10 88

10_87

10_89

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10 88 Quick Notes

Square quasi-symmetrical depiction.

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_20,14,1,13 X₈₃₉₄ X_2,9,3,10 X_14,7,15,8 X_18,15,19,16 X_12,6,13,5 X_10,18,11,17 X_16,12,17,11 X_6,19,7,20
Gauss code	1, -4, 3, -1, 7, -10, 5, -3, 4, -8, 9, -7, 2, -5, 6, -9, 8, -6, 10, -2
Dowker-Thistlethwaite code	4 8 12 14 2 16 20 18 10 6
Conway Notation	[.21.21]

Three dimensional invariants

Symmetry type	Negative amphicheiral
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-6]
Hyperbolic Volume	15.6466
A-Polynomial	See Data:10 88/A-polynomial

[edit Notes for 10 88'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 88's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+8 t^2-24 t+35-24 t^{-1} +8 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 101, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-8 q^3+13 q^2-16 q+17-16 q^{-1} +13 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -2 z^4-a^4 z^2+2 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} -3 z^2+a^2+ a^{-2} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+6 z^8 a^{-2} +12 z^8+7 a^3 z^7+14 a z^7+14 z^7 a^{-1} +7 z^7 a^{-3} +4 a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} +4 z^6 a^{-4} -12 z^6+a^5 z^5-11 a^3 z^5-32 a z^5-32 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-10 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -8 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+19 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +3 a^4 z^2+7 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} +8 z^2-a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^2- a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-3 q^{10}+3 q^8-2 q^4+3 q^2-3+3 q^{-2} -2 q^{-4} +3 q^{-8} -3 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-15 q^{70}+4 q^{68}+21 q^{66}-53 q^{64}+91 q^{62}-111 q^{60}+94 q^{58}-33 q^{56}-75 q^{54}+204 q^{52}-299 q^{50}+314 q^{48}-218 q^{46}+18 q^{44}+223 q^{42}-417 q^{40}+487 q^{38}-379 q^{36}+134 q^{34}+154 q^{32}-373 q^{30}+418 q^{28}-277 q^{26}+21 q^{24}+235 q^{22}-365 q^{20}+303 q^{18}-66 q^{16}-243 q^{14}+490 q^{12}-562 q^{10}+415 q^8-96 q^6-286 q^4+588 q^2-699+588 q^{-2} -286 q^{-4} -96 q^{-6} +415 q^{-8} -562 q^{-10} +490 q^{-12} -243 q^{-14} -66 q^{-16} +303 q^{-18} -365 q^{-20} +235 q^{-22} +21 q^{-24} -277 q^{-26} +418 q^{-28} -373 q^{-30} +154 q^{-32} +134 q^{-34} -379 q^{-36} +487 q^{-38} -417 q^{-40} +223 q^{-42} +18 q^{-44} -218 q^{-46} +314 q^{-48} -299 q^{-50} +204 q^{-52} -75 q^{-54} -33 q^{-56} +94 q^{-58} -111 q^{-60} +91 q^{-62} -53 q^{-64} +21 q^{-66} +4 q^{-68} -15 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_88.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+3 q^9-4 q^7+5 q^5-3 q^3+q+ q^{-1} -3 q^{-3} +5 q^{-5} -4 q^{-7} +3 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-3 q^{30}+11 q^{26}-14 q^{24}-10 q^{22}+37 q^{20}-18 q^{18}-37 q^{16}+54 q^{14}-53 q^{10}+36 q^8+21 q^6-37 q^4-q^2+29- q^{-2} -37 q^{-4} +21 q^{-6} +36 q^{-8} -53 q^{-10} +54 q^{-14} -37 q^{-16} -18 q^{-18} +37 q^{-20} -10 q^{-22} -14 q^{-24} +11 q^{-26} -3 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+3 q^{61}-7 q^{57}-2 q^{55}+18 q^{53}+13 q^{51}-44 q^{49}-36 q^{47}+70 q^{45}+93 q^{43}-88 q^{41}-184 q^{39}+77 q^{37}+291 q^{35}-15 q^{33}-384 q^{31}-101 q^{29}+441 q^{27}+232 q^{25}-426 q^{23}-355 q^{21}+345 q^{19}+441 q^{17}-229 q^{15}-461 q^{13}+93 q^{11}+432 q^9+41 q^7-362 q^5-164 q^3+271 q+271 q^{-1} -164 q^{-3} -362 q^{-5} +41 q^{-7} +432 q^{-9} +93 q^{-11} -461 q^{-13} -229 q^{-15} +441 q^{-17} +345 q^{-19} -355 q^{-21} -426 q^{-23} +232 q^{-25} +441 q^{-27} -101 q^{-29} -384 q^{-31} -15 q^{-33} +291 q^{-35} +77 q^{-37} -184 q^{-39} -88 q^{-41} +93 q^{-43} +70 q^{-45} -36 q^{-47} -44 q^{-49} +13 q^{-51} +18 q^{-53} -2 q^{-55} -7 q^{-57} +3 q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-3 q^{102}+7 q^{98}-2 q^{96}-2 q^{94}-21 q^{92}+4 q^{90}+54 q^{88}+13 q^{86}-29 q^{84}-144 q^{82}-46 q^{80}+234 q^{78}+234 q^{76}+24 q^{74}-554 q^{72}-518 q^{70}+359 q^{68}+966 q^{66}+786 q^{64}-922 q^{62}-1844 q^{60}-517 q^{58}+1658 q^{56}+2747 q^{54}+35 q^{52}-3124 q^{50}-2894 q^{48}+724 q^{46}+4612 q^{44}+2698 q^{42}-2489 q^{40}-5095 q^{38}-2009 q^{36}+4299 q^{34}+5074 q^{32}+92 q^{30}-5038 q^{28}-4396 q^{26}+1926 q^{24}+5220 q^{22}+2525 q^{20}-2979 q^{18}-4789 q^{16}-623 q^{14}+3557 q^{12}+3551 q^{10}-574 q^8-3759 q^6-2380 q^4+1504 q^2+3699+1504 q^{-2} -2380 q^{-4} -3759 q^{-6} -574 q^{-8} +3551 q^{-10} +3557 q^{-12} -623 q^{-14} -4789 q^{-16} -2979 q^{-18} +2525 q^{-20} +5220 q^{-22} +1926 q^{-24} -4396 q^{-26} -5038 q^{-28} +92 q^{-30} +5074 q^{-32} +4299 q^{-34} -2009 q^{-36} -5095 q^{-38} -2489 q^{-40} +2698 q^{-42} +4612 q^{-44} +724 q^{-46} -2894 q^{-48} -3124 q^{-50} +35 q^{-52} +2747 q^{-54} +1658 q^{-56} -517 q^{-58} -1844 q^{-60} -922 q^{-62} +786 q^{-64} +966 q^{-66} +359 q^{-68} -518 q^{-70} -554 q^{-72} +24 q^{-74} +234 q^{-76} +234 q^{-78} -46 q^{-80} -144 q^{-82} -29 q^{-84} +13 q^{-86} +54 q^{-88} +4 q^{-90} -21 q^{-92} -2 q^{-94} -2 q^{-96} +7 q^{-98} -3 q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+3 q^{153}-7 q^{149}+2 q^{147}+6 q^{145}+5 q^{143}+4 q^{141}-14 q^{139}-41 q^{137}-10 q^{135}+70 q^{133}+105 q^{131}+43 q^{129}-135 q^{127}-298 q^{125}-225 q^{123}+222 q^{121}+732 q^{119}+682 q^{117}-159 q^{115}-1354 q^{113}-1787 q^{111}-496 q^{109}+2095 q^{107}+3755 q^{105}+2268 q^{103}-2201 q^{101}-6450 q^{99}-5989 q^{97}+635 q^{95}+9175 q^{93}+11773 q^{91}+3865 q^{89}-10226 q^{87}-18885 q^{85}-12188 q^{83}+7671 q^{81}+25415 q^{79}+23813 q^{77}-4 q^{75}-28588 q^{73}-36643 q^{71}-13038 q^{69}+26035 q^{67}+47593 q^{65}+29531 q^{63}-16857 q^{61}-53210 q^{59}-46175 q^{57}+1965 q^{55}+51702 q^{53}+59244 q^{51}+15546 q^{49}-43098 q^{47}-65778 q^{45}-32065 q^{43}+29307 q^{41}+65010 q^{39}+44420 q^{37}-13624 q^{35}-57872 q^{33}-50838 q^{31}-1066 q^{29}+46663 q^{27}+51692 q^{25}+12623 q^{23}-34147 q^{21}-48336 q^{19}-20550 q^{17}+22334 q^{15}+42967 q^{13}+25616 q^{11}-12308 q^9-37451 q^7-29220 q^5+3904 q^3+32846 q+32846 q^{-1} +3904 q^{-3} -29220 q^{-5} -37451 q^{-7} -12308 q^{-9} +25616 q^{-11} +42967 q^{-13} +22334 q^{-15} -20550 q^{-17} -48336 q^{-19} -34147 q^{-21} +12623 q^{-23} +51692 q^{-25} +46663 q^{-27} -1066 q^{-29} -50838 q^{-31} -57872 q^{-33} -13624 q^{-35} +44420 q^{-37} +65010 q^{-39} +29307 q^{-41} -32065 q^{-43} -65778 q^{-45} -43098 q^{-47} +15546 q^{-49} +59244 q^{-51} +51702 q^{-53} +1965 q^{-55} -46175 q^{-57} -53210 q^{-59} -16857 q^{-61} +29531 q^{-63} +47593 q^{-65} +26035 q^{-67} -13038 q^{-69} -36643 q^{-71} -28588 q^{-73} -4 q^{-75} +23813 q^{-77} +25415 q^{-79} +7671 q^{-81} -12188 q^{-83} -18885 q^{-85} -10226 q^{-87} +3865 q^{-89} +11773 q^{-91} +9175 q^{-93} +635 q^{-95} -5989 q^{-97} -6450 q^{-99} -2201 q^{-101} +2268 q^{-103} +3755 q^{-105} +2095 q^{-107} -496 q^{-109} -1787 q^{-111} -1354 q^{-113} -159 q^{-115} +682 q^{-117} +732 q^{-119} +222 q^{-121} -225 q^{-123} -298 q^{-125} -135 q^{-127} +43 q^{-129} +105 q^{-131} +70 q^{-133} -10 q^{-135} -41 q^{-137} -14 q^{-139} +4 q^{-141} +5 q^{-143} +6 q^{-145} +2 q^{-147} -7 q^{-149} +3 q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-3 q^{10}+3 q^8-2 q^4+3 q^2-3+3 q^{-2} -2 q^{-4} +3 q^{-8} -3 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }[/math]

2,0

[math]\displaystyle{ q^{42}-q^{40}-3 q^{38}+q^{36}+7 q^{34}+q^{32}-14 q^{30}-3 q^{28}+21 q^{26}+4 q^{24}-25 q^{22}-5 q^{20}+28 q^{18}+8 q^{16}-32 q^{14}+q^{12}+26 q^{10}-6 q^8-15 q^6+7 q^4+7 q^2-10+7 q^{-2} +7 q^{-4} -15 q^{-6} -6 q^{-8} +26 q^{-10} + q^{-12} -32 q^{-14} +8 q^{-16} +28 q^{-18} -5 q^{-20} -25 q^{-22} +4 q^{-24} +21 q^{-26} -3 q^{-28} -14 q^{-30} + q^{-32} +7 q^{-34} + q^{-36} -3 q^{-38} - q^{-40} + q^{-42} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-3 q^{32}+q^{30}+7 q^{28}-14 q^{26}+4 q^{24}+22 q^{22}-30 q^{20}+5 q^{18}+35 q^{16}-41 q^{14}+2 q^{12}+35 q^{10}-30 q^8-6 q^6+22 q^4-2 q^2-10-2 q^{-2} +22 q^{-4} -6 q^{-6} -30 q^{-8} +35 q^{-10} +2 q^{-12} -41 q^{-14} +35 q^{-16} +5 q^{-18} -30 q^{-20} +22 q^{-22} +4 q^{-24} -14 q^{-26} +7 q^{-28} + q^{-30} -3 q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}+q^{19}+2 q^{15}-3 q^{13}+4 q^{11}-2 q^9+2 q^7-2 q^5+2 q^3-q- q^{-1} +2 q^{-3} -2 q^{-5} +2 q^{-7} -2 q^{-9} +4 q^{-11} -3 q^{-13} +2 q^{-15} + q^{-19} - q^{-21} }[/math]

B2 Invariants.

Weight

Invariant

0,1

[math]\displaystyle{ -q^{34}+3 q^{32}-7 q^{30}+13 q^{28}-22 q^{26}+32 q^{24}-40 q^{22}+48 q^{20}-49 q^{18}+45 q^{16}-33 q^{14}+16 q^{12}+7 q^{10}-32 q^8+56 q^6-76 q^4+90 q^2-96+90 q^{-2} -76 q^{-4} +56 q^{-6} -32 q^{-8} +7 q^{-10} +16 q^{-12} -33 q^{-14} +45 q^{-16} -49 q^{-18} +48 q^{-20} -40 q^{-22} +32 q^{-24} -22 q^{-26} +13 q^{-28} -7 q^{-30} +3 q^{-32} - q^{-34} }[/math]

1,0

[math]\displaystyle{ q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+10 q^{46}-18 q^{42}-13 q^{40}+19 q^{38}+31 q^{36}-4 q^{34}-43 q^{32}-19 q^{30}+40 q^{28}+42 q^{26}-21 q^{24}-53 q^{22}-4 q^{20}+49 q^{18}+22 q^{16}-36 q^{14}-31 q^{12}+22 q^{10}+33 q^8-11 q^6-33 q^4+4 q^2+35+4 q^{-2} -33 q^{-4} -11 q^{-6} +33 q^{-8} +22 q^{-10} -31 q^{-12} -36 q^{-14} +22 q^{-16} +49 q^{-18} -4 q^{-20} -53 q^{-22} -21 q^{-24} +42 q^{-26} +40 q^{-28} -19 q^{-30} -43 q^{-32} -4 q^{-34} +31 q^{-36} +19 q^{-38} -13 q^{-40} -18 q^{-42} +10 q^{-46} +4 q^{-48} -3 q^{-50} -3 q^{-52} + q^{-56} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-15 q^{70}+4 q^{68}+21 q^{66}-53 q^{64}+91 q^{62}-111 q^{60}+94 q^{58}-33 q^{56}-75 q^{54}+204 q^{52}-299 q^{50}+314 q^{48}-218 q^{46}+18 q^{44}+223 q^{42}-417 q^{40}+487 q^{38}-379 q^{36}+134 q^{34}+154 q^{32}-373 q^{30}+418 q^{28}-277 q^{26}+21 q^{24}+235 q^{22}-365 q^{20}+303 q^{18}-66 q^{16}-243 q^{14}+490 q^{12}-562 q^{10}+415 q^8-96 q^6-286 q^4+588 q^2-699+588 q^{-2} -286 q^{-4} -96 q^{-6} +415 q^{-8} -562 q^{-10} +490 q^{-12} -243 q^{-14} -66 q^{-16} +303 q^{-18} -365 q^{-20} +235 q^{-22} +21 q^{-24} -277 q^{-26} +418 q^{-28} -373 q^{-30} +154 q^{-32} +134 q^{-34} -379 q^{-36} +487 q^{-38} -417 q^{-40} +223 q^{-42} +18 q^{-44} -218 q^{-46} +314 q^{-48} -299 q^{-50} +204 q^{-52} -75 q^{-54} -33 q^{-56} +94 q^{-58} -111 q^{-60} +91 q^{-62} -53 q^{-64} +21 q^{-66} +4 q^{-68} -15 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/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 88"];

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

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

Out[4]= [math]\displaystyle{ -t^3+8 t^2-24 t+35-24 t^{-1} +8 t^{-2} - t^{-3} }[/math]

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

Out[5]= [math]\displaystyle{ -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]= { 101, 0 }

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

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

Out[8]= [math]\displaystyle{ -q^5+4 q^4-8 q^3+13 q^2-16 q+17-16 q^{-1} +13 q^{-2} -8 q^{-3} +4 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+2 z^4 a^{-2} -2 z^4-a^4 z^2+2 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} -3 z^2+a^2+ a^{-2} -1 }[/math]

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

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

Out[10]= [math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+6 z^8 a^{-2} +12 z^8+7 a^3 z^7+14 a z^7+14 z^7 a^{-1} +7 z^7 a^{-3} +4 a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} +4 z^6 a^{-4} -12 z^6+a^5 z^5-11 a^3 z^5-32 a z^5-32 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-10 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -8 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+19 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +3 a^4 z^2+7 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} +8 z^2-a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^2- a^{-2} -1 }[/math]

Vassiliev invariants

V₂ and V₃:

(-1, 0)

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{ 0 }[/math]

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

[math]\displaystyle{ -\frac{14}{3} }[/math]

[math]\displaystyle{ -\frac{34}{3} }[/math]

[math]\displaystyle{ 0 }[/math]

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

[math]\displaystyle{ 0 }[/math]

[math]\displaystyle{ \frac{56}{3} }[/math]

[math]\displaystyle{ \frac{136}{3} }[/math]

[math]\displaystyle{ \frac{449}{30} }[/math]

[math]\displaystyle{ -\frac{326}{5} }[/math]

[math]\displaystyle{ \frac{4978}{45} }[/math]

[math]\displaystyle{ -\frac{257}{18} }[/math]

[math]\displaystyle{ \frac{449}{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 88. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

5

1

-4

5

8

3

5

3

8

5

-3

1

9

8

1

-1

8

9

1

-3

5

8

-3

-5

3

8

5

-7

1

5

-4

-9

3

-11

1

-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=-5 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math]	[math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math]	[math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math]	[math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=0 }[/math]	[math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math]	[math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math]	[math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math]	[math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=3 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math]	[math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=4 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=5 }[/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, 88]]
Out[2]=	10
In[3]:=	PD[Knot[10, 88]]
Out[3]=	PD[X[4, 2, 5, 1], X[20, 14, 1, 13], X[8, 3, 9, 4], X[2, 9, 3, 10], X[14, 7, 15, 8], X[18, 15, 19, 16], X[12, 6, 13, 5], X[10, 18, 11, 17], X[16, 12, 17, 11], X[6, 19, 7, 20]]
In[4]:=	GaussCode[Knot[10, 88]]
Out[4]=	GaussCode[1, -4, 3, -1, 7, -10, 5, -3, 4, -8, 9, -7, 2, -5, 6, -9, 8, -6, 10, -2]
In[5]:=	BR[Knot[10, 88]]
Out[5]=	BR[5, {-1, 2, -1, -3, 2, -3, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[10, 88]][t]
Out[6]=	-3 8 24 2 3 35 - t + -- - -- - 24 t + 8 t - t 2 t t
In[7]:=	Conway[Knot[10, 88]][z]
Out[7]=	2 4 6 1 - z + 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 88]}
In[9]:=	{KnotDet[Knot[10, 88]], KnotSignature[Knot[10, 88]]}
Out[9]=	{101, 0}
In[10]:=	J=Jones[Knot[10, 88]][q]
Out[10]=	-5 4 8 13 16 2 3 4 5 17 - q + -- - -- + -- - -- - 16 q + 13 q - 8 q + 4 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[10, 88]}
In[12]:=	A2Invariant[Knot[10, 88]][q]
Out[12]=	-16 -14 2 3 3 2 3 2 4 8 -3 - q + q + --- - --- + -- - -- + -- + 3 q - 2 q + 3 q - 12 10 8 4 2 q q q q q 10 12 14 16 3 q + 2 q + q - q
In[13]:=	Kauffman[Knot[10, 88]][a, z]
Out[13]=	2 2 -2 2 z 4 z 3 2 3 z 7 z -1 - a - a - -- - --- - 4 a z - a z + 8 z + ---- + ---- + 3 a 4 2 a a a 3 3 3 2 2 4 2 z 6 z 19 z 3 3 3 5 3 7 a z + 3 a z - -- + ---- + ----- + 19 a z + 6 a z - a z - 5 3 a a a 4 4 5 5 5 4 6 z 10 z 2 4 4 4 z 11 z 32 z 8 z - ---- - ----- - 10 a z - 6 a z + -- - ----- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 4 z 2 z 2 6 32 a z - 11 a z + a z - 12 z + ---- - ---- - 2 a z + 4 2 a a 7 7 8 4 6 7 z 14 z 7 3 7 8 6 z 2 8 4 a z + ---- + ----- + 14 a z + 7 a z + 12 z + ---- + 6 a z + 3 a 2 a a 9 2 z 9 ---- + 2 a z a
In[14]:=	{Vassiliev[2][Knot[10, 88]], Vassiliev[3][Knot[10, 88]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 88]][q, t]
Out[15]=	9 1 3 1 5 3 8 5 - + 9 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 8 8 3 3 2 5 2 5 3 7 3 ---- + --- + 8 q t + 8 q t + 5 q t + 8 q t + 3 q t + 5 q t + 3 q t q t 7 4 9 4 11 5 q t + 3 q t + q t

10 88

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