8 3

8_2

8_4

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

Visit 8 3's page at Knotilus!

Visit 8 3's page at the original Knot Atlas!

8 3 Quick Notes

8 3 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_14,10,15,9 X_10,5,11,6 X_12,3,13,4 X_4,11,5,12 X_2,13,3,14 X_16,8,1,7 X_8,16,9,15
Gauss code	1, -6, 4, -5, 3, -1, 7, -8, 2, -3, 5, -4, 6, -2, 8, -7
Dowker-Thistlethwaite code	6 12 10 16 14 4 2 8
Conway Notation	[44]

Three dimensional invariants

Symmetry type	Fully amphicheiral
Unknotting number	2
3-genus	1
Bridge index	2
Super bridge index	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,6\}}
Nakanishi index	1
Maximal Thurston-Bennequin number	[-5][-5]
Hyperbolic Volume	5.23868
A-Polynomial	See Data:8 3/A-polynomial

[edit Notes for 8 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Topological 4 genus	$1$
Concordance genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Rasmussen s-Invariant	0

[edit Notes for 8 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -4 t+9-4 t^{-1} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-4 z^2}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q^{4}-q^{3}+2q^{2}-3q+3-3q^{-1}+2q^{-2}-q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^4-z^2 a^2-2 z^2-1-z^2 a^{-2} + a^{-4} }
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a z^7+z^7 a^{-1} +a^2 z^6+z^6 a^{-2} +2 z^6+a^3 z^5-4 a z^5-4 z^5 a^{-1} +z^5 a^{-3} +a^4 z^4-2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -6 z^4-2 a^3 z^3+8 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} -3 a^4 z^2+a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +8 z^2-4 a z-4 z a^{-1} +a^4+ a^{-4} -1}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{14}+q^{12}+q^8-q^4-1- q^{-4} + q^{-8} + q^{-12} + q^{-14} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}+q^{62}-q^{60}+q^{58}+q^{56}-q^{54}+2 q^{52}-q^{50}+2 q^{48}-q^{46}+q^{42}-2 q^{40}+4 q^{38}-3 q^{36}+q^{34}+q^{32}-2 q^{30}+3 q^{28}-2 q^{26}+q^{24}+2 q^{22}-2 q^{20}+q^{18}-2 q^{14}+4 q^{12}-4 q^{10}+q^8-3 q^4+3 q^2-5+3 q^{-2} -3 q^{-4} + q^{-8} -4 q^{-10} +4 q^{-12} -2 q^{-14} + q^{-18} -2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-26} +3 q^{-28} -2 q^{-30} + q^{-32} + q^{-34} -3 q^{-36} +4 q^{-38} -2 q^{-40} + q^{-42} - q^{-46} +2 q^{-48} - q^{-50} +2 q^{-52} - q^{-54} + q^{-56} + q^{-58} - q^{-60} + q^{-62} + q^{-66} }

Further Quantum Invariants

Further quantum knot invariants for 8_3.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^9+q^5-q^3- q^{-3} + q^{-5} + q^{-9} }
2	$q^{26}+q^{20}-q^{18}-2q^{16}+q^{14}-2q^{10}+2q^{8}+q^{6}-q^{4}+q^{2}+1+q^{-2}-q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-14}-2q^{-16}-q^{-18}+q^{-20}+q^{-26}$
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{51}-q^{41}-q^{39}+q^{35}-q^{33}-2 q^{31}+3 q^{27}+3 q^{25}-2 q^{23}-2 q^{21}+2 q^{19}+4 q^{17}-2 q^{15}-3 q^{13}+q^{11}+3 q^9-2 q^5-2 q^{-5} +3 q^{-9} + q^{-11} -3 q^{-13} -2 q^{-15} +4 q^{-17} +2 q^{-19} -2 q^{-21} -2 q^{-23} +3 q^{-25} +3 q^{-27} -2 q^{-31} - q^{-33} + q^{-35} - q^{-39} - q^{-41} + q^{-51} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}-q^{76}-q^{72}+q^{68}-q^{66}-2 q^{62}-q^{60}+3 q^{58}+2 q^{56}+3 q^{54}-2 q^{52}-3 q^{50}-q^{48}+q^{46}+7 q^{44}+2 q^{42}-3 q^{40}-6 q^{38}-5 q^{36}+6 q^{34}+6 q^{32}+q^{30}-6 q^{28}-9 q^{26}+4 q^{24}+6 q^{22}+3 q^{20}-3 q^{18}-7 q^{16}+q^{14}+3 q^{12}+3 q^{10}-2 q^6+q^4+q^2+1+ q^{-2} + q^{-4} -2 q^{-6} +3 q^{-10} +3 q^{-12} + q^{-14} -7 q^{-16} -3 q^{-18} +3 q^{-20} +6 q^{-22} +4 q^{-24} -9 q^{-26} -6 q^{-28} + q^{-30} +6 q^{-32} +6 q^{-34} -5 q^{-36} -6 q^{-38} -3 q^{-40} +2 q^{-42} +7 q^{-44} + q^{-46} - q^{-48} -3 q^{-50} -2 q^{-52} +3 q^{-54} +2 q^{-56} +3 q^{-58} - q^{-60} -2 q^{-62} - q^{-66} + q^{-68} - q^{-72} - q^{-76} + q^{-84} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{125}-q^{117}-q^{115}+q^{107}-2 q^{103}-q^{101}+q^{97}+3 q^{95}+3 q^{93}-3 q^{89}-3 q^{87}-2 q^{85}+3 q^{83}+5 q^{81}+5 q^{79}+q^{77}-5 q^{75}-8 q^{73}-6 q^{71}+8 q^{67}+10 q^{65}+5 q^{63}-6 q^{61}-14 q^{59}-11 q^{57}+12 q^{53}+16 q^{51}+7 q^{49}-11 q^{47}-18 q^{45}-10 q^{43}+6 q^{41}+18 q^{39}+16 q^{37}-3 q^{35}-15 q^{33}-12 q^{31}+11 q^{27}+12 q^{25}+2 q^{23}-7 q^{21}-8 q^{19}-2 q^{17}+4 q^{15}+4 q^{13}+2 q^{11}-q^9-3 q^7-3 q^{-7} - q^{-9} +2 q^{-11} +4 q^{-13} +4 q^{-15} -2 q^{-17} -8 q^{-19} -7 q^{-21} +2 q^{-23} +12 q^{-25} +11 q^{-27} -12 q^{-31} -15 q^{-33} -3 q^{-35} +16 q^{-37} +18 q^{-39} +6 q^{-41} -10 q^{-43} -18 q^{-45} -11 q^{-47} +7 q^{-49} +16 q^{-51} +12 q^{-53} -11 q^{-57} -14 q^{-59} -6 q^{-61} +5 q^{-63} +10 q^{-65} +8 q^{-67} -6 q^{-71} -8 q^{-73} -5 q^{-75} + q^{-77} +5 q^{-79} +5 q^{-81} +3 q^{-83} -2 q^{-85} -3 q^{-87} -3 q^{-89} +3 q^{-93} +3 q^{-95} + q^{-97} - q^{-101} -2 q^{-103} + q^{-107} - q^{-115} - q^{-117} + q^{-125} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{14}+q^{12}+q^8-q^4-1- q^{-4} + q^{-8} + q^{-12} + q^{-14} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}+2 q^{32}-2 q^{30}+4 q^{28}-2 q^{26}+4 q^{24}-6 q^{22}+5 q^{20}-6 q^{18}+4 q^{16}-6 q^{14}-q^{12}-6 q^8+8 q^6-8 q^4+16 q^2-6+16 q^{-2} -8 q^{-4} +8 q^{-6} -6 q^{-8} - q^{-12} -6 q^{-14} +4 q^{-16} -6 q^{-18} +5 q^{-20} -6 q^{-22} +4 q^{-24} -2 q^{-26} +4 q^{-28} -2 q^{-30} +2 q^{-32} + q^{-36} }
2,0	$q^{36}+q^{34}+q^{32}+q^{28}+q^{26}-q^{24}-3q^{22}-2q^{20}-2q^{14}+2q^{10}+q^{4}+2q^{2}+2+2q^{-2}+q^{-4}+2q^{-10}-2q^{-14}-2q^{-20}-3q^{-22}-q^{-24}+q^{-26}+q^{-28}+q^{-32}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}+q^{24}+q^{22}+q^{18}+2 q^{16}-2 q^{14}-q^{12}-3 q^8-q^6+q^4+2 q^2+2+2 q^{-2} + q^{-4} - q^{-6} -3 q^{-8} - q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} + q^{-22} + q^{-24} + q^{-28} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{19}+q^{17}+q^{15}+q^{11}-q^5-q- q^{-1} - q^{-5} + q^{-11} + q^{-15} + q^{-17} + q^{-19} }

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}+q^{24}-q^{22}+2 q^{20}-q^{18}+2 q^{16}+q^{12}-q^8+q^6-3 q^4+2 q^2-4+2 q^{-2} -3 q^{-4} + q^{-6} - q^{-8} + q^{-12} +2 q^{-16} - q^{-18} +2 q^{-20} - q^{-22} + q^{-24} + q^{-28} }
1,0	$q^{46}+q^{38}+q^{36}-q^{32}+q^{28}+2q^{26}-q^{24}-2q^{22}-q^{20}+q^{18}-2q^{14}-q^{12}+q^{8}+q^{2}+3+q^{-2}+q^{-8}-q^{-12}-2q^{-14}+q^{-18}-q^{-20}-2q^{-22}-q^{-24}+2q^{-26}+q^{-28}-q^{-32}+q^{-36}+q^{-38}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}+q^{62}-q^{60}+q^{58}+q^{56}-q^{54}+2 q^{52}-q^{50}+2 q^{48}-q^{46}+q^{42}-2 q^{40}+4 q^{38}-3 q^{36}+q^{34}+q^{32}-2 q^{30}+3 q^{28}-2 q^{26}+q^{24}+2 q^{22}-2 q^{20}+q^{18}-2 q^{14}+4 q^{12}-4 q^{10}+q^8-3 q^4+3 q^2-5+3 q^{-2} -3 q^{-4} + q^{-8} -4 q^{-10} +4 q^{-12} -2 q^{-14} + q^{-18} -2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-26} +3 q^{-28} -2 q^{-30} + q^{-32} + q^{-34} -3 q^{-36} +4 q^{-38} -2 q^{-40} + q^{-42} - q^{-46} +2 q^{-48} - q^{-50} +2 q^{-52} - q^{-54} + q^{-56} + q^{-58} - q^{-60} + q^{-62} + q^{-66} }

.

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 3"];

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

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

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -4 t+9-4 t^{-1} }

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

Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-4 z^2}

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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 17, 0 }

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

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

Out[8]= $q^{4}-q^{3}+2q^{2}-3q+3-3q^{-1}+2q^{-2}-q^{-3}+q^{-4}$

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

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

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^4-z^2 a^2-2 z^2-1-z^2 a^{-2} + a^{-4} }

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

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

Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a z^7+z^7 a^{-1} +a^2 z^6+z^6 a^{-2} +2 z^6+a^3 z^5-4 a z^5-4 z^5 a^{-1} +z^5 a^{-3} +a^4 z^4-2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -6 z^4-2 a^3 z^3+8 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} -3 a^4 z^2+a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +8 z^2-4 a z-4 z a^{-1} +a^4+ a^{-4} -1}

Vassiliev invariants

V₂ and V₃:

(-4, 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -16}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 128}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{520}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{200}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

-{\frac {2048}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{8320}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{3200}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{37502}{15}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{6728}{15}}

-{\frac {96968}{45}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2174}{9}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{7262}{15}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over $r$ ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of 8 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

0

5

2

1

3

1

-1

1

2

0

-1

2

0

-3

1

-1

-5

1

2

1

-7

0

-9

1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[8, 3]]
Out[2]=	8
In[3]:=	PD[Knot[8, 3]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 10, 15, 9], X[10, 5, 11, 6], X[12, 3, 13, 4], X[4, 11, 5, 12], X[2, 13, 3, 14], X[16, 8, 1, 7], X[8, 16, 9, 15]]
In[4]:=	GaussCode[Knot[8, 3]]
Out[4]=	GaussCode[1, -6, 4, -5, 3, -1, 7, -8, 2, -3, 5, -4, 6, -2, 8, -7]
In[5]:=	BR[Knot[8, 3]]
Out[5]=	BR[5, {-1, -1, -2, 1, 3, -2, 3, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[8, 3]][t]
Out[6]=	4 9 - - - 4 t t
In[7]:=	Conway[Knot[8, 3]][z]
Out[7]=	2 1 - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 3], Knot[10, 1]}
In[9]:=	{KnotDet[Knot[8, 3]], KnotSignature[Knot[8, 3]]}
Out[9]=	{17, 0}
In[10]:=	J=Jones[Knot[8, 3]][q]
Out[10]=	-4 -3 2 3 2 3 4 3 + q - q + -- - - - 3 q + 2 q - q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 3]}
In[12]:=	A2Invariant[Knot[8, 3]][q]
Out[12]=	-14 -12 -8 -4 4 8 12 14 -1 + q + q + q - q - q + q + q + q
In[13]:=	Kauffman[Knot[8, 3]][a, z]
Out[13]=	2 2 -4 4 4 z 2 3 z z 2 2 4 2 -1 + a + a - --- - 4 a z + 8 z - ---- + -- + a z - 3 a z - a 4 2 a a 3 3 4 4 2 z 8 z 3 3 3 4 z 2 z 2 4 4 4 ---- + ---- + 8 a z - 2 a z - 6 z + -- - ---- - 2 a z + a z + 3 a 4 2 a a a 5 5 6 7 z 4 z 5 3 5 6 z 2 6 z 7 -- - ---- - 4 a z + a z + 2 z + -- + a z + -- + a z 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[8, 3]], Vassiliev[3][Knot[8, 3]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[8, 3]][q, t]
Out[15]=	2 1 1 2 1 2 3 5 2 - + 2 q + ----- + ----- + ----- + ---- + --- + 2 q t + q t + 2 q t + q 9 4 5 3 5 2 3 q t q t q t q t q t 5 3 9 4 q t + q t

8 3

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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