8 12

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8 11.gif

8_11

8 13.gif

8_13

8 12.gif Visit 8 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 12's page at Knotilus!

Visit 8 12's page at the original Knot Atlas!

8 12 Quick Notes




In symmetric decorative form

Knot presentations

Planar diagram presentation X4251 X10,8,11,7 X8394 X2,9,3,10 X14,6,15,5 X16,11,1,12 X12,15,13,16 X6,14,7,13
Gauss code 1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6
Dowker-Thistlethwaite code 4 8 14 10 2 16 6 12
Conway Notation [2222]

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-5]
Hyperbolic Volume 8.93586
A-Polynomial See Data:8 12/A-polynomial

[edit Notes for 8 12's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ 2 }[/math]
Rasmussen s-Invariant 0

[edit Notes for 8 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^2-7 t+13-7 t^{-1} + t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 29, 0 }
Jones polynomial [math]\displaystyle{ q^4-2 q^3+4 q^2-5 q+5-5 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^4-2 z^2 a^2-a^2+z^4+z^2+1-2 z^2 a^{-2} - a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 a z^5+2 z^5 a^{-1} +2 z^5 a^{-3} +a^4 z^4-a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -4 z^4-3 a^3 z^3-3 a z^3-3 z^3 a^{-1} -3 z^3 a^{-3} -2 a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +a^3 z+z a^{-3} +a^4+a^2+ a^{-2} + a^{-4} +1 }[/math]
The A2 invariant [math]\displaystyle{ q^{14}+q^{12}-q^{10}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{66}-q^{64}+3 q^{62}-3 q^{60}+2 q^{58}-3 q^{54}+9 q^{52}-11 q^{50}+12 q^{48}-8 q^{46}+10 q^{42}-17 q^{40}+23 q^{38}-18 q^{36}+8 q^{34}+4 q^{32}-16 q^{30}+17 q^{28}-13 q^{26}+3 q^{24}+6 q^{22}-12 q^{20}+9 q^{18}-12 q^{14}+21 q^{12}-23 q^{10}+15 q^8-q^6-14 q^4+27 q^2-29+27 q^{-2} -14 q^{-4} - q^{-6} +15 q^{-8} -23 q^{-10} +21 q^{-12} -12 q^{-14} +9 q^{-18} -12 q^{-20} +6 q^{-22} +3 q^{-24} -13 q^{-26} +17 q^{-28} -16 q^{-30} +4 q^{-32} +8 q^{-34} -18 q^{-36} +23 q^{-38} -17 q^{-40} +10 q^{-42} -8 q^{-46} +12 q^{-48} -11 q^{-50} +9 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 8_12.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^9-q^7+2 q^5-q^3- q^{-3} +2 q^{-5} - q^{-7} + q^{-9} }[/math]
2 [math]\displaystyle{ q^{26}-q^{24}-q^{22}+4 q^{20}-2 q^{18}-5 q^{16}+7 q^{14}-7 q^{10}+5 q^8+2 q^6-4 q^4+q^2+3+ q^{-2} -4 q^{-4} +2 q^{-6} +5 q^{-8} -7 q^{-10} +7 q^{-14} -5 q^{-16} -2 q^{-18} +4 q^{-20} - q^{-22} - q^{-24} + q^{-26} }[/math]
3 [math]\displaystyle{ q^{51}-q^{49}-q^{47}+q^{45}+3 q^{43}-2 q^{41}-7 q^{39}+2 q^{37}+12 q^{35}+q^{33}-16 q^{31}-7 q^{29}+20 q^{27}+12 q^{25}-20 q^{23}-17 q^{21}+16 q^{19}+22 q^{17}-12 q^{15}-20 q^{13}+7 q^{11}+18 q^9-2 q^7-13 q^5-4 q^3+9 q+9 q^{-1} -4 q^{-3} -13 q^{-5} -2 q^{-7} +18 q^{-9} +7 q^{-11} -20 q^{-13} -12 q^{-15} +22 q^{-17} +16 q^{-19} -17 q^{-21} -20 q^{-23} +12 q^{-25} +20 q^{-27} -7 q^{-29} -16 q^{-31} + q^{-33} +12 q^{-35} +2 q^{-37} -7 q^{-39} -2 q^{-41} +3 q^{-43} + q^{-45} - q^{-47} - q^{-49} + q^{-51} }[/math]
4 [math]\displaystyle{ q^{84}-q^{82}-q^{80}+q^{78}+3 q^{74}-4 q^{72}-5 q^{70}+3 q^{68}+5 q^{66}+14 q^{64}-9 q^{62}-22 q^{60}-7 q^{58}+11 q^{56}+44 q^{54}+4 q^{52}-40 q^{50}-42 q^{48}-7 q^{46}+77 q^{44}+44 q^{42}-32 q^{40}-76 q^{38}-49 q^{36}+76 q^{34}+80 q^{32}+5 q^{30}-78 q^{28}-82 q^{26}+44 q^{24}+81 q^{22}+33 q^{20}-50 q^{18}-75 q^{16}+8 q^{14}+52 q^{12}+42 q^{10}-15 q^8-49 q^6-20 q^4+18 q^2+41+18 q^{-2} -20 q^{-4} -49 q^{-6} -15 q^{-8} +42 q^{-10} +52 q^{-12} +8 q^{-14} -75 q^{-16} -50 q^{-18} +33 q^{-20} +81 q^{-22} +44 q^{-24} -82 q^{-26} -78 q^{-28} +5 q^{-30} +80 q^{-32} +76 q^{-34} -49 q^{-36} -76 q^{-38} -32 q^{-40} +44 q^{-42} +77 q^{-44} -7 q^{-46} -42 q^{-48} -40 q^{-50} +4 q^{-52} +44 q^{-54} +11 q^{-56} -7 q^{-58} -22 q^{-60} -9 q^{-62} +14 q^{-64} +5 q^{-66} +3 q^{-68} -5 q^{-70} -4 q^{-72} +3 q^{-74} + q^{-78} - q^{-80} - q^{-82} + q^{-84} }[/math]
5 [math]\displaystyle{ q^{125}-q^{123}-q^{121}+q^{119}+q^{113}-2 q^{111}-4 q^{109}+3 q^{107}+7 q^{105}+5 q^{103}-13 q^{99}-19 q^{97}-6 q^{95}+23 q^{93}+39 q^{91}+22 q^{89}-23 q^{87}-67 q^{85}-59 q^{83}+8 q^{81}+95 q^{79}+114 q^{77}+28 q^{75}-105 q^{73}-174 q^{71}-96 q^{69}+88 q^{67}+233 q^{65}+180 q^{63}-46 q^{61}-258 q^{59}-266 q^{57}-31 q^{55}+253 q^{53}+337 q^{51}+116 q^{49}-221 q^{47}-367 q^{45}-192 q^{43}+157 q^{41}+366 q^{39}+250 q^{37}-93 q^{35}-333 q^{33}-262 q^{31}+25 q^{29}+273 q^{27}+260 q^{25}+20 q^{23}-207 q^{21}-230 q^{19}-56 q^{17}+140 q^{15}+196 q^{13}+81 q^{11}-80 q^9-157 q^7-104 q^5+27 q^3+127 q+127 q^{-1} +27 q^{-3} -104 q^{-5} -157 q^{-7} -80 q^{-9} +81 q^{-11} +196 q^{-13} +140 q^{-15} -56 q^{-17} -230 q^{-19} -207 q^{-21} +20 q^{-23} +260 q^{-25} +273 q^{-27} +25 q^{-29} -262 q^{-31} -333 q^{-33} -93 q^{-35} +250 q^{-37} +366 q^{-39} +157 q^{-41} -192 q^{-43} -367 q^{-45} -221 q^{-47} +116 q^{-49} +337 q^{-51} +253 q^{-53} -31 q^{-55} -266 q^{-57} -258 q^{-59} -46 q^{-61} +180 q^{-63} +233 q^{-65} +88 q^{-67} -96 q^{-69} -174 q^{-71} -105 q^{-73} +28 q^{-75} +114 q^{-77} +95 q^{-79} +8 q^{-81} -59 q^{-83} -67 q^{-85} -23 q^{-87} +22 q^{-89} +39 q^{-91} +23 q^{-93} -6 q^{-95} -19 q^{-97} -13 q^{-99} +5 q^{-103} +7 q^{-105} +3 q^{-107} -4 q^{-109} -2 q^{-111} + q^{-113} + q^{-119} - q^{-121} - q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{14}+q^{12}-q^{10}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
1,1 [math]\displaystyle{ q^{36}-2 q^{34}+6 q^{32}-10 q^{30}+19 q^{28}-30 q^{26}+42 q^{24}-54 q^{22}+64 q^{20}-70 q^{18}+62 q^{16}-46 q^{14}+23 q^{12}+10 q^{10}-44 q^8+80 q^6-106 q^4+124 q^2-130+124 q^{-2} -106 q^{-4} +80 q^{-6} -44 q^{-8} +10 q^{-10} +23 q^{-12} -46 q^{-14} +62 q^{-16} -70 q^{-18} +64 q^{-20} -54 q^{-22} +42 q^{-24} -30 q^{-26} +19 q^{-28} -10 q^{-30} +6 q^{-32} -2 q^{-34} + q^{-36} }[/math]
2,0 [math]\displaystyle{ q^{36}+q^{34}-2 q^{30}+3 q^{26}-4 q^{22}-q^{20}+4 q^{18}+2 q^{16}-5 q^{14}+4 q^{10}-q^8-2 q^6+q^4+2 q^2+2 q^{-2} + q^{-4} -2 q^{-6} - q^{-8} +4 q^{-10} -5 q^{-14} +2 q^{-16} +4 q^{-18} - q^{-20} -4 q^{-22} +3 q^{-26} -2 q^{-30} + q^{-34} + q^{-36} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{28}-q^{26}+q^{24}+3 q^{22}-3 q^{20}+q^{18}+5 q^{16}-6 q^{14}+4 q^{10}-5 q^8-q^6+3 q^4+q^2+ q^{-2} +3 q^{-4} - q^{-6} -5 q^{-8} +4 q^{-10} -6 q^{-14} +5 q^{-16} + q^{-18} -3 q^{-20} +3 q^{-22} + q^{-24} - q^{-26} + q^{-28} }[/math]
1,0,0 [math]\displaystyle{ q^{19}+q^{17}+q^{15}-q^{13}+q^{11}-q^9-q^5+q^3+ q^{-3} - q^{-5} - q^{-9} + q^{-11} - q^{-13} + q^{-15} + q^{-17} + q^{-19} }[/math]

B2 Invariants.

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

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{66}-q^{64}+3 q^{62}-3 q^{60}+2 q^{58}-3 q^{54}+9 q^{52}-11 q^{50}+12 q^{48}-8 q^{46}+10 q^{42}-17 q^{40}+23 q^{38}-18 q^{36}+8 q^{34}+4 q^{32}-16 q^{30}+17 q^{28}-13 q^{26}+3 q^{24}+6 q^{22}-12 q^{20}+9 q^{18}-12 q^{14}+21 q^{12}-23 q^{10}+15 q^8-q^6-14 q^4+27 q^2-29+27 q^{-2} -14 q^{-4} - q^{-6} +15 q^{-8} -23 q^{-10} +21 q^{-12} -12 q^{-14} +9 q^{-18} -12 q^{-20} +6 q^{-22} +3 q^{-24} -13 q^{-26} +17 q^{-28} -16 q^{-30} +4 q^{-32} +8 q^{-34} -18 q^{-36} +23 q^{-38} -17 q^{-40} +10 q^{-42} -8 q^{-46} +12 q^{-48} -11 q^{-50} +9 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }[/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["8 12"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^2-7 t+13-7 t^{-1} + t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 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]= { 29, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^4-2 q^3+4 q^2-5 q+5-5 q^{-1} +4 q^{-2} -2 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{ a^4-2 z^2 a^2-a^2+z^4+z^2+1-2 z^2 a^{-2} - a^{-2} + a^{-4} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 a z^5+2 z^5 a^{-1} +2 z^5 a^{-3} +a^4 z^4-a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -4 z^4-3 a^3 z^3-3 a z^3-3 z^3 a^{-1} -3 z^3 a^{-3} -2 a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +a^3 z+z a^{-3} +a^4+a^2+ a^{-2} + a^{-4} +1 }[/math]

Vassiliev invariants

V2 and V3: (-3, 0)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ -12 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 82 }[/math] [math]\displaystyle{ 30 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -288 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -984 }[/math] [math]\displaystyle{ -360 }[/math] [math]\displaystyle{ -\frac{8031}{10} }[/math] [math]\displaystyle{ \frac{1226}{15} }[/math] [math]\displaystyle{ -\frac{9262}{15} }[/math] [math]\displaystyle{ \frac{479}{6} }[/math] [math]\displaystyle{ -\frac{1311}{10} }[/math]

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 8 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234χ
9        11
7       1 -1
5      31 2
3     21  -1
1    33   0
-1   33    0
-3  12     -1
-5 13      2
-7 1       -1
-91        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[8, 12]]
Out[2]=  
8
In[3]:=
PD[Knot[8, 12]]
Out[3]=  
PD[X[4, 2, 5, 1], X[10, 8, 11, 7], X[8, 3, 9, 4], X[2, 9, 3, 10], 
  X[14, 6, 15, 5], X[16, 11, 1, 12], X[12, 15, 13, 16], X[6, 14, 7, 13]]
In[4]:=
GaussCode[Knot[8, 12]]
Out[4]=  
GaussCode[1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6]
In[5]:=
BR[Knot[8, 12]]
Out[5]=  
BR[5, {-1, 2, -1, -3, 2, 4, -3, 4}]
In[6]:=
alex = Alexander[Knot[8, 12]][t]
Out[6]=  
      -2   7          2

13 + t   - - - 7 t + t


           t
In[7]:=
Conway[Knot[8, 12]][z]
Out[7]=  
       2    4
1 - 3 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[8, 12]}
In[9]:=
{KnotDet[Knot[8, 12]], KnotSignature[Knot[8, 12]]}
Out[9]=  
{29, 0}
In[10]:=
J=Jones[Knot[8, 12]][q]
Out[10]=  
     -4   2    4    5            2      3    4

5 + q   - -- + -- - - - 5 q + 4 q  - 2 q  + q



          3    2   q


          q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[8, 12]}
In[12]:=
A2Invariant[Knot[8, 12]][q]
Out[12]=  
      -14    -12    -10    -8    -4    -2    2    4    8    10    12

-1 + q    + q    - q    + q   - q   + q   + q  - q  + q  - q   + q   + 



  14


  q
In[13]:=
Kauffman[Knot[8, 12]][a, z]
Out[13]=  
                                         2      2

    -4    -2    2    4   z     3     2 z    2 z       2  2      4  2



1 + a   + a   + a  + a  + -- + a  z - ---- - ---- - 2 a  z  - 2 a  z  - 



                          3            4      2
                         a            a      a

    3      3                              4    4
 3 z    3 z         3      3  3      4   z    z     2  4    4  4
 ---- - ---- - 3 a z  - 3 a  z  - 4 z  + -- - -- - a  z  + a  z  + 
   3     a                                4    2
  a                                      a    a

    5      5                                6              7
 2 z    2 z         5      3  5      6   2 z       2  6   z       7
 ---- + ---- + 2 a z  + 2 a  z  + 4 z  + ---- + 2 a  z  + -- + a z
   3     a                                 2              a


   a                                       a
In[14]:=
{Vassiliev[2][Knot[8, 12]], Vassiliev[3][Knot[8, 12]]}
Out[14]=  
{0, 0}
In[15]:=
Kh[Knot[8, 12]][q, t]
Out[15]=  
3           1       1       1       3       1      2      3

- + 3 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + 
q          9  4    7  3    5  3    5  2    3  2    3     q t



         q  t    q  t    q  t    q  t    q  t    q  t

    3      3  2      5  2    5  3    7  3    9  4


  2 q  t + q  t  + 3 q  t  + q  t  + q  t  + q  t
Retrieved from "https://katlas.org/index.php?title=8_12&oldid=36134"