9 14

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-9 t+15-9 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 37, 0 }
Jones polynomial	[math]\displaystyle{ q^6-2 q^5+3 q^4-5 q^3+6 q^2-6 q+6-4 q^{-1} +3 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{-2} +z^4-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +5 z^7 a^{-3} +2 z^7 a^{-5} +3 z^6 a^{-2} +z^6 a^{-6} +4 z^6+4 a z^5-4 z^5 a^{-1} -16 z^5 a^{-3} -8 z^5 a^{-5} +3 a^2 z^4-12 z^4 a^{-2} -9 z^4 a^{-4} -4 z^4 a^{-6} -4 z^4+a^3 z^3-3 a z^3+2 z^3 a^{-1} +15 z^3 a^{-3} +9 z^3 a^{-5} -2 a^2 z^2+8 z^2 a^{-2} +10 z^2 a^{-4} +4 z^2 a^{-6} -2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{10}+q^8+q^6-q^4+2 q^2+ q^{-2} + q^{-4} + q^{-8} -2 q^{-10} - q^{-12} - q^{-16} + q^{-18} + q^{-20} }[/math]
The G2 invariant	[math]\displaystyle{ q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+8 q^{38}-10 q^{36}+11 q^{34}-8 q^{32}+2 q^{30}+4 q^{28}-10 q^{26}+16 q^{24}-16 q^{22}+14 q^{20}-8 q^{18}-q^{16}+11 q^{14}-15 q^{12}+17 q^{10}-12 q^8+3 q^6+6 q^4-10 q^2+10-2 q^{-2} -6 q^{-4} +15 q^{-6} -16 q^{-8} +9 q^{-10} +5 q^{-12} -19 q^{-14} +30 q^{-16} -28 q^{-18} +18 q^{-20} -16 q^{-24} +28 q^{-26} -30 q^{-28} +23 q^{-30} -10 q^{-32} -6 q^{-34} +16 q^{-36} -19 q^{-38} +15 q^{-40} -5 q^{-42} -7 q^{-44} +11 q^{-46} -13 q^{-48} +5 q^{-50} +5 q^{-52} -16 q^{-54} +21 q^{-56} -19 q^{-58} +6 q^{-60} +8 q^{-62} -20 q^{-64} +25 q^{-66} -21 q^{-68} +11 q^{-70} + q^{-72} -10 q^{-74} +16 q^{-76} -14 q^{-78} +10 q^{-80} -2 q^{-82} -2 q^{-84} +3 q^{-86} -4 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_14.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^7+2 q^5-q^3+2 q+ q^{-5} -2 q^{-7} + q^{-9} - q^{-11} + q^{-13} }[/math]
2	[math]\displaystyle{ q^{20}-2 q^{18}-q^{16}+4 q^{14}-4 q^{12}+7 q^8-5 q^6-q^4+7 q^2-3-3 q^{-2} +3 q^{-4} +2 q^{-6} -3 q^{-8} -2 q^{-10} +5 q^{-12} - q^{-14} -6 q^{-16} +5 q^{-18} +2 q^{-20} -6 q^{-22} +4 q^{-24} +4 q^{-26} -5 q^{-28} +3 q^{-32} -2 q^{-34} - q^{-36} + q^{-38} }[/math]
3	[math]\displaystyle{ -q^{39}+2 q^{37}+q^{35}-2 q^{33}-2 q^{31}+q^{29}+3 q^{27}-5 q^{25}+6 q^{21}-10 q^{17}+3 q^{15}+14 q^{13}-q^{11}-17 q^9+19 q^5+5 q^3-16 q-9 q^{-1} +10 q^{-3} +11 q^{-5} -2 q^{-7} -14 q^{-9} -3 q^{-11} +12 q^{-13} +10 q^{-15} -11 q^{-17} -13 q^{-19} +8 q^{-21} +16 q^{-23} -6 q^{-25} -16 q^{-27} +2 q^{-29} +18 q^{-31} +2 q^{-33} -17 q^{-35} -6 q^{-37} +16 q^{-39} +12 q^{-41} -13 q^{-43} -15 q^{-45} +5 q^{-47} +16 q^{-49} - q^{-51} -14 q^{-53} -4 q^{-55} +10 q^{-57} +7 q^{-59} -5 q^{-61} -6 q^{-63} +2 q^{-65} +4 q^{-67} -2 q^{-71} - q^{-73} + q^{-75} }[/math]
4	[math]\displaystyle{ q^{64}-2 q^{62}-q^{60}+2 q^{58}+5 q^{54}-4 q^{52}-2 q^{50}-6 q^{46}+11 q^{44}-2 q^{42}+3 q^{40}-21 q^{36}+4 q^{34}+6 q^{32}+26 q^{30}+10 q^{28}-43 q^{26}-22 q^{24}+4 q^{22}+56 q^{20}+40 q^{18}-48 q^{16}-56 q^{14}-24 q^{12}+60 q^{10}+70 q^8-16 q^6-54 q^4-55 q^2+21+63 q^{-2} +24 q^{-4} -14 q^{-6} -52 q^{-8} -26 q^{-10} +19 q^{-12} +41 q^{-14} +30 q^{-16} -25 q^{-18} -48 q^{-20} -17 q^{-22} +41 q^{-24} +47 q^{-26} -3 q^{-28} -53 q^{-30} -36 q^{-32} +39 q^{-34} +51 q^{-36} +8 q^{-38} -57 q^{-40} -48 q^{-42} +34 q^{-44} +54 q^{-46} +29 q^{-48} -47 q^{-50} -64 q^{-52} +8 q^{-54} +45 q^{-56} +56 q^{-58} -14 q^{-60} -63 q^{-62} -29 q^{-64} +8 q^{-66} +62 q^{-68} +30 q^{-70} -28 q^{-72} -41 q^{-74} -34 q^{-76} +32 q^{-78} +44 q^{-80} +15 q^{-82} -17 q^{-84} -47 q^{-86} -6 q^{-88} +21 q^{-90} +27 q^{-92} +12 q^{-94} -25 q^{-96} -17 q^{-98} -4 q^{-100} +12 q^{-102} +16 q^{-104} -4 q^{-106} -6 q^{-108} -6 q^{-110} +6 q^{-114} + q^{-116} -2 q^{-120} - q^{-122} + q^{-124} }[/math]
5	[math]\displaystyle{ -q^{95}+2 q^{93}+q^{91}-2 q^{89}-3 q^{85}-2 q^{83}+3 q^{81}+7 q^{79}+3 q^{77}-q^{75}-8 q^{73}-12 q^{71}-4 q^{69}+13 q^{67}+26 q^{65}+10 q^{63}-15 q^{61}-36 q^{59}-38 q^{57}+9 q^{55}+66 q^{53}+62 q^{51}+q^{49}-80 q^{47}-112 q^{45}-33 q^{43}+111 q^{41}+167 q^{39}+71 q^{37}-114 q^{35}-226 q^{33}-137 q^{31}+104 q^{29}+277 q^{27}+209 q^{25}-60 q^{23}-298 q^{21}-276 q^{19}-8 q^{17}+276 q^{15}+324 q^{13}+90 q^{11}-216 q^9-326 q^7-162 q^5+119 q^3+282 q+208 q^{-1} -17 q^{-3} -202 q^{-5} -216 q^{-7} -66 q^{-9} +103 q^{-11} +184 q^{-13} +132 q^{-15} -13 q^{-17} -142 q^{-19} -158 q^{-21} -53 q^{-23} +90 q^{-25} +172 q^{-27} +104 q^{-29} -65 q^{-31} -169 q^{-33} -121 q^{-35} +46 q^{-37} +177 q^{-39} +134 q^{-41} -48 q^{-43} -189 q^{-45} -148 q^{-47} +46 q^{-49} +209 q^{-51} +168 q^{-53} -43 q^{-55} -223 q^{-57} -200 q^{-59} +18 q^{-61} +233 q^{-63} +234 q^{-65} +20 q^{-67} -213 q^{-69} -260 q^{-71} -80 q^{-73} +172 q^{-75} +272 q^{-77} +137 q^{-79} -105 q^{-81} -250 q^{-83} -190 q^{-85} +19 q^{-87} +202 q^{-89} +217 q^{-91} +59 q^{-93} -122 q^{-95} -199 q^{-97} -131 q^{-99} +32 q^{-101} +157 q^{-103} +159 q^{-105} +50 q^{-107} -79 q^{-109} -149 q^{-111} -112 q^{-113} +3 q^{-115} +104 q^{-117} +127 q^{-119} +61 q^{-121} -42 q^{-123} -107 q^{-125} -95 q^{-127} -16 q^{-129} +65 q^{-131} +93 q^{-133} +51 q^{-135} -18 q^{-137} -65 q^{-139} -62 q^{-141} -15 q^{-143} +34 q^{-145} +51 q^{-147} +27 q^{-149} -7 q^{-151} -29 q^{-153} -28 q^{-155} -6 q^{-157} +14 q^{-159} +17 q^{-161} +7 q^{-163} -2 q^{-165} -8 q^{-167} -8 q^{-169} +4 q^{-173} +3 q^{-175} + q^{-177} -2 q^{-181} - q^{-183} + q^{-185} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{10}+q^8+q^6-q^4+2 q^2+ q^{-2} + q^{-4} + q^{-8} -2 q^{-10} - q^{-12} - q^{-16} + q^{-18} + q^{-20} }[/math]
1,1	[math]\displaystyle{ q^{28}-4 q^{26}+8 q^{24}-12 q^{22}+18 q^{20}-28 q^{18}+34 q^{16}-38 q^{14}+43 q^{12}-48 q^{10}+50 q^8-44 q^6+46 q^4-32 q^2+22-24 q^{-4} +50 q^{-6} -80 q^{-8} +102 q^{-10} -118 q^{-12} +126 q^{-14} -126 q^{-16} +108 q^{-18} -91 q^{-20} +62 q^{-22} -30 q^{-24} +34 q^{-28} -50 q^{-30} +70 q^{-32} -76 q^{-34} +71 q^{-36} -62 q^{-38} +48 q^{-40} -36 q^{-42} +21 q^{-44} -12 q^{-46} +6 q^{-48} -2 q^{-50} + q^{-52} }[/math]
2,0	[math]\displaystyle{ q^{26}-q^{24}-2 q^{22}+q^{20}+2 q^{18}-q^{16}-4 q^{14}+3 q^{12}+5 q^{10}-3 q^8-2 q^6+6 q^4+4 q^2-2-2 q^{-2} +2 q^{-4} - q^{-8} + q^{-10} -3 q^{-14} +2 q^{-16} + q^{-18} -5 q^{-20} -3 q^{-22} +2 q^{-24} +3 q^{-26} -2 q^{-28} +5 q^{-32} +4 q^{-34} -2 q^{-36} -2 q^{-38} + q^{-40} + q^{-42} -2 q^{-44} -3 q^{-46} + q^{-50} + q^{-52} }[/math]

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{22}+2 q^{20}-3 q^{18}+4 q^{16}-5 q^{14}+6 q^{12}-6 q^{10}+6 q^8-4 q^6+3 q^4+q^2-3+7 q^{-2} -8 q^{-4} +12 q^{-6} -11 q^{-8} +12 q^{-10} -10 q^{-12} +8 q^{-14} -6 q^{-16} + q^{-18} -4 q^{-22} +4 q^{-24} -6 q^{-26} +6 q^{-28} -6 q^{-30} +5 q^{-32} -3 q^{-34} +3 q^{-36} - q^{-38} + q^{-40} }[/math]
1,0	[math]\displaystyle{ q^{36}-2 q^{32}-2 q^{30}+q^{28}+4 q^{26}+q^{24}-4 q^{22}-4 q^{20}+2 q^{18}+6 q^{16}+2 q^{14}-5 q^{12}-4 q^{10}+3 q^8+7 q^6-4 q^2+6 q^{-2} +3 q^{-4} -3 q^{-6} -4 q^{-8} +2 q^{-10} +3 q^{-12} -2 q^{-14} -5 q^{-16} +3 q^{-20} - q^{-22} -5 q^{-24} - q^{-26} +6 q^{-28} +4 q^{-30} -3 q^{-32} -6 q^{-34} +2 q^{-36} +7 q^{-38} +3 q^{-40} -5 q^{-42} -5 q^{-44} +2 q^{-46} +5 q^{-48} -4 q^{-52} -2 q^{-54} +2 q^{-56} +2 q^{-58} - q^{-60} - q^{-62} + q^{-66} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+8 q^{38}-10 q^{36}+11 q^{34}-8 q^{32}+2 q^{30}+4 q^{28}-10 q^{26}+16 q^{24}-16 q^{22}+14 q^{20}-8 q^{18}-q^{16}+11 q^{14}-15 q^{12}+17 q^{10}-12 q^8+3 q^6+6 q^4-10 q^2+10-2 q^{-2} -6 q^{-4} +15 q^{-6} -16 q^{-8} +9 q^{-10} +5 q^{-12} -19 q^{-14} +30 q^{-16} -28 q^{-18} +18 q^{-20} -16 q^{-24} +28 q^{-26} -30 q^{-28} +23 q^{-30} -10 q^{-32} -6 q^{-34} +16 q^{-36} -19 q^{-38} +15 q^{-40} -5 q^{-42} -7 q^{-44} +11 q^{-46} -13 q^{-48} +5 q^{-50} +5 q^{-52} -16 q^{-54} +21 q^{-56} -19 q^{-58} +6 q^{-60} +8 q^{-62} -20 q^{-64} +25 q^{-66} -21 q^{-68} +11 q^{-70} + q^{-72} -10 q^{-74} +16 q^{-76} -14 q^{-78} +10 q^{-80} -2 q^{-82} -2 q^{-84} +3 q^{-86} -4 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }[/math]

.

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["9 14"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-9 t+15-9 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 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]=

{ 37, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-2 q^5+3 q^4-5 q^3+6 q^2-6 q+6-4 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^4 a^{-2} +z^4-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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