-3 q^{35}-19 q^{33}-q^{31}+32 q^{29}+16 q^{27}-45 q^{25}-44 q^{23}+50 q^{21}+79 q^{19}-35 q^{17}-115 q^{15}+3 q^{13}+142 q^{11}+42 q^9-147 q^7-91 q^5+129 q^3+134 q-102 q^{-1} -154 q^{-3} +61 q^{-5} +165 q^{-7} -19 q^{-9} -156 q^{-11} -18 q^{-13} +137 q^{-15} +54 q^{-17} -106 q^{-19} -87 q^{-21} +62 q^{-23} +119 q^{-25} -17 q^{-27} -137 q^{-29} -42 q^{-31} +145 q^{-33} +91 q^{-35} -128 q^{-37} -134 q^{-39} +103 q^{-41} +152 q^{-43} -64 q^{-45} -143 q^{-47} +23 q^{-49} +121 q^{-51} - q^{-53} -85 q^{-55} -11 q^{-57} +51 q^{-59} +12 q^{-61} -27 q^{-63} -8 q^{-65} +14 q^{-67} +2 q^{-69} -5 q^{-71} +2 q^{-75} -2 q^{-79} + q^{-81} }[/math]
4	[math]\displaystyle{ q^{76}-2 q^{74}-2 q^{72}+3 q^{70}+3 q^{68}+8 q^{66}-10 q^{64}-19 q^{62}-q^{60}+14 q^{58}+53 q^{56}-q^{54}-67 q^{52}-64 q^{50}-13 q^{48}+157 q^{46}+117 q^{44}-58 q^{42}-213 q^{40}-229 q^{38}+174 q^{36}+366 q^{34}+219 q^{32}-227 q^{30}-624 q^{28}-168 q^{26}+432 q^{24}+724 q^{22}+204 q^{20}-792 q^{18}-775 q^{16}-15 q^{14}+977 q^{12}+909 q^{10}-399 q^8-1123 q^6-743 q^4+676 q^2+1331+282 q^{-2} -940 q^{-4} -1202 q^{-6} +105 q^{-8} +1243 q^{-10} +766 q^{-12} -499 q^{-14} -1218 q^{-16} -332 q^{-18} +871 q^{-20} +921 q^{-22} -84 q^{-24} -976 q^{-26} -609 q^{-28} +410 q^{-30} +928 q^{-32} +334 q^{-34} -601 q^{-36} -855 q^{-38} -183 q^{-40} +800 q^{-42} +823 q^{-44} +3 q^{-46} -971 q^{-48} -901 q^{-50} +374 q^{-52} +1141 q^{-54} +764 q^{-56} -682 q^{-58} -1373 q^{-60} -296 q^{-62} +941 q^{-64} +1238 q^{-66} -68 q^{-68} -1213 q^{-70} -735 q^{-72} +345 q^{-74} +1079 q^{-76} +379 q^{-78} -622 q^{-80} -646 q^{-82} -105 q^{-84} +558 q^{-86} +371 q^{-88} -157 q^{-90} -293 q^{-92} -175 q^{-94} +169 q^{-96} +168 q^{-98} -8 q^{-100} -65 q^{-102} -83 q^{-104} +34 q^{-106} +40 q^{-108} -2 q^{-110} -21 q^{-114} +7 q^{-116} +5 q^{-118} -4 q^{-120} +4 q^{-122} -3 q^{-124} +2 q^{-126} -2 q^{-130} + q^{-132} }[/math]
5	[math]\displaystyle{ q^{115}-2 q^{113}-2 q^{111}+3 q^{109}+3 q^{107}+3 q^{105}+q^{103}-10 q^{101}-19 q^{99}-q^{97}+23 q^{95}+35 q^{93}+27 q^{91}-23 q^{89}-85 q^{87}-89 q^{85}+9 q^{83}+138 q^{81}+196 q^{79}+96 q^{77}-158 q^{75}-378 q^{73}-322 q^{71}+72 q^{69}+543 q^{67}+695 q^{65}+268 q^{63}-568 q^{61}-1177 q^{59}-914 q^{57}+266 q^{55}+1540 q^{53}+1828 q^{51}+587 q^{49}-1513 q^{47}-2818 q^{45}-1968 q^{43}+793 q^{41}+3448 q^{39}+3691 q^{37}+794 q^{35}-3334 q^{33}-5328 q^{31}-3066 q^{29}+2184 q^{27}+6311 q^{25}+5617 q^{23}+15 q^{21}-6259 q^{19}-7859 q^{17}-2875 q^{15}+5033 q^{13}+9235 q^{11}+5829 q^9-2791 q^7-9491 q^5-8329 q^3+86 q+8648 q^{-1} +9869 q^{-3} +2601 q^{-5} -6974 q^{-7} -10438 q^{-9} -4742 q^{-11} +4983 q^{-13} +10058 q^{-15} +6127 q^{-17} -3010 q^{-19} -9090 q^{-21} -6793 q^{-23} +1407 q^{-25} +7849 q^{-27} +6891 q^{-29} -169 q^{-31} -6607 q^{-33} -6724 q^{-35} -809 q^{-37} +5491 q^{-39} +6538 q^{-41} +1725 q^{-43} -4415 q^{-45} -6478 q^{-47} -2842 q^{-49} +3226 q^{-51} +6503 q^{-53} +4267 q^{-55} -1664 q^{-57} -6445 q^{-59} -5982 q^{-61} -366 q^{-63} +5954 q^{-65} +7714 q^{-67} +2969 q^{-69} -4836 q^{-71} -9102 q^{-73} -5758 q^{-75} +2879 q^{-77} +9654 q^{-79} +8445 q^{-81} -279 q^{-83} -9174 q^{-85} -10342 q^{-87} -2584 q^{-89} +7509 q^{-91} +11130 q^{-93} +5151 q^{-95} -5068 q^{-97} -10600 q^{-99} -6856 q^{-101} +2367 q^{-103} +8890 q^{-105} +7465 q^{-107} +29 q^{-109} -6579 q^{-111} -6955 q^{-113} -1623 q^{-115} +4154 q^{-117} +5651 q^{-119} +2375 q^{-121} -2159 q^{-123} -4050 q^{-125} -2356 q^{-127} +802 q^{-129} +2556 q^{-131} +1882 q^{-133} -61 q^{-135} -1413 q^{-137} -1291 q^{-139} -223 q^{-141} +697 q^{-143} +767 q^{-145} +230 q^{-147} -284 q^{-149} -397 q^{-151} -173 q^{-153} +101 q^{-155} +194 q^{-157} +89 q^{-159} -34 q^{-161} -73 q^{-163} -41 q^{-165} +4 q^{-167} +27 q^{-169} +22 q^{-171} -5 q^{-173} -12 q^{-175} -2 q^{-179} - q^{-181} +5 q^{-183} + q^{-185} -3 q^{-187} +2 q^{-189} -2 q^{-193} + q^{-195} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{10}-q^6+2 q^4-2 q^2+2 q^{-2} - q^{-4} +4 q^{-6} - q^{-8} + q^{-10} - q^{-12} -3 q^{-14} +2 q^{-16} - q^{-18} + q^{-22} }[/math]
2,0	[math]\displaystyle{ q^{28}-2 q^{24}-2 q^{22}+3 q^{20}+5 q^{18}-4 q^{16}-6 q^{14}+6 q^{12}+9 q^{10}-6 q^8-11 q^6+7 q^4+12 q^2-9-9 q^{-2} +11 q^{-4} +3 q^{-6} -7 q^{-8} +7 q^{-12} -2 q^{-14} +10 q^{-18} -3 q^{-20} -12 q^{-22} +5 q^{-24} +8 q^{-26} -15 q^{-28} -8 q^{-30} +13 q^{-32} +8 q^{-34} -8 q^{-36} -4 q^{-38} +10 q^{-40} +3 q^{-42} -6 q^{-44} -2 q^{-46} + q^{-48} - q^{-52} + q^{-56} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{20}-2 q^{18}+2 q^{16}+3 q^{14}-8 q^{12}+7 q^{10}+5 q^8-16 q^6+13 q^4+6 q^2-21+14 q^{-2} +9 q^{-4} -18 q^{-6} +8 q^{-8} +11 q^{-10} -5 q^{-12} -4 q^{-14} +3 q^{-16} +7 q^{-18} -13 q^{-20} -7 q^{-22} +20 q^{-24} -13 q^{-26} -10 q^{-28} +24 q^{-30} -9 q^{-32} -10 q^{-34} +15 q^{-36} -3 q^{-38} -7 q^{-40} +5 q^{-42} -2 q^{-46} + q^{-48} }[/math]
1,0,0	[math]\displaystyle{ q^{13}+q^9-q^7+2 q^5-3 q^3+q-2 q^{-1} +2 q^{-3} +3 q^{-7} +3 q^{-9} + q^{-13} -3 q^{-15} -4 q^{-19} +2 q^{-21} - q^{-23} + q^{-25} + q^{-29} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{20}-2 q^{18}+6 q^{16}-9 q^{14}+14 q^{12}-19 q^{10}+23 q^8-26 q^6+25 q^4-22 q^2+13-4 q^{-2} -7 q^{-4} +22 q^{-6} -32 q^{-8} +45 q^{-10} -47 q^{-12} +50 q^{-14} -45 q^{-16} +37 q^{-18} -27 q^{-20} +13 q^{-22} -2 q^{-24} -11 q^{-26} +18 q^{-28} -24 q^{-30} +25 q^{-32} -24 q^{-34} +21 q^{-36} -15 q^{-38} +11 q^{-40} -7 q^{-42} +4 q^{-44} -2 q^{-46} + q^{-48} }[/math]
1,0	[math]\displaystyle{ q^{34}-2 q^{30}-2 q^{28}+4 q^{26}+6 q^{24}-3 q^{22}-11 q^{20}-3 q^{18}+15 q^{16}+13 q^{14}-12 q^{12}-22 q^{10}+2 q^8+27 q^6+12 q^4-23 q^2-22+11 q^{-2} +26 q^{-4} +2 q^{-6} -23 q^{-8} -8 q^{-10} +17 q^{-12} +13 q^{-14} -12 q^{-16} -11 q^{-18} +10 q^{-20} +15 q^{-22} -7 q^{-24} -18 q^{-26} +2 q^{-28} +19 q^{-30} +2 q^{-32} -22 q^{-34} -11 q^{-36} +19 q^{-38} +19 q^{-40} -13 q^{-42} -25 q^{-44} +2 q^{-46} +27 q^{-48} +11 q^{-50} -17 q^{-52} -19 q^{-54} +5 q^{-56} +18 q^{-58} +5 q^{-60} -9 q^{-62} -9 q^{-64} + q^{-66} +6 q^{-68} +2 q^{-70} -2 q^{-72} -2 q^{-74} + q^{-78} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{46}-2 q^{44}+6 q^{42}-10 q^{40}+12 q^{38}-10 q^{36}-q^{34}+23 q^{32}-44 q^{30}+64 q^{28}-63 q^{26}+33 q^{24}+18 q^{22}-83 q^{20}+135 q^{18}-149 q^{16}+112 q^{14}-28 q^{12}-76 q^{10}+158 q^8-183 q^6+145 q^4-55 q^2-52+123 q^{-2} -140 q^{-4} +86 q^{-6} +9 q^{-8} -96 q^{-10} +143 q^{-12} -111 q^{-14} +24 q^{-16} +88 q^{-18} -182 q^{-20} +220 q^{-22} -177 q^{-24} +68 q^{-26} +74 q^{-28} -192 q^{-30} +256 q^{-32} -225 q^{-34} +126 q^{-36} +6 q^{-38} -123 q^{-40} +179 q^{-42} -167 q^{-44} +85 q^{-46} +20 q^{-48} -97 q^{-50} +117 q^{-52} -74 q^{-54} -17 q^{-56} +101 q^{-58} -146 q^{-60} +127 q^{-62} -63 q^{-64} -31 q^{-66} +113 q^{-68} -154 q^{-70} +149 q^{-72} -93 q^{-74} +23 q^{-76} +41 q^{-78} -84 q^{-80} +92 q^{-82} -79 q^{-84} +52 q^{-86} -16 q^{-88} -10 q^{-90} +27 q^{-92} -32 q^{-94} +28 q^{-96} -19 q^{-98} +11 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }[/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 59"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-7 t^2+18 t-23+18 t^{-1} -7 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 75, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^7-3 q^6+6 q^5-10 q^4+12 q^3-12 q^2+12 q-9+6 q^{-1} -3 q^{-2} + q^{-3} }[/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} +3 z^4 a^{-2} -2 z^4 a^{-4} -2 z^4+a^2 z^2+5 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -4 z^2+a^2+4 a^{-2} -3 a^{-4} + a^{-6} -2 }[/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} +4 z^8 a^{-4} +3 z^8+3 a z^7+8 z^7 a^{-1} +11 z^7 a^{-3} +6 z^7 a^{-5} +a^2 z^6-9 z^6 a^{-2} +z^6 a^{-4} +5 z^6 a^{-6} -4 z^6-9 a z^5-27 z^5 a^{-1} -28 z^5 a^{-3} -7 z^5 a^{-5} +3 z^5 a^{-7} -3 a^2 z^4-8 z^4 a^{-2} -11 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} -6 z^4+8 a z^3+21 z^3 a^{-1} +20 z^3 a^{-3} +4 z^3 a^{-5} -3 z^3 a^{-7} +3 a^2 z^2+11 z^2 a^{-2} +10 z^2 a^{-4} +3 z^2 a^{-6} -z^2 a^{-8} +8 z^2-2 a z-5 z a^{-1} -4 z a^{-3} +z a^{-7} -a^2-4 a^{-2} -3 a^{-4} - a^{-6} -2 }[/math]