10 102: Difference between revisions

Revision as of 20:16, 28 August 2005

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10 102 Quick Notes

10 102 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_16,10,17,9 X_10,3,11,4 X_2,15,3,16 X_14,5,15,6 X_18,8,19,7 X_4,11,5,12 X_8,18,9,17 X_20,14,1,13 X_12,20,13,19
Gauss code	1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -10, 9, -5, 4, -2, 8, -6, 10, -9
Dowker-Thistlethwaite code	6 10 14 18 16 4 20 2 8 12
Conway Notation	[3:2:20]

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-5][-7]
Hyperbolic Volume	13.7273
A-Polynomial	See Data:10 102/A-polynomial

[edit Notes for 10 102'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 10 102's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+8 t^2-16 t+21-16 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^6-4 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 73, 0 }
Jones polynomial	[math]\displaystyle{ q^6-3 q^5+6 q^4-9 q^3+11 q^2-12 q+12-9 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -3 z^2+a^2- a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+6 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +5 a^2 z^6-24 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -7 z^6+3 a^3 z^5-9 a z^5-17 z^5 a^{-1} -14 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-6 a^2 z^4+21 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} +3 z^4-3 a^3 z^3+7 a z^3+16 z^3 a^{-1} +13 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+3 a^2 z^2-8 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -2 z a^{-5} -a^2+ a^{-2} + a^{-4} }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}-q^{10}+q^8+q^6-2 q^4+3 q^2-1+ q^{-2} -2 q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+11 q^{52}-18 q^{50}+26 q^{48}-28 q^{46}+19 q^{44}-4 q^{42}-19 q^{40}+42 q^{38}-59 q^{36}+68 q^{34}-61 q^{32}+33 q^{30}+15 q^{28}-64 q^{26}+110 q^{24}-123 q^{22}+100 q^{20}-42 q^{18}-40 q^{16}+108 q^{14}-134 q^{12}+108 q^{10}-29 q^8-57 q^6+110 q^4-105 q^2+39+57 q^{-2} -141 q^{-4} +164 q^{-6} -116 q^{-8} +14 q^{-10} +107 q^{-12} -193 q^{-14} +216 q^{-16} -165 q^{-18} +60 q^{-20} +55 q^{-22} -151 q^{-24} +192 q^{-26} -166 q^{-28} +88 q^{-30} +14 q^{-32} -100 q^{-34} +135 q^{-36} -108 q^{-38} +29 q^{-40} +61 q^{-42} -125 q^{-44} +126 q^{-46} -65 q^{-48} -34 q^{-50} +131 q^{-52} -171 q^{-54} +148 q^{-56} -68 q^{-58} -33 q^{-60} +111 q^{-62} -142 q^{-64} +125 q^{-66} -69 q^{-68} +6 q^{-70} +42 q^{-72} -63 q^{-74} +57 q^{-76} -36 q^{-78} +16 q^{-80} + q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_102.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-2 q^7+3 q^5-3 q^3+3 q- q^{-3} +2 q^{-5} -3 q^{-7} +3 q^{-9} -2 q^{-11} + q^{-13} }[/math]
2	[math]\displaystyle{ q^{26}-2 q^{24}+5 q^{20}-7 q^{18}+q^{16}+13 q^{14}-17 q^{12}-3 q^{10}+25 q^8-17 q^6-13 q^4+24 q^2-2-15 q^{-2} +6 q^{-4} +13 q^{-6} -9 q^{-8} -12 q^{-10} +20 q^{-12} + q^{-14} -24 q^{-16} +16 q^{-18} +13 q^{-20} -24 q^{-22} +4 q^{-24} +17 q^{-26} -12 q^{-28} -5 q^{-30} +8 q^{-32} - q^{-34} -2 q^{-36} + q^{-38} }[/math]
3	[math]\displaystyle{ q^{51}-2 q^{49}+2 q^{45}+q^{43}-3 q^{41}-q^{39}+6 q^{37}-5 q^{35}-10 q^{33}+14 q^{31}+24 q^{29}-23 q^{27}-53 q^{25}+28 q^{23}+91 q^{21}-13 q^{19}-129 q^{17}-24 q^{15}+148 q^{13}+70 q^{11}-140 q^9-107 q^7+99 q^5+134 q^3-46 q-131 q^{-1} -6 q^{-3} +111 q^{-5} +57 q^{-7} -87 q^{-9} -90 q^{-11} +58 q^{-13} +118 q^{-15} -35 q^{-17} -136 q^{-19} +5 q^{-21} +148 q^{-23} +29 q^{-25} -150 q^{-27} -67 q^{-29} +135 q^{-31} +105 q^{-33} -98 q^{-35} -133 q^{-37} +47 q^{-39} +140 q^{-41} +4 q^{-43} -116 q^{-45} -47 q^{-47} +77 q^{-49} +65 q^{-51} -36 q^{-53} -56 q^{-55} +3 q^{-57} +36 q^{-59} +10 q^{-61} -17 q^{-63} -8 q^{-65} +5 q^{-67} +4 q^{-69} - q^{-71} -2 q^{-73} + q^{-75} }[/math]
4	[math]\displaystyle{ q^{84}-2 q^{82}+2 q^{78}-2 q^{76}+5 q^{74}-5 q^{72}-q^{70}+q^{68}-11 q^{66}+21 q^{64}+5 q^{62}+5 q^{60}-18 q^{58}-67 q^{56}+25 q^{54}+70 q^{52}+104 q^{50}-17 q^{48}-250 q^{46}-139 q^{44}+118 q^{42}+426 q^{40}+261 q^{38}-418 q^{36}-636 q^{34}-223 q^{32}+708 q^{30}+945 q^{28}-70 q^{26}-1017 q^{24}-1010 q^{22}+343 q^{20}+1386 q^{18}+739 q^{16}-626 q^{14}-1428 q^{12}-481 q^{10}+947 q^8+1150 q^6+222 q^4-987 q^2-928+87 q^{-2} +845 q^{-4} +724 q^{-6} -234 q^{-8} -813 q^{-10} -493 q^{-12} +369 q^{-14} +818 q^{-16} +262 q^{-18} -620 q^{-20} -779 q^{-22} +102 q^{-24} +874 q^{-26} +588 q^{-28} -530 q^{-30} -1064 q^{-32} -171 q^{-34} +923 q^{-36} +1002 q^{-38} -243 q^{-40} -1256 q^{-42} -685 q^{-44} +591 q^{-46} +1318 q^{-48} +423 q^{-50} -934 q^{-52} -1115 q^{-54} -211 q^{-56} +1022 q^{-58} +978 q^{-60} -69 q^{-62} -868 q^{-64} -845 q^{-66} +174 q^{-68} +789 q^{-70} +566 q^{-72} -105 q^{-74} -684 q^{-76} -390 q^{-78} +129 q^{-80} +441 q^{-82} +324 q^{-84} -144 q^{-86} -277 q^{-88} -183 q^{-90} +58 q^{-92} +199 q^{-94} +72 q^{-96} -25 q^{-98} -93 q^{-100} -47 q^{-102} +32 q^{-104} +26 q^{-106} +19 q^{-108} -11 q^{-110} -14 q^{-112} +2 q^{-114} + q^{-116} +4 q^{-118} - q^{-120} -2 q^{-122} + q^{-124} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}-q^{10}+q^8+q^6-2 q^4+3 q^2-1+ q^{-2} -2 q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }[/math]
1,1	[math]\displaystyle{ q^{36}-4 q^{34}+10 q^{32}-20 q^{30}+36 q^{28}-58 q^{26}+90 q^{24}-128 q^{22}+171 q^{20}-218 q^{18}+278 q^{16}-348 q^{14}+407 q^{12}-458 q^{10}+496 q^8-480 q^6+399 q^4-242 q^2+24+242 q^{-2} -534 q^{-4} +798 q^{-6} -1008 q^{-8} +1140 q^{-10} -1165 q^{-12} +1098 q^{-14} -934 q^{-16} +698 q^{-18} -408 q^{-20} +96 q^{-22} +188 q^{-24} -420 q^{-26} +583 q^{-28} -656 q^{-30} +638 q^{-32} -556 q^{-34} +445 q^{-36} -324 q^{-38} +208 q^{-40} -122 q^{-42} +66 q^{-44} -30 q^{-46} +12 q^{-48} -4 q^{-50} + q^{-52} }[/math]
2,0	[math]\displaystyle{ q^{32}-q^{30}-q^{28}+3 q^{26}-4 q^{22}+2 q^{20}+7 q^{18}-2 q^{16}-10 q^{14}+4 q^{12}+12 q^{10}-9 q^8-9 q^6+9 q^4+4 q^2-5-3 q^{-2} +8 q^{-4} -2 q^{-6} -4 q^{-8} +7 q^{-10} + q^{-12} -7 q^{-14} +4 q^{-16} +9 q^{-18} -8 q^{-20} -5 q^{-22} +6 q^{-24} +6 q^{-26} -7 q^{-28} -7 q^{-30} +8 q^{-32} +3 q^{-34} -5 q^{-36} -3 q^{-38} +2 q^{-40} +3 q^{-42} - q^{-44} - q^{-46} + q^{-48} }[/math]

A3 Invariants.

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

A4 Invariants.

Weight

Invariant

0,1,0,0

[math]\displaystyle{ q^{34}-q^{32}-q^{30}+3 q^{28}-4 q^{24}+q^{22}+7 q^{20}-9 q^{16}+3 q^{14}+13 q^{12}-7 q^{10}-12 q^8+14 q^6+7 q^4-12 q^2+1+15 q^{-2} -3 q^{-4} -12 q^{-6} +8 q^{-8} +5 q^{-10} -19 q^{-12} -3 q^{-14} +17 q^{-16} -11 q^{-18} -12 q^{-20} +17 q^{-22} +9 q^{-24} -12 q^{-26} - q^{-28} +14 q^{-30} -10 q^{-34} +3 q^{-36} +6 q^{-38} -6 q^{-40} -2 q^{-42} +4 q^{-44} - q^{-46} - q^{-48} + q^{-50} }[/math]

1,0,0,0

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

B2 Invariants.

Weight

Invariant

0,1

[math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-7 q^{22}+10 q^{20}-14 q^{18}+19 q^{16}-21 q^{14}+24 q^{12}-22 q^{10}+18 q^8-10 q^6+12 q^2-24+34 q^{-2} -42 q^{-4} +46 q^{-6} -46 q^{-8} +41 q^{-10} -32 q^{-12} +22 q^{-14} -9 q^{-16} -3 q^{-18} +13 q^{-20} -20 q^{-22} +23 q^{-24} -24 q^{-26} +23 q^{-28} -19 q^{-30} +14 q^{-32} -9 q^{-34} +5 q^{-36} -2 q^{-38} + q^{-40} }[/math]

1,0

[math]\displaystyle{ q^{46}-2 q^{42}-2 q^{40}+2 q^{38}+6 q^{36}+q^{34}-8 q^{32}-8 q^{30}+5 q^{28}+16 q^{26}+4 q^{24}-16 q^{22}-16 q^{20}+9 q^{18}+24 q^{16}+3 q^{14}-23 q^{12}-13 q^{10}+17 q^8+20 q^6-9 q^4-20 q^2+3+20 q^{-2} +3 q^{-4} -17 q^{-6} -7 q^{-8} +13 q^{-10} +9 q^{-12} -12 q^{-14} -12 q^{-16} +10 q^{-18} +15 q^{-20} -7 q^{-22} -21 q^{-24} +24 q^{-28} +12 q^{-30} -20 q^{-32} -22 q^{-34} +11 q^{-36} +26 q^{-38} +3 q^{-40} -21 q^{-42} -12 q^{-44} +13 q^{-46} +14 q^{-48} -4 q^{-50} -11 q^{-52} -2 q^{-54} +6 q^{-56} +3 q^{-58} -2 q^{-60} -2 q^{-62} + q^{-66} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{38}-2 q^{36}+2 q^{34}-3 q^{32}+6 q^{30}-8 q^{28}+8 q^{26}-10 q^{24}+16 q^{22}-16 q^{20}+15 q^{18}-17 q^{16}+21 q^{14}-15 q^{12}+11 q^{10}-9 q^8+5 q^6+7 q^4-10 q^2+17-24 q^{-2} +31 q^{-4} -32 q^{-6} +33 q^{-8} -39 q^{-10} +33 q^{-12} -32 q^{-14} +26 q^{-16} -24 q^{-18} +14 q^{-20} -7 q^{-22} +3 q^{-24} +5 q^{-26} -9 q^{-28} +18 q^{-30} -16 q^{-32} +19 q^{-34} -19 q^{-36} +20 q^{-38} -17 q^{-40} +13 q^{-42} -12 q^{-44} +8 q^{-46} -5 q^{-48} +3 q^{-50} -2 q^{-52} + q^{-54} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+11 q^{52}-18 q^{50}+26 q^{48}-28 q^{46}+19 q^{44}-4 q^{42}-19 q^{40}+42 q^{38}-59 q^{36}+68 q^{34}-61 q^{32}+33 q^{30}+15 q^{28}-64 q^{26}+110 q^{24}-123 q^{22}+100 q^{20}-42 q^{18}-40 q^{16}+108 q^{14}-134 q^{12}+108 q^{10}-29 q^8-57 q^6+110 q^4-105 q^2+39+57 q^{-2} -141 q^{-4} +164 q^{-6} -116 q^{-8} +14 q^{-10} +107 q^{-12} -193 q^{-14} +216 q^{-16} -165 q^{-18} +60 q^{-20} +55 q^{-22} -151 q^{-24} +192 q^{-26} -166 q^{-28} +88 q^{-30} +14 q^{-32} -100 q^{-34} +135 q^{-36} -108 q^{-38} +29 q^{-40} +61 q^{-42} -125 q^{-44} +126 q^{-46} -65 q^{-48} -34 q^{-50} +131 q^{-52} -171 q^{-54} +148 q^{-56} -68 q^{-58} -33 q^{-60} +111 q^{-62} -142 q^{-64} +125 q^{-66} -69 q^{-68} +6 q^{-70} +42 q^{-72} -63 q^{-74} +57 q^{-76} -36 q^{-78} +16 q^{-80} + q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 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.

In[3]:= K = Knot["10 102"];

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

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

Out[4]= [math]\displaystyle{ -2 t^3+8 t^2-16 t+21-16 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]

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

Out[5]= [math]\displaystyle{ -2 z^6-4 z^4-2 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]= { 73, 0 }

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

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

Out[8]= [math]\displaystyle{ q^6-3 q^5+6 q^4-9 q^3+11 q^2-12 q+12-9 q^{-1} +6 q^{-2} -3 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{ -z^6 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -3 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{ 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+6 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +5 a^2 z^6-24 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -7 z^6+3 a^3 z^5-9 a z^5-17 z^5 a^{-1} -14 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-6 a^2 z^4+21 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} +3 z^4-3 a^3 z^3+7 a z^3+16 z^3 a^{-1} +13 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+3 a^2 z^2-8 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -2 z a^{-5} -a^2+ a^{-2} + a^{-4} }[/math]

Vassiliev invariants

V₂ and V₃:

(-2, -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{ -8 }[/math]

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

[math]\displaystyle{ \frac{212}{3} }[/math]

[math]\displaystyle{ \frac{148}{3} }[/math]

[math]\displaystyle{ 64 }[/math]

[math]\displaystyle{ \frac{496}{3} }[/math]

[math]\displaystyle{ \frac{64}{3} }[/math]

[math]\displaystyle{ 88 }[/math]

[math]\displaystyle{ -\frac{256}{3} }[/math]

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

[math]\displaystyle{ -\frac{1696}{3} }[/math]

[math]\displaystyle{ -\frac{1184}{3} }[/math]

[math]\displaystyle{ -\frac{5431}{15} }[/math]

[math]\displaystyle{ \frac{5044}{15} }[/math]

[math]\displaystyle{ -\frac{37924}{45} }[/math]

[math]\displaystyle{ \frac{1831}{9} }[/math]

[math]\displaystyle{ -\frac{3511}{15} }[/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 10 102. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

2

-2

9

4

1

3

7

5

2

-3

5

6

4

2

3

6

5

-1

1

6

0

-1

4

7

3

-3

2

5

-3

-5

1

4

3

-7

2

-2

-9

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}^{2}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math]	[math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math]	[math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math]	[math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math]	[math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=3 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math]	[math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=4 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=5 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=6 }[/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, 102]]
Out[2]=	10
In[3]:=	PD[Knot[10, 102]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 10, 17, 9], X[10, 3, 11, 4], X[2, 15, 3, 16], X[14, 5, 15, 6], X[18, 8, 19, 7], X[4, 11, 5, 12], X[8, 18, 9, 17], X[20, 14, 1, 13], X[12, 20, 13, 19]]
In[4]:=	GaussCode[Knot[10, 102]]
Out[4]=	GaussCode[1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -10, 9, -5, 4, -2, 8, -6, 10, -9]
In[5]:=	BR[Knot[10, 102]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, 2, -1, 2, 2, 3, 3}]
In[6]:=	alex = Alexander[Knot[10, 102]][t]
Out[6]=	2 8 16 2 3 21 - -- + -- - -- - 16 t + 8 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 102]][z]
Out[7]=	2 4 6 1 - 2 z - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 102]}
In[9]:=	{KnotDet[Knot[10, 102]], KnotSignature[Knot[10, 102]]}
Out[9]=	{73, 0}
In[10]:=	J=Jones[Knot[10, 102]][q]
Out[10]=	-4 3 6 9 2 3 4 5 6 12 + q - -- + -- - - - 12 q + 11 q - 9 q + 6 q - 3 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 102]}
In[12]:=	A2Invariant[Knot[10, 102]][q]
Out[12]=	-12 -10 -8 -6 2 3 2 6 8 10 -1 + q - q + q + q - -- + -- + q - 2 q + 2 q - 2 q + 4 2 q q 12 14 16 18 q + q - q + q
In[13]:=	Kauffman[Knot[10, 102]][a, z]
Out[13]=	2 2 2 -4 -2 2 2 z 4 z 4 z 2 2 z 4 z 8 z a + a - a - --- - --- - --- - 2 a z + 2 z + ---- - ---- - ---- + 5 3 a 6 4 2 a a a a a 3 3 3 2 2 4 2 7 z 13 z 16 z 3 3 3 4 3 a z - a z + ---- + ----- + ----- + 7 a z - 3 a z + 3 z - 5 3 a a a 4 4 4 5 5 5 3 z 8 z 21 z 2 4 4 4 9 z 14 z 17 z ---- + ---- + ----- - 6 a z + a z - ---- - ----- - ----- - 6 4 2 5 3 a a a a a a 6 6 6 7 7 5 3 5 6 z 11 z 24 z 2 6 3 z z 9 a z + 3 a z - 7 z + -- - ----- - ----- + 5 a z + ---- + -- + 6 4 2 5 3 a a a a a 7 8 8 9 9 4 z 7 8 4 z 9 z 2 z 2 z ---- + 6 a z + 5 z + ---- + ---- + ---- + ---- a 4 2 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 102]], Vassiliev[3][Knot[10, 102]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[10, 102]][q, t]
Out[15]=	7 1 2 1 4 2 5 4 - + 6 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 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 7 4 9 4 6 q t + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + 9 5 11 5 13 6 q t + 2 q t + q t

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+<!-- -->
+<!--  -->
+<!-- -->
 <!-- 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=102|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-7,5,-1,6,-8,2,-3,7,-10,9,-5,4,-2,8,-6,10,-9/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=10|k=102|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-7,5,-1,6,-8,2,-3,7,-10,9,-5,4,-2,8,-6,10,-9/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>-7</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>2</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>-2</td></tr>
 <tr align=center><td>-9</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 144: / Line 136: @@
   q  t  + 2 q   t  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

10 102: Difference between revisions

Revision as of 20:16, 28 August 2005

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