8 11

8_10

8_12

Visit 8 11's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 11's page at Knotilus!

Visit 8 11's page at the original Knot Atlas!

8 11 Quick Notes

8 11 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_5,12,6,13 X_3,11,4,10 X_11,3,12,2 X_9,16,10,1 X_15,6,16,7 X_7,14,8,15 X_13,8,14,9
Gauss code	-1, 4, -3, 1, -2, 6, -7, 8, -5, 3, -4, 2, -8, 7, -6, 5
Dowker-Thistlethwaite code	4 10 12 14 16 2 8 6
Conway Notation	[3212]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	2
Bridge index	2
Super bridge index	[math]\displaystyle{ \{4,5\} }[/math]
Nakanishi index	1
Maximal Thurston-Bennequin number	[-9][-1]
Hyperbolic Volume	8.28632
A-Polynomial	See Data:8 11/A-polynomial

[edit Notes for 8 11'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{ 1 }[/math]
Rasmussen s-Invariant	-2

[edit Notes for 8 11's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+7 t-9+7 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 27, -2 }
Jones polynomial	[math]\displaystyle{ q-2+4 q^{-1} -4 q^{-2} +5 q^{-3} -5 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6+a^6-z^4 a^4-2 z^2 a^4-2 a^4-z^4 a^2-z^2 a^2+a^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8-2 z^2 a^8+2 z^5 a^7-4 z^3 a^7+2 z a^7+2 z^6 a^6-3 z^4 a^6+2 z^2 a^6-a^6+z^7 a^5+z^5 a^5-3 z^3 a^5+3 z a^5+4 z^6 a^4-7 z^4 a^4+6 z^2 a^4-2 a^4+z^7 a^3+z^5 a^3-2 z^3 a^3+z a^3+2 z^6 a^2-2 z^4 a^2-a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}+q^{16}-2 q^{14}-q^{12}-q^{10}+2 q^6+2 q^2+ q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-q^{104}-3 q^{102}+7 q^{100}-8 q^{98}+8 q^{96}-6 q^{94}+2 q^{92}+7 q^{90}-13 q^{88}+16 q^{86}-13 q^{84}+7 q^{82}+3 q^{80}-10 q^{78}+13 q^{76}-9 q^{74}+8 q^{72}+4 q^{70}-10 q^{68}+7 q^{66}-2 q^{64}-8 q^{62}+14 q^{60}-18 q^{58}+10 q^{56}-3 q^{54}-9 q^{52}+18 q^{50}-25 q^{48}+19 q^{46}-14 q^{44}-q^{42}+11 q^{40}-17 q^{38}+16 q^{36}-9 q^{34}+4 q^{32}+6 q^{30}-9 q^{28}+7 q^{26}-7 q^{22}+14 q^{20}-11 q^{18}+5 q^{16}+6 q^{14}-11 q^{12}+17 q^{10}-14 q^8+9 q^6-2 q^4-7 q^2+10-9 q^{-2} +8 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_11.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-q^{13}+q^{11}-2 q^9+q^5+2 q- q^{-1} + q^{-3} }[/math]
2	[math]\displaystyle{ q^{42}-q^{40}-q^{38}+3 q^{36}-2 q^{34}-4 q^{32}+5 q^{30}-5 q^{26}+6 q^{24}+2 q^{22}-4 q^{20}+q^{16}-4 q^{12}+2 q^{10}+4 q^8-5 q^6+q^4+6 q^2-4- q^{-2} +4 q^{-4} - q^{-6} - q^{-8} + q^{-10} }[/math]
3	[math]\displaystyle{ q^{81}-q^{79}-q^{77}+q^{75}+2 q^{73}-2 q^{71}-5 q^{69}+2 q^{67}+8 q^{65}-11 q^{61}-2 q^{59}+14 q^{57}+8 q^{55}-15 q^{53}-11 q^{51}+12 q^{49}+14 q^{47}-13 q^{45}-15 q^{43}+8 q^{41}+13 q^{39}-q^{37}-11 q^{35}+6 q^{31}+6 q^{29}-4 q^{27}-9 q^{25}-q^{23}+14 q^{21}+3 q^{19}-16 q^{17}-6 q^{15}+16 q^{13}+10 q^{11}-14 q^9-12 q^7+11 q^5+14 q^3-7 q-11 q^{-1} +2 q^{-3} +10 q^{-5} + q^{-7} -6 q^{-9} - q^{-11} +3 q^{-13} + q^{-15} - q^{-17} - q^{-19} + q^{-21} }[/math]
4	[math]\displaystyle{ q^{132}-q^{130}-q^{128}+q^{126}+2 q^{122}-4 q^{120}-3 q^{118}+4 q^{116}+3 q^{114}+8 q^{112}-8 q^{110}-13 q^{108}+q^{106}+9 q^{104}+25 q^{102}-5 q^{100}-28 q^{98}-20 q^{96}+3 q^{94}+49 q^{92}+19 q^{90}-28 q^{88}-47 q^{86}-23 q^{84}+56 q^{82}+47 q^{80}-7 q^{78}-54 q^{76}-47 q^{74}+43 q^{72}+53 q^{70}+12 q^{68}-39 q^{66}-49 q^{64}+15 q^{62}+36 q^{60}+22 q^{58}-16 q^{56}-33 q^{54}-7 q^{52}+17 q^{50}+24 q^{48}+6 q^{46}-15 q^{44}-31 q^{42}-5 q^{40}+30 q^{38}+29 q^{36}-2 q^{34}-52 q^{32}-25 q^{30}+32 q^{28}+51 q^{26}+18 q^{24}-60 q^{22}-43 q^{20}+16 q^{18}+54 q^{16}+40 q^{14}-43 q^{12}-48 q^{10}-11 q^8+35 q^6+49 q^4-13 q^2-29-23 q^{-2} +6 q^{-4} +32 q^{-6} +5 q^{-8} -6 q^{-10} -16 q^{-12} -6 q^{-14} +12 q^{-16} +3 q^{-18} +2 q^{-20} -4 q^{-22} -3 q^{-24} +3 q^{-26} + q^{-30} - q^{-32} - q^{-34} + q^{-36} }[/math]
5	[math]\displaystyle{ q^{195}-q^{193}-q^{191}+q^{189}-2 q^{181}-2 q^{179}+4 q^{177}+6 q^{175}+q^{173}-4 q^{171}-9 q^{169}-8 q^{167}+4 q^{165}+18 q^{163}+18 q^{161}-q^{159}-24 q^{157}-35 q^{155}-16 q^{153}+25 q^{151}+58 q^{149}+43 q^{147}-15 q^{145}-74 q^{143}-84 q^{141}-17 q^{139}+82 q^{137}+126 q^{135}+65 q^{133}-65 q^{131}-161 q^{129}-120 q^{127}+33 q^{125}+172 q^{123}+175 q^{121}+18 q^{119}-170 q^{117}-209 q^{115}-64 q^{113}+138 q^{111}+220 q^{109}+109 q^{107}-107 q^{105}-212 q^{103}-124 q^{101}+62 q^{99}+182 q^{97}+130 q^{95}-23 q^{93}-146 q^{91}-124 q^{89}+3 q^{87}+103 q^{85}+101 q^{83}+25 q^{81}-67 q^{79}-93 q^{77}-33 q^{75}+37 q^{73}+71 q^{71}+61 q^{69}-5 q^{67}-69 q^{65}-76 q^{63}-24 q^{61}+61 q^{59}+107 q^{57}+56 q^{55}-63 q^{53}-136 q^{51}-92 q^{49}+49 q^{47}+164 q^{45}+130 q^{43}-34 q^{41}-179 q^{39}-170 q^{37}+5 q^{35}+181 q^{33}+199 q^{31}+37 q^{29}-161 q^{27}-211 q^{25}-79 q^{23}+118 q^{21}+208 q^{19}+115 q^{17}-67 q^{15}-176 q^{13}-131 q^{11}+9 q^9+131 q^7+134 q^5+28 q^3-77 q-106 q^{-1} -52 q^{-3} +28 q^{-5} +76 q^{-7} +55 q^{-9} - q^{-11} -40 q^{-13} -43 q^{-15} -13 q^{-17} +16 q^{-19} +27 q^{-21} +16 q^{-23} -5 q^{-25} -13 q^{-27} -9 q^{-29} +3 q^{-33} +5 q^{-35} +2 q^{-37} -3 q^{-39} - q^{-41} + q^{-43} + q^{-49} - q^{-51} - q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}+q^{16}-2 q^{14}-q^{12}-q^{10}+2 q^6+2 q^2+ q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{60}-2 q^{58}+4 q^{56}-8 q^{54}+15 q^{52}-20 q^{50}+26 q^{48}-38 q^{46}+43 q^{44}-44 q^{42}+40 q^{40}-34 q^{38}+20 q^{36}+4 q^{34}-26 q^{32}+50 q^{30}-66 q^{28}+84 q^{26}-88 q^{24}+86 q^{22}-79 q^{20}+56 q^{18}-38 q^{16}+14 q^{14}+6 q^{12}-26 q^{10}+42 q^8-44 q^6+44 q^4-40 q^2+36-22 q^{-2} +17 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{56}+q^{48}-q^{46}-4 q^{44}-q^{42}+2 q^{40}-3 q^{36}+2 q^{34}+5 q^{32}+3 q^{30}-q^{28}+2 q^{26}-4 q^{22}-q^{20}-q^{18}-2 q^{16}-q^{14}+3 q^{12}-2 q^8+q^6+4 q^4-q^2-2+3 q^{-2} +3 q^{-4} - q^{-8} + q^{-12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{48}-q^{46}+q^{42}-4 q^{40}+4 q^{36}-3 q^{34}+2 q^{32}+7 q^{30}-2 q^{28}+q^{24}-3 q^{22}-4 q^{20}-3 q^{18}+q^{16}-q^{14}-2 q^{12}+5 q^{10}+2 q^8-4 q^6+5 q^4+2 q^2-2+3 q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
1,0,0	[math]\displaystyle{ q^{29}+q^{25}+q^{21}-2 q^{19}-q^{17}-2 q^{15}-q^{13}+q^9+2 q^7+2 q^3+ q^{-1} + q^{-5} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{62}-q^{58}+q^{56}+q^{54}-3 q^{52}-3 q^{50}+q^{48}+q^{46}-3 q^{44}+7 q^{40}+4 q^{38}+q^{36}+5 q^{34}+4 q^{32}-4 q^{30}-4 q^{28}-3 q^{26}-6 q^{24}-7 q^{22}-q^{20}+2 q^{18}-3 q^{16}+q^{14}+6 q^{12}+2 q^{10}-2 q^8+3 q^6+4 q^4+q^2+2 q^{-2} +2 q^{-4} + q^{-10} }[/math]
1,0,0,0	[math]\displaystyle{ q^{36}+q^{32}+q^{30}+q^{26}-2 q^{24}-q^{22}-2 q^{20}-2 q^{18}-q^{16}+q^{12}+q^{10}+2 q^8+2 q^4+1+ q^{-2} + q^{-6} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{48}-q^{46}+2 q^{44}-3 q^{42}+4 q^{40}-4 q^{38}+4 q^{36}-3 q^{34}+2 q^{32}-q^{30}-2 q^{28}+4 q^{26}-7 q^{24}+7 q^{22}-8 q^{20}+7 q^{18}-7 q^{16}+5 q^{14}-2 q^{12}+q^{10}+2 q^8-2 q^6+5 q^4-4 q^2+4-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }[/math]
1,0	[math]\displaystyle{ q^{78}-q^{74}-q^{72}+q^{70}+2 q^{68}-q^{66}-4 q^{64}-2 q^{62}+3 q^{60}+4 q^{58}-q^{56}-4 q^{54}+5 q^{50}+4 q^{48}-2 q^{46}-2 q^{44}+2 q^{42}+3 q^{40}-2 q^{38}-4 q^{36}-q^{34}+2 q^{32}-q^{30}-4 q^{28}-2 q^{26}+2 q^{24}+2 q^{22}-2 q^{20}-2 q^{18}+2 q^{16}+5 q^{14}-4 q^{10}-2 q^8+5 q^6+4 q^4-q^2-3+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{66}-q^{64}+q^{62}-2 q^{60}+2 q^{58}-4 q^{56}+2 q^{54}-3 q^{52}+4 q^{50}-2 q^{48}+3 q^{46}+q^{44}+4 q^{42}+3 q^{40}-2 q^{38}+3 q^{36}-5 q^{34}+4 q^{32}-9 q^{30}+2 q^{28}-9 q^{26}+4 q^{24}-5 q^{22}+3 q^{20}-3 q^{18}+3 q^{16}+2 q^{14}+q^{12}+2 q^{10}-2 q^8+5 q^6-2 q^4+4 q^2-2+4 q^{-2} - q^{-4} +2 q^{-6} - q^{-8} + q^{-10} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-q^{104}-3 q^{102}+7 q^{100}-8 q^{98}+8 q^{96}-6 q^{94}+2 q^{92}+7 q^{90}-13 q^{88}+16 q^{86}-13 q^{84}+7 q^{82}+3 q^{80}-10 q^{78}+13 q^{76}-9 q^{74}+8 q^{72}+4 q^{70}-10 q^{68}+7 q^{66}-2 q^{64}-8 q^{62}+14 q^{60}-18 q^{58}+10 q^{56}-3 q^{54}-9 q^{52}+18 q^{50}-25 q^{48}+19 q^{46}-14 q^{44}-q^{42}+11 q^{40}-17 q^{38}+16 q^{36}-9 q^{34}+4 q^{32}+6 q^{30}-9 q^{28}+7 q^{26}-7 q^{22}+14 q^{20}-11 q^{18}+5 q^{16}+6 q^{14}-11 q^{12}+17 q^{10}-14 q^8+9 q^6-2 q^4-7 q^2+10-9 q^{-2} +8 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/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["8 11"];

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

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

Out[4]= [math]\displaystyle{ -2 t^2+7 t-9+7 t^{-1} -2 t^{-2} }[/math]

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

Out[5]= [math]\displaystyle{ -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]= { 27, -2 }

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

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

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

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

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

Out[9]= [math]\displaystyle{ z^2 a^6+a^6-z^4 a^4-2 z^2 a^4-2 a^4-z^4 a^2-z^2 a^2+a^2+z^2+1 }[/math]

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

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

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

Vassiliev invariants

V₂ and V₃:

(-1, 2)

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{ 16 }[/math]

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

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

[math]\displaystyle{ \frac{62}{3} }[/math]

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

[math]\displaystyle{ -\frac{416}{3} }[/math]

[math]\displaystyle{ -\frac{128}{3} }[/math]

[math]\displaystyle{ -80 }[/math]

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

[math]\displaystyle{ 128 }[/math]

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

[math]\displaystyle{ -\frac{248}{3} }[/math]

[math]\displaystyle{ \frac{14849}{30} }[/math]

[math]\displaystyle{ -\frac{578}{15} }[/math]

[math]\displaystyle{ \frac{11698}{45} }[/math]

[math]\displaystyle{ \frac{1567}{18} }[/math]

[math]\displaystyle{ -\frac{31}{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]-2 is the signature of 8 11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

-1

3

1

2

-3

2

0

-5

3

2

1

-7

2

0

-9

1

3

-2

-11

1

2

1

-13

1

-1

-15

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[8, 11]]
Out[2]=	8
In[3]:=	PD[Knot[8, 11]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[9, 16, 10, 1], X[15, 6, 16, 7], X[7, 14, 8, 15], X[13, 8, 14, 9]]
In[4]:=	GaussCode[Knot[8, 11]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 6, -7, 8, -5, 3, -4, 2, -8, 7, -6, 5]
In[5]:=	BR[Knot[8, 11]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[8, 11]][t]
Out[6]=	2 7 2 -9 - -- + - + 7 t - 2 t 2 t t
In[7]:=	Conway[Knot[8, 11]][z]
Out[7]=	2 4 1 - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 11], Knot[10, 147], Knot[11, NonAlternating, 122]}
In[9]:=	{KnotDet[Knot[8, 11]], KnotSignature[Knot[8, 11]]}
Out[9]=	{27, -2}
In[10]:=	J=Jones[Knot[8, 11]][q]
Out[10]=	-7 2 3 5 5 4 4 -2 + q - -- + -- - -- + -- - -- + - + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 11]}
In[12]:=	A2Invariant[Knot[8, 11]][q]
Out[12]=	-22 -16 2 -12 -10 2 2 4 q + q - --- - q - q + -- + -- + q 14 6 2 q q q
In[13]:=	Kauffman[Knot[8, 11]][a, z]
Out[13]=	2 4 6 3 5 7 2 4 2 1 - a - 2 a - a + a z + 3 a z + 2 a z - 2 z + 6 a z + 6 2 8 2 3 3 3 5 3 7 3 4 2 a z - 2 a z - 3 a z - 2 a z - 3 a z - 4 a z + z - 2 4 4 4 6 4 8 4 5 3 5 5 5 2 a z - 7 a z - 3 a z + a z + 2 a z + a z + a z + 7 5 2 6 4 6 6 6 3 7 5 7 2 a z + 2 a z + 4 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 11]], Vassiliev[3][Knot[8, 11]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[8, 11]][q, t]
Out[15]=	2 3 1 1 1 2 1 3 2 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 2 3 2 2 t 3 2 ----- + ----- + ---- + ---- + - + q t + q t 7 2 5 2 5 3 q q t q t q t q t

8 11

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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