9 27

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9 26.gif

9_26

9 28.gif

9_28

9 27.gif Visit 9 27's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 27's page at Knotilus!

Visit 9 27's page at the original Knot Atlas!

9 27 Quick Notes


9 27 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X11,1,12,18 X5,13,6,12 X13,17,14,16 X7,14,8,15 X15,6,16,7 X17,9,18,8 X9,2,10,3
Gauss code -1, 9, -2, 1, -4, 7, -6, 8, -9, 2, -3, 4, -5, 6, -7, 5, -8, 3
Dowker-Thistlethwaite code 4 10 12 14 2 18 16 6 8
Conway Notation [212112]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-5]
Hyperbolic Volume 11.
A-Polynomial See Data:9 27/A-polynomial

[edit Notes for 9 27's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 27's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6-z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 49, 0 }
Jones polynomial [math]\displaystyle{ q^4-3 q^3+5 q^2-7 q+9-8 q^{-1} +7 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6+2 a^2 z^4+z^4 a^{-2} -4 z^4-a^4 z^2+5 a^2 z^2+2 z^2 a^{-2} -6 z^2-a^4+3 a^2+ a^{-2} -2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+6 a^2 z^6+4 z^6 a^{-2} +7 z^6+a^5 z^5-4 a^3 z^5-8 a z^5+3 z^5 a^{-3} -7 a^4 z^4-17 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -16 z^4-2 a^5 z^3-2 a^3 z^3-4 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+12 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +12 z^2+a^5 z+2 a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^4-3 a^2- a^{-2} -2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{16}+q^{12}-q^{10}+2 q^8+2 q^2-1+2 q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-4 q^{70}-6 q^{68}+19 q^{66}-29 q^{64}+33 q^{62}-29 q^{60}+6 q^{58}+22 q^{56}-50 q^{54}+65 q^{52}-56 q^{50}+32 q^{48}+6 q^{46}-42 q^{44}+61 q^{42}-58 q^{40}+33 q^{38}+3 q^{36}-32 q^{34}+41 q^{32}-25 q^{30}-q^{28}+36 q^{26}-53 q^{24}+50 q^{22}-24 q^{20}-20 q^{18}+64 q^{16}-89 q^{14}+88 q^{12}-53 q^{10}+8 q^8+45 q^6-81 q^4+87 q^2-67+26 q^{-2} +16 q^{-4} -47 q^{-6} +48 q^{-8} -25 q^{-10} -5 q^{-12} +31 q^{-14} -41 q^{-16} +24 q^{-18} + q^{-20} -35 q^{-22} +58 q^{-24} -59 q^{-26} +43 q^{-28} -9 q^{-30} -22 q^{-32} +45 q^{-34} -51 q^{-36} +45 q^{-38} -28 q^{-40} +7 q^{-42} +11 q^{-44} -23 q^{-46} +24 q^{-48} -18 q^{-50} +13 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_27.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{11}+2 q^9-2 q^7+2 q^5-q^3+q+2 q^{-1} -2 q^{-3} +2 q^{-5} -2 q^{-7} + q^{-9} }[/math]
2 [math]\displaystyle{ q^{32}-2 q^{30}-2 q^{28}+7 q^{26}-2 q^{24}-9 q^{22}+11 q^{20}+2 q^{18}-15 q^{16}+9 q^{14}+7 q^{12}-13 q^{10}+3 q^8+8 q^6-4 q^4-4 q^2+5+9 q^{-2} -11 q^{-4} -3 q^{-6} +15 q^{-8} -10 q^{-10} -7 q^{-12} +13 q^{-14} -4 q^{-16} -5 q^{-18} +6 q^{-20} - q^{-22} -2 q^{-24} + q^{-26} }[/math]
3 [math]\displaystyle{ -q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-7 q^{55}+2 q^{53}+16 q^{51}+q^{49}-23 q^{47}-11 q^{45}+29 q^{43}+26 q^{41}-31 q^{39}-42 q^{37}+26 q^{35}+55 q^{33}-13 q^{31}-65 q^{29}-q^{27}+67 q^{25}+14 q^{23}-63 q^{21}-25 q^{19}+52 q^{17}+34 q^{15}-39 q^{13}-36 q^{11}+22 q^9+39 q^7-4 q^5-36 q^3-18 q+34 q^{-1} +42 q^{-3} -27 q^{-5} -55 q^{-7} +13 q^{-9} +68 q^{-11} -2 q^{-13} -69 q^{-15} -12 q^{-17} +62 q^{-19} +20 q^{-21} -48 q^{-23} -23 q^{-25} +34 q^{-27} +21 q^{-29} -21 q^{-31} -16 q^{-33} +11 q^{-35} +11 q^{-37} -7 q^{-39} -5 q^{-41} +3 q^{-43} +3 q^{-45} - q^{-47} -2 q^{-49} + q^{-51} }[/math]
4 [math]\displaystyle{ q^{104}-2 q^{102}-2 q^{100}+3 q^{98}+3 q^{96}+7 q^{94}-9 q^{92}-16 q^{90}-q^{88}+11 q^{86}+41 q^{84}-2 q^{82}-47 q^{80}-41 q^{78}-7 q^{76}+100 q^{74}+62 q^{72}-42 q^{70}-117 q^{68}-108 q^{66}+117 q^{64}+173 q^{62}+62 q^{60}-141 q^{58}-259 q^{56}+16 q^{54}+223 q^{52}+230 q^{50}-46 q^{48}-346 q^{46}-148 q^{44}+156 q^{42}+333 q^{40}+100 q^{38}-305 q^{36}-255 q^{34}+30 q^{32}+318 q^{30}+195 q^{28}-190 q^{26}-261 q^{24}-69 q^{22}+225 q^{20}+214 q^{18}-58 q^{16}-208 q^{14}-135 q^{12}+100 q^{10}+195 q^8+88 q^6-124 q^4-197 q^2-61+154 q^{-2} +254 q^{-4} -2 q^{-6} -232 q^{-8} -236 q^{-10} +52 q^{-12} +355 q^{-14} +151 q^{-16} -171 q^{-18} -343 q^{-20} -94 q^{-22} +321 q^{-24} +243 q^{-26} -34 q^{-28} -299 q^{-30} -185 q^{-32} +176 q^{-34} +204 q^{-36} +72 q^{-38} -160 q^{-40} -163 q^{-42} +51 q^{-44} +96 q^{-46} +80 q^{-48} -49 q^{-50} -85 q^{-52} +7 q^{-54} +23 q^{-56} +41 q^{-58} -9 q^{-60} -29 q^{-62} +4 q^{-64} +12 q^{-68} -2 q^{-70} -8 q^{-72} +3 q^{-74} +3 q^{-78} - q^{-80} -2 q^{-82} + q^{-84} }[/math]
5 [math]\displaystyle{ -q^{155}+2 q^{153}+2 q^{151}-3 q^{149}-3 q^{147}-3 q^{145}+9 q^{141}+16 q^{139}+q^{137}-20 q^{135}-29 q^{133}-19 q^{131}+20 q^{129}+61 q^{127}+62 q^{125}-11 q^{123}-97 q^{121}-121 q^{119}-47 q^{117}+108 q^{115}+220 q^{113}+161 q^{111}-78 q^{109}-309 q^{107}-326 q^{105}-59 q^{103}+348 q^{101}+541 q^{99}+291 q^{97}-284 q^{95}-722 q^{93}-609 q^{91}+62 q^{89}+811 q^{87}+963 q^{85}+295 q^{83}-745 q^{81}-1264 q^{79}-732 q^{77}+497 q^{75}+1431 q^{73}+1192 q^{71}-123 q^{69}-1433 q^{67}-1558 q^{65}-315 q^{63}+1258 q^{61}+1785 q^{59}+742 q^{57}-978 q^{55}-1843 q^{53}-1070 q^{51}+644 q^{49}+1750 q^{47}+1273 q^{45}-324 q^{43}-1556 q^{41}-1345 q^{39}+50 q^{37}+1315 q^{35}+1314 q^{33}+149 q^{31}-1043 q^{29}-1228 q^{27}-316 q^{25}+799 q^{23}+1121 q^{21}+450 q^{19}-541 q^{17}-1013 q^{15}-623 q^{13}+280 q^{11}+929 q^9+805 q^7+27 q^5-818 q^3-1032 q-389 q^{-1} +672 q^{-3} +1265 q^{-5} +782 q^{-7} -438 q^{-9} -1410 q^{-11} -1218 q^{-13} +110 q^{-15} +1477 q^{-17} +1587 q^{-19} +274 q^{-21} -1350 q^{-23} -1852 q^{-25} -701 q^{-27} +1098 q^{-29} +1925 q^{-31} +1056 q^{-33} -717 q^{-35} -1801 q^{-37} -1283 q^{-39} +301 q^{-41} +1502 q^{-43} +1336 q^{-45} +71 q^{-47} -1114 q^{-49} -1217 q^{-51} -315 q^{-53} +702 q^{-55} +984 q^{-57} +430 q^{-59} -366 q^{-61} -710 q^{-63} -417 q^{-65} +137 q^{-67} +449 q^{-69} +335 q^{-71} -3 q^{-73} -256 q^{-75} -239 q^{-77} -36 q^{-79} +127 q^{-81} +138 q^{-83} +47 q^{-85} -51 q^{-87} -81 q^{-89} -32 q^{-91} +23 q^{-93} +34 q^{-95} +17 q^{-97} -5 q^{-99} -13 q^{-101} -10 q^{-103} +3 q^{-105} +8 q^{-107} - q^{-109} -3 q^{-111} +3 q^{-119} - q^{-121} -2 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{16}+q^{12}-q^{10}+2 q^8+2 q^2-1+2 q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
1,1 [math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+52 q^{36}-86 q^{34}+130 q^{32}-180 q^{30}+222 q^{28}-248 q^{26}+258 q^{24}-236 q^{22}+176 q^{20}-88 q^{18}-24 q^{16}+144 q^{14}-267 q^{12}+376 q^{10}-450 q^8+492 q^6-483 q^4+440 q^2-352+244 q^{-2} -120 q^{-4} -4 q^{-6} +102 q^{-8} -176 q^{-10} +222 q^{-12} -238 q^{-14} +228 q^{-16} -196 q^{-18} +162 q^{-20} -126 q^{-22} +88 q^{-24} -58 q^{-26} +36 q^{-28} -20 q^{-30} +10 q^{-32} -4 q^{-34} + q^{-36} }[/math]
2,0 [math]\displaystyle{ q^{42}-2 q^{38}-2 q^{36}+2 q^{34}+4 q^{32}-3 q^{30}-3 q^{28}+4 q^{26}+4 q^{24}-5 q^{22}-5 q^{20}+5 q^{18}+2 q^{16}-6 q^{14}+6 q^{10}-2 q^8-q^6+6 q^4+q^2-2+3 q^{-2} +5 q^{-4} -7 q^{-6} -5 q^{-8} +7 q^{-10} +2 q^{-12} -7 q^{-14} +6 q^{-18} -3 q^{-22} +2 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{34}-2 q^{32}+q^{30}+3 q^{28}-7 q^{26}+3 q^{24}+4 q^{22}-12 q^{20}+5 q^{18}+8 q^{16}-10 q^{14}+5 q^{12}+10 q^{10}-5 q^8-q^6+4 q^4+q^2-4-4 q^{-2} +9 q^{-4} -5 q^{-6} -8 q^{-8} +13 q^{-10} -2 q^{-12} -8 q^{-14} +10 q^{-16} -6 q^{-20} +4 q^{-22} -2 q^{-26} + q^{-28} }[/math]
1,0,0 [math]\displaystyle{ -q^{21}-q^{17}+q^{15}-q^{13}+3 q^{11}+2 q^7+q^3- q^{-1} + q^{-3} -2 q^{-5} + q^{-7} - q^{-9} +2 q^{-11} - q^{-13} + q^{-15} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+7 q^{28}-9 q^{26}+11 q^{24}-12 q^{22}+12 q^{20}-9 q^{18}+6 q^{16}-5 q^{12}+12 q^{10}-17 q^8+21 q^6-22 q^4+23 q^2-20+16 q^{-2} -9 q^{-4} +3 q^{-6} +2 q^{-8} -7 q^{-10} +10 q^{-12} -12 q^{-14} +12 q^{-16} -10 q^{-18} +8 q^{-20} -6 q^{-22} +4 q^{-24} -2 q^{-26} + q^{-28} }[/math]
1,0 [math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+5 q^{46}-2 q^{44}-8 q^{42}-3 q^{40}+9 q^{38}+8 q^{36}-7 q^{34}-13 q^{32}+13 q^{28}+7 q^{26}-9 q^{24}-9 q^{22}+5 q^{20}+11 q^{18}-8 q^{14}-q^{12}+8 q^{10}+3 q^8-7 q^6-5 q^4+7 q^2+7-5 q^{-2} -9 q^{-4} +4 q^{-6} +10 q^{-8} - q^{-10} -12 q^{-12} -4 q^{-14} +11 q^{-16} +9 q^{-18} -6 q^{-20} -12 q^{-22} +11 q^{-26} +6 q^{-28} -5 q^{-30} -7 q^{-32} +5 q^{-36} +2 q^{-38} -2 q^{-40} -2 q^{-42} + q^{-46} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{46}-2 q^{44}+3 q^{42}-4 q^{40}+6 q^{38}-8 q^{36}+7 q^{34}-10 q^{32}+9 q^{30}-11 q^{28}+6 q^{26}-6 q^{24}+6 q^{22}-q^{18}+9 q^{16}-4 q^{14}+15 q^{12}-14 q^{10}+16 q^8-16 q^6+18 q^4-19 q^2+12-15 q^{-2} +10 q^{-4} -6 q^{-6} + q^{-8} - q^{-10} -2 q^{-12} +9 q^{-14} -6 q^{-16} +8 q^{-18} -9 q^{-20} +11 q^{-22} -7 q^{-24} +6 q^{-26} -7 q^{-28} +5 q^{-30} -3 q^{-32} +2 q^{-34} -2 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-4 q^{70}-6 q^{68}+19 q^{66}-29 q^{64}+33 q^{62}-29 q^{60}+6 q^{58}+22 q^{56}-50 q^{54}+65 q^{52}-56 q^{50}+32 q^{48}+6 q^{46}-42 q^{44}+61 q^{42}-58 q^{40}+33 q^{38}+3 q^{36}-32 q^{34}+41 q^{32}-25 q^{30}-q^{28}+36 q^{26}-53 q^{24}+50 q^{22}-24 q^{20}-20 q^{18}+64 q^{16}-89 q^{14}+88 q^{12}-53 q^{10}+8 q^8+45 q^6-81 q^4+87 q^2-67+26 q^{-2} +16 q^{-4} -47 q^{-6} +48 q^{-8} -25 q^{-10} -5 q^{-12} +31 q^{-14} -41 q^{-16} +24 q^{-18} + q^{-20} -35 q^{-22} +58 q^{-24} -59 q^{-26} +43 q^{-28} -9 q^{-30} -22 q^{-32} +45 q^{-34} -51 q^{-36} +45 q^{-38} -28 q^{-40} +7 q^{-42} +11 q^{-44} -23 q^{-46} +24 q^{-48} -18 q^{-50} +13 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 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["9 27"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6-z^4+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]= { 49, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^4-3 q^3+5 q^2-7 q+9-8 q^{-1} +7 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^6+2 a^2 z^4+z^4 a^{-2} -4 z^4-a^4 z^2+5 a^2 z^2+2 z^2 a^{-2} -6 z^2-a^4+3 a^2+ a^{-2} -2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+6 a^2 z^6+4 z^6 a^{-2} +7 z^6+a^5 z^5-4 a^3 z^5-8 a z^5+3 z^5 a^{-3} -7 a^4 z^4-17 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -16 z^4-2 a^5 z^3-2 a^3 z^3-4 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+12 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +12 z^2+a^5 z+2 a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^4-3 a^2- a^{-2} -2 }[/math]

Vassiliev invariants

V2 and V3: (0, -1)
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{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{16}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ -24 }[/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 9 27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -5-4-3-2-101234χ
9         11
7        2 -2
5       31 2
3      42  -2
1     53   2
-1    45    1
-3   34     -1
-5  24      2
-7 13       -2
-9 2        2
-111         -1

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[9, 27]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 27]]
Out[3]=  
PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 18], X[5, 13, 6, 12], 

 X[13, 17, 14, 16], X[7, 14, 8, 15], X[15, 6, 16, 7], X[17, 9, 18, 8], 



  X[9, 2, 10, 3]]
In[4]:=
GaussCode[Knot[9, 27]]
Out[4]=  
GaussCode[-1, 9, -2, 1, -4, 7, -6, 8, -9, 2, -3, 4, -5, 6, -7, 5, -8, 3]
In[5]:=
BR[Knot[9, 27]]
Out[5]=  
BR[4, {-1, -1, 2, -1, 2, 2, -3, 2, -3}]
In[6]:=
alex = Alexander[Knot[9, 27]][t]
Out[6]=  
      -3   5    11             2    3

15 - t   + -- - -- - 11 t + 5 t  - t



           2   t


           t
In[7]:=
Conway[Knot[9, 27]][z]
Out[7]=  
     4    6
1 - z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 27], Knot[11, NonAlternating, 4], 
  Knot[11, NonAlternating, 21], Knot[11, NonAlternating, 172]}
In[9]:=
{KnotDet[Knot[9, 27]], KnotSignature[Knot[9, 27]]}
Out[9]=  
{49, 0}
In[10]:=
J=Jones[Knot[9, 27]][q]
Out[10]=  
     -5   3    5    7    8            2      3    4

9 - q   + -- - -- + -- - - - 7 q + 5 q  - 3 q  + q



          4    3    2   q


          q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 27], Knot[11, NonAlternating, 83]}
In[12]:=
A2Invariant[Knot[9, 27]][q]
Out[12]=  
      -16    -12    -10   2    2       2      4    8    10    12

-1 - q    + q    - q    + -- + -- + 2 q  - 2 q  + q  - q   + q



                          8    2


                          q    q
In[13]:=
Kauffman[Knot[9, 27]][a, z]
Out[13]=  
                                                                   2

     -2      2    4   z    2 z              3      5         2   z



-2 - a   - 3 a  - a  + -- + --- + 2 a z + 2 a  z + a  z + 12 z  - -- + 



                       3    a                                     4
                      a                                          a

    2                           3      3
 3 z        2  2      4  2   4 z    4 z       3  3      5  3       4
 ---- + 12 a  z  + 4 a  z  - ---- - ---- - 2 a  z  - 2 a  z  - 16 z  + 
   2                           3     a
  a                           a

  4      4                           5
 z    5 z        2  4      4  4   3 z         5      3  5    5  5
 -- - ---- - 17 a  z  - 7 a  z  + ---- - 8 a z  - 4 a  z  + a  z  + 
  4     2                           3
 a     a                           a

           6                          7
    6   4 z       2  6      4  6   3 z         7      3  7    8    2  8
 7 z  + ---- + 6 a  z  + 3 a  z  + ---- + 6 a z  + 3 a  z  + z  + a  z
          2                         a


          a
In[14]:=
{Vassiliev[2][Knot[9, 27]], Vassiliev[3][Knot[9, 27]]}
Out[14]=  
{0, -1}
In[15]:=
Kh[Knot[9, 27]][q, t]
Out[15]=  
5           1        2       1       3       2       4       3

- + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
q          11  5    9  4    7  4    7  3    5  3    5  2    3  2



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

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

  9  4


  q  t
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