10 114: Difference between revisions

Revision as of 20:06, 28 August 2005

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10 114 Quick Notes

10 114 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X₈₃₉₄ X_18,13,19,14 X_20,11,1,12 X_12,19,13,20 X_2,16,3,15 X_4,17,5,18 X_10,6,11,5 X_14,7,15,8 X_16,10,17,9
Gauss code	1, -6, 2, -7, 8, -1, 9, -2, 10, -8, 4, -5, 3, -9, 6, -10, 7, -3, 5, -4
Dowker-Thistlethwaite code	6 8 10 14 16 20 18 2 4 12
Conway Notation	[8*30]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-7][-5]
Hyperbolic Volume	15.3049
A-Polynomial	See Data:10 114/A-polynomial

[edit Notes for 10 114'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{ 3 }[/math]
Rasmussen s-Invariant	0

[edit Notes for 10 114's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+10 t^2-21 t+27-21 t^{-1} +10 t^{-2} -2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^6-2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 93, 0 }
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+8 q^2-12 q+15-15 q^{-1} +15 q^{-2} -11 q^{-3} +7 q^{-4} -4 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^6-z^6+a^4 z^4-2 a^2 z^4+z^4 a^{-2} -2 z^4+a^4 z^2+z^2 a^{-2} -z^2-a^4+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 a^3 z^9+3 a z^9+6 a^4 z^8+14 a^2 z^8+8 z^8+4 a^5 z^7+2 a^3 z^7+8 a z^7+10 z^7 a^{-1} +a^6 z^6-17 a^4 z^6-35 a^2 z^6+8 z^6 a^{-2} -9 z^6-11 a^5 z^5-21 a^3 z^5-27 a z^5-13 z^5 a^{-1} +4 z^5 a^{-3} -2 a^6 z^4+14 a^4 z^4+26 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +z^4+7 a^5 z^3+18 a^3 z^3+18 a z^3+5 z^3 a^{-1} -2 z^3 a^{-3} -3 a^4 z^2-5 a^2 z^2+2 z^2 a^{-2} -2 a^3 z-3 a z-z a^{-1} -a^4-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{18}-2 q^{16}-3 q^{10}+4 q^8+2 q^4+2 q^2-2+3 q^{-2} -3 q^{-4} + q^{-6} + q^{-8} -2 q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-3 q^{92}+7 q^{90}-14 q^{88}+17 q^{86}-18 q^{84}+7 q^{82}+23 q^{80}-59 q^{78}+102 q^{76}-118 q^{74}+87 q^{72}-8 q^{70}-116 q^{68}+233 q^{66}-288 q^{64}+240 q^{62}-90 q^{60}-114 q^{58}+292 q^{56}-369 q^{54}+304 q^{52}-124 q^{50}-102 q^{48}+262 q^{46}-301 q^{44}+192 q^{42}+8 q^{40}-188 q^{38}+283 q^{36}-240 q^{34}+79 q^{32}+135 q^{30}-323 q^{28}+407 q^{26}-345 q^{24}+160 q^{22}+105 q^{20}-340 q^{18}+471 q^{16}-440 q^{14}+265 q^{12}-9 q^{10}-238 q^8+371 q^6-346 q^4+186 q^2+42-216 q^{-2} +264 q^{-4} -170 q^{-6} -16 q^{-8} +194 q^{-10} -288 q^{-12} +259 q^{-14} -123 q^{-16} -57 q^{-18} +212 q^{-20} -287 q^{-22} +265 q^{-24} -164 q^{-26} +35 q^{-28} +77 q^{-30} -152 q^{-32} +168 q^{-34} -142 q^{-36} +92 q^{-38} -29 q^{-40} -22 q^{-42} +51 q^{-44} -63 q^{-46} +54 q^{-48} -34 q^{-50} +16 q^{-52} + q^{-54} -8 q^{-56} +10 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_114.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-3 q^{11}+3 q^9-4 q^7+4 q^5+3 q^{-1} -4 q^{-3} +4 q^{-5} -3 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{38}-3 q^{36}-q^{34}+12 q^{32}-8 q^{30}-17 q^{28}+27 q^{26}+4 q^{24}-39 q^{22}+24 q^{20}+24 q^{18}-43 q^{16}+6 q^{14}+33 q^{12}-23 q^{10}-14 q^8+24 q^6+9 q^4-26 q^2+38 q^{-2} -24 q^{-4} -26 q^{-6} +43 q^{-8} -8 q^{-10} -30 q^{-12} +25 q^{-14} +2 q^{-16} -14 q^{-18} +8 q^{-20} + q^{-22} -3 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{75}-3 q^{73}-q^{71}+8 q^{69}+7 q^{67}-16 q^{65}-30 q^{63}+23 q^{61}+66 q^{59}-2 q^{57}-112 q^{55}-54 q^{53}+140 q^{51}+142 q^{49}-124 q^{47}-235 q^{45}+52 q^{43}+303 q^{41}+54 q^{39}-318 q^{37}-163 q^{35}+279 q^{33}+257 q^{31}-217 q^{29}-302 q^{27}+125 q^{25}+321 q^{23}-50 q^{21}-307 q^{19}-25 q^{17}+276 q^{15}+96 q^{13}-226 q^{11}-167 q^9+162 q^7+241 q^5-70 q^3-291 q-46 q^{-1} +317 q^{-3} +160 q^{-5} -286 q^{-7} -261 q^{-9} +216 q^{-11} +304 q^{-13} -116 q^{-15} -291 q^{-17} +27 q^{-19} +230 q^{-21} +25 q^{-23} -151 q^{-25} -37 q^{-27} +81 q^{-29} +30 q^{-31} -41 q^{-33} -15 q^{-35} +23 q^{-37} +2 q^{-39} -9 q^{-41} -2 q^{-43} +5 q^{-45} + q^{-47} -3 q^{-49} + q^{-51} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{18}-2 q^{16}-3 q^{10}+4 q^8+2 q^4+2 q^2-2+3 q^{-2} -3 q^{-4} + q^{-6} + q^{-8} -2 q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{52}-6 q^{50}+20 q^{48}-52 q^{46}+119 q^{44}-240 q^{42}+424 q^{40}-670 q^{38}+971 q^{36}-1274 q^{34}+1510 q^{32}-1624 q^{30}+1560 q^{28}-1284 q^{26}+786 q^{24}-128 q^{22}-620 q^{20}+1380 q^{18}-2070 q^{16}+2604 q^{14}-2922 q^{12}+2992 q^{10}-2792 q^8+2358 q^6-1732 q^4+1018 q^2-276-380 q^{-2} +881 q^{-4} -1196 q^{-6} +1332 q^{-8} -1326 q^{-10} +1202 q^{-12} -1022 q^{-14} +830 q^{-16} -640 q^{-18} +468 q^{-20} -328 q^{-22} +220 q^{-24} -134 q^{-26} +73 q^{-28} -38 q^{-30} +18 q^{-32} -6 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{48}-2 q^{46}-2 q^{44}+5 q^{42}+3 q^{40}-4 q^{38}-7 q^{36}+7 q^{34}+12 q^{32}-13 q^{30}-12 q^{28}+10 q^{26}+9 q^{24}-12 q^{22}-12 q^{20}+16 q^{18}+8 q^{16}-14 q^{14}+11 q^{10}-6 q^8-q^6+14 q^4-2 q^2-7+8 q^{-2} +13 q^{-4} -17 q^{-6} -15 q^{-8} +19 q^{-10} +6 q^{-12} -18 q^{-14} -2 q^{-16} +14 q^{-18} +3 q^{-20} -9 q^{-22} -2 q^{-24} +6 q^{-26} - q^{-28} -2 q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{40}-3 q^{38}+q^{36}+5 q^{34}-11 q^{32}+10 q^{30}+9 q^{28}-24 q^{26}+17 q^{24}+7 q^{22}-34 q^{20}+17 q^{18}+12 q^{16}-26 q^{14}+13 q^{12}+17 q^{10}-8 q^8-3 q^6+5 q^4+12 q^2-15-9 q^{-2} +31 q^{-4} -19 q^{-6} -16 q^{-8} +33 q^{-10} -14 q^{-12} -16 q^{-14} +22 q^{-16} -5 q^{-18} -10 q^{-20} +9 q^{-22} -3 q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ q^{23}-2 q^{21}+q^{19}-3 q^{17}+q^{15}-3 q^{13}+4 q^{11}+3 q^7+q^5+q^3+q-2 q^{-1} +3 q^{-3} -3 q^{-5} +2 q^{-7} -2 q^{-9} +2 q^{-11} -2 q^{-13} + q^{-15} }[/math]

A4 Invariants.

Weight

Invariant

0,1,0,0

[math]\displaystyle{ q^{50}-2 q^{48}-2 q^{46}+6 q^{44}-3 q^{42}-8 q^{40}+12 q^{38}+9 q^{36}-15 q^{34}-2 q^{32}+19 q^{30}-7 q^{28}-31 q^{26}+3 q^{24}+23 q^{22}-21 q^{20}-16 q^{18}+36 q^{16}+12 q^{14}-21 q^{12}+15 q^{10}+20 q^8-19 q^6-12 q^4+20 q^2+1-25 q^{-2} +10 q^{-4} +26 q^{-6} -18 q^{-8} -15 q^{-10} +20 q^{-12} +6 q^{-14} -18 q^{-16} -3 q^{-18} +12 q^{-20} +2 q^{-22} -8 q^{-24} +6 q^{-28} -2 q^{-30} -2 q^{-32} + q^{-34} }[/math]

1,0,0,0

[math]\displaystyle{ q^{28}-2 q^{26}+q^{24}-2 q^{22}-2 q^{20}+q^{18}-3 q^{16}+4 q^{14}+3 q^{10}+2 q^8+q^6+q^4+1-2 q^{-2} +3 q^{-4} -3 q^{-6} +2 q^{-8} - q^{-10} - q^{-12} +2 q^{-14} -2 q^{-16} + q^{-18} }[/math]

B2 Invariants.

Weight

Invariant

0,1

[math]\displaystyle{ q^{40}-3 q^{38}+7 q^{36}-13 q^{34}+21 q^{32}-30 q^{30}+35 q^{28}-40 q^{26}+39 q^{24}-35 q^{22}+24 q^{20}-9 q^{18}-10 q^{16}+32 q^{14}-49 q^{12}+67 q^{10}-74 q^8+79 q^6-71 q^4+60 q^2-43+23 q^{-2} -3 q^{-4} -15 q^{-6} +28 q^{-8} -37 q^{-10} +40 q^{-12} -38 q^{-14} +32 q^{-16} -25 q^{-18} +18 q^{-20} -11 q^{-22} +6 q^{-24} -3 q^{-26} + q^{-28} }[/math]

1,0

[math]\displaystyle{ q^{66}-3 q^{62}-3 q^{60}+4 q^{58}+9 q^{56}-2 q^{54}-16 q^{52}-6 q^{50}+23 q^{48}+21 q^{46}-17 q^{44}-35 q^{42}+q^{40}+40 q^{38}+18 q^{36}-36 q^{34}-37 q^{32}+15 q^{30}+40 q^{28}+2 q^{26}-36 q^{24}-13 q^{22}+27 q^{20}+22 q^{18}-17 q^{16}-19 q^{14}+15 q^{12}+26 q^{10}-9 q^8-28 q^6+3 q^4+32 q^2+6-33 q^{-2} -18 q^{-4} +30 q^{-6} +30 q^{-8} -20 q^{-10} -40 q^{-12} +2 q^{-14} +40 q^{-16} +17 q^{-18} -27 q^{-20} -29 q^{-22} +8 q^{-24} +27 q^{-26} +8 q^{-28} -15 q^{-30} -14 q^{-32} +3 q^{-34} +10 q^{-36} +3 q^{-38} -3 q^{-40} -3 q^{-42} + q^{-46} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{54}-3 q^{52}+4 q^{50}-7 q^{48}+12 q^{46}-17 q^{44}+22 q^{42}-25 q^{40}+31 q^{38}-32 q^{36}+30 q^{34}-31 q^{32}+26 q^{30}-23 q^{28}+7 q^{26}-2 q^{24}-8 q^{22}+22 q^{20}-33 q^{18}+46 q^{16}-45 q^{14}+60 q^{12}-60 q^{10}+59 q^8-54 q^6+53 q^4-43 q^2+29-21 q^{-2} +13 q^{-4} +4 q^{-6} -14 q^{-8} +17 q^{-10} -25 q^{-12} +33 q^{-14} -31 q^{-16} +26 q^{-18} -29 q^{-20} +27 q^{-22} -18 q^{-24} +14 q^{-26} -14 q^{-28} +10 q^{-30} -4 q^{-32} +3 q^{-34} -3 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{94}-3 q^{92}+7 q^{90}-14 q^{88}+17 q^{86}-18 q^{84}+7 q^{82}+23 q^{80}-59 q^{78}+102 q^{76}-118 q^{74}+87 q^{72}-8 q^{70}-116 q^{68}+233 q^{66}-288 q^{64}+240 q^{62}-90 q^{60}-114 q^{58}+292 q^{56}-369 q^{54}+304 q^{52}-124 q^{50}-102 q^{48}+262 q^{46}-301 q^{44}+192 q^{42}+8 q^{40}-188 q^{38}+283 q^{36}-240 q^{34}+79 q^{32}+135 q^{30}-323 q^{28}+407 q^{26}-345 q^{24}+160 q^{22}+105 q^{20}-340 q^{18}+471 q^{16}-440 q^{14}+265 q^{12}-9 q^{10}-238 q^8+371 q^6-346 q^4+186 q^2+42-216 q^{-2} +264 q^{-4} -170 q^{-6} -16 q^{-8} +194 q^{-10} -288 q^{-12} +259 q^{-14} -123 q^{-16} -57 q^{-18} +212 q^{-20} -287 q^{-22} +265 q^{-24} -164 q^{-26} +35 q^{-28} +77 q^{-30} -152 q^{-32} +168 q^{-34} -142 q^{-36} +92 q^{-38} -29 q^{-40} -22 q^{-42} +51 q^{-44} -63 q^{-46} +54 q^{-48} -34 q^{-50} +16 q^{-52} + q^{-54} -8 q^{-56} +10 q^{-58} -10 q^{-60} +6 q^{-62} -3 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.

In[3]:= K = Knot["10 114"];

In[4]:= Alexander[K][t]

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

Out[4]= [math]\displaystyle{ -2 t^3+10 t^2-21 t+27-21 t^{-1} +10 t^{-2} -2 t^{-3} }[/math]

In[5]:= Conway[K][z]

Out[5]= [math]\displaystyle{ -2 z^6-2 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]= { 93, 0 }

In[8]:= Jones[K][q]

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

Out[8]= [math]\displaystyle{ q^4-4 q^3+8 q^2-12 q+15-15 q^{-1} +15 q^{-2} -11 q^{-3} +7 q^{-4} -4 q^{-5} + q^{-6} }[/math]

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= [math]\displaystyle{ -a^2 z^6-z^6+a^4 z^4-2 a^2 z^4+z^4 a^{-2} -2 z^4+a^4 z^2+z^2 a^{-2} -z^2-a^4+2 a^2 }[/math]

In[10]:= Kauffman[K][a, z]

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

Out[10]= [math]\displaystyle{ 3 a^3 z^9+3 a z^9+6 a^4 z^8+14 a^2 z^8+8 z^8+4 a^5 z^7+2 a^3 z^7+8 a z^7+10 z^7 a^{-1} +a^6 z^6-17 a^4 z^6-35 a^2 z^6+8 z^6 a^{-2} -9 z^6-11 a^5 z^5-21 a^3 z^5-27 a z^5-13 z^5 a^{-1} +4 z^5 a^{-3} -2 a^6 z^4+14 a^4 z^4+26 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +z^4+7 a^5 z^3+18 a^3 z^3+18 a z^3+5 z^3 a^{-1} -2 z^3 a^{-3} -3 a^4 z^2-5 a^2 z^2+2 z^2 a^{-2} -2 a^3 z-3 a z-z a^{-1} -a^4-2 a^2 }[/math]

Vassiliev invariants

V₂ and V₃:

(1, -1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

[math]\displaystyle{ 4 }[/math]

[math]\displaystyle{ -8 }[/math]

[math]\displaystyle{ 8 }[/math]

[math]\displaystyle{ \frac{110}{3} }[/math]

[math]\displaystyle{ \frac{58}{3} }[/math]

[math]\displaystyle{ -32 }[/math]

[math]\displaystyle{ -\frac{272}{3} }[/math]

[math]\displaystyle{ -\frac{32}{3} }[/math]

[math]\displaystyle{ -40 }[/math]

[math]\displaystyle{ \frac{32}{3} }[/math]

[math]\displaystyle{ 32 }[/math]

[math]\displaystyle{ \frac{440}{3} }[/math]

[math]\displaystyle{ \frac{232}{3} }[/math]

[math]\displaystyle{ \frac{8191}{30} }[/math]

[math]\displaystyle{ -\frac{1462}{15} }[/math]

[math]\displaystyle{ \frac{9302}{45} }[/math]

[math]\displaystyle{ \frac{1217}{18} }[/math]

[math]\displaystyle{ \frac{1951}{30} }[/math]

V_2,1 through V_6,9 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 10 114. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

3

-3

5

1

4

3

7

3

-4

1

8

5

3

-1

8

0

-3

7

0

-5

4

8

4

-7

3

7

-4

-9

1

4

3

-11

3

-3

-13

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=-6 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-3 }[/math]	[math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math]	[math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math]	[math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=0 }[/math]	[math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math]	[math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math]	[math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math]	[math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=3 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/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[10, 114]]
Out[2]=	10
In[3]:=	PD[Knot[10, 114]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[18, 13, 19, 14], X[20, 11, 1, 12], X[12, 19, 13, 20], X[2, 16, 3, 15], X[4, 17, 5, 18], X[10, 6, 11, 5], X[14, 7, 15, 8], X[16, 10, 17, 9]]
In[4]:=	GaussCode[Knot[10, 114]]
Out[4]=	GaussCode[1, -6, 2, -7, 8, -1, 9, -2, 10, -8, 4, -5, 3, -9, 6, -10, 7, -3, 5, -4]
In[5]:=	BR[Knot[10, 114]]
Out[5]=	BR[4, {-1, -1, -2, 1, 3, -2, 3, -2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[10, 114]][t]
Out[6]=	2 10 21 2 3 27 - -- + -- - -- - 21 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 114]][z]
Out[7]=	2 4 6 1 + z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 114], Knot[11, Alternating, 93]}
In[9]:=	{KnotDet[Knot[10, 114]], KnotSignature[Knot[10, 114]]}
Out[9]=	{93, 0}
In[10]:=	J=Jones[Knot[10, 114]][q]
Out[10]=	-6 4 7 11 15 15 2 3 4 15 + q - -- + -- - -- + -- - -- - 12 q + 8 q - 4 q + q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 114]}
In[12]:=	A2Invariant[Knot[10, 114]][q]
Out[12]=	-18 2 3 4 2 2 2 4 6 8 10 -2 + q - --- - --- + -- + -- + -- + 3 q - 3 q + q + q - 2 q + 16 10 8 4 2 q q q q q 12 q
In[13]:=	Kauffman[Knot[10, 114]][a, z]
Out[13]=	2 3 2 4 z 3 2 z 2 2 4 2 2 z -2 a - a - - - 3 a z - 2 a z + ---- - 5 a z - 3 a z - ---- + a 2 3 a a 3 4 4 5 z 3 3 3 5 3 4 z 8 z 2 4 ---- + 18 a z + 18 a z + 7 a z + z + -- - ---- + 26 a z + a 4 2 a a 5 5 4 4 6 4 4 z 13 z 5 3 5 5 5 14 a z - 2 a z + ---- - ----- - 27 a z - 21 a z - 11 a z - 3 a a 6 7 6 8 z 2 6 4 6 6 6 10 z 7 9 z + ---- - 35 a z - 17 a z + a z + ----- + 8 a z + 2 a a 3 7 5 7 8 2 8 4 8 9 3 9 2 a z + 4 a z + 8 z + 14 a z + 6 a z + 3 a z + 3 a z
In[14]:=	{Vassiliev[2][Knot[10, 114]], Vassiliev[3][Knot[10, 114]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[10, 114]][q, t]
Out[15]=	8 1 3 1 4 3 7 4 - + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 8 7 7 8 3 3 2 5 2 ----- + ----- + ---- + --- + 5 q t + 7 q t + 3 q t + 5 q t + 5 2 3 2 3 q t q t q t q t 5 3 7 3 9 4 q t + 3 q t + q t

@@ Line 1: / Line 1: @@
 <!--  -->
+<!-- -->
+<!--  -->
+<!-- -->
 <!-- provide an anchor so we can return to the top of the page -->
 <span id="top"></span>
+<!-- -->
 <!-- this relies on transclusion for next and previous links -->
 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=10|k=114|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-6,2,-7,8,-1,9,-2,10,-8,4,-5,3,-9,6,-10,7,-3,5,-4/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=10|k=114|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-6,2,-7,8,-1,9,-2,10,-8,4,-5,3,-9,6,-10,7,-3,5,-4/goTop.html}}
-|{{:{{PAGENAME}} Quick Notes}}
-|}
 <br style="clear:both" />
@@ Line 24: / Line 21: @@
 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=13.3333%><table cellpadding=0 cellspacing=0>
@@ Line 48: / Line 41: @@
 <tr align=center><td>-11</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-3</td></tr>
 <tr align=center><td>-13</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 143: / Line 135: @@
   q  t  + 3 q  t  + q  t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

10 114: Difference between revisions

Revision as of 20:06, 28 August 2005

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