8 13

8_12

8_14

Visit 8 13's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 13's page at Knotilus! Visit 8 13's page at the original Knot Atlas! 8 13 Quick Notes

8 13 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X15,7,16,6 X7,15,8,14 X13,9,14,8 X9,2,10,3
Gauss code	-1, 8, -2, 1, -4, 5, -6, 7, -8, 2, -3, 4, -7, 6, -5, 3
Dowker-Thistlethwaite code	4 10 12 14 2 16 8 6
Conway Notation	[31112]

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 [-4][-6] Hyperbolic Volume 8.53123 A-Polynomial See Data:8 13/A-polynomial

[edit Notes for 8 13'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{ 2 }[/math] Rasmussen s-Invariant 0

[edit Notes for 8 13's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-7 t+11-7 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	{ 29, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+2 q^4-3 q^3+5 q^2-5 q+5-4 q^{-1} +3 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{-2} +z^4-a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +z^2+2 a^{-2} - a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +5 z^6 a^{-2} +2 z^6 a^{-4} +3 z^6+4 a z^5+4 z^5 a^{-1} +z^5 a^{-3} +z^5 a^{-5} +3 a^2 z^4-11 z^4 a^{-2} -6 z^4 a^{-4} -2 z^4+a^3 z^3-4 a z^3-9 z^3 a^{-1} -7 z^3 a^{-3} -3 z^3 a^{-5} -2 a^2 z^2+7 z^2 a^{-2} +5 z^2 a^{-4} +a z+3 z a^{-1} +4 z a^{-3} +2 z a^{-5} -2 a^{-2} - a^{-4} }[/math]
The A2 invariant	[math]\displaystyle{ -q^{10}+q^8+q^6-q^4+q^2-1+ q^{-2} + q^{-4} + q^{-6} +2 q^{-8} - q^{-10} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+9 q^{38}-11 q^{36}+12 q^{34}-8 q^{32}+6 q^{28}-13 q^{26}+18 q^{24}-16 q^{22}+10 q^{20}-8 q^{16}+14 q^{14}-10 q^{12}+4 q^{10}+2 q^8-10 q^6+9 q^4-3 q^2-7+17 q^{-2} -22 q^{-4} +19 q^{-6} -6 q^{-8} -9 q^{-10} +19 q^{-12} -26 q^{-14} +26 q^{-16} -14 q^{-18} +3 q^{-20} +11 q^{-22} -16 q^{-24} +21 q^{-26} -10 q^{-28} + q^{-30} +6 q^{-32} -9 q^{-34} +8 q^{-36} -6 q^{-40} +14 q^{-42} -15 q^{-44} +9 q^{-46} + q^{-48} -13 q^{-50} +18 q^{-52} -19 q^{-54} +11 q^{-56} -4 q^{-58} -6 q^{-60} +11 q^{-62} -13 q^{-64} +10 q^{-66} -4 q^{-68} - q^{-70} +2 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_13.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^7+2 q^5-q^3+q+2 q^{-5} - q^{-7} + q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{20}-2 q^{18}-q^{16}+5 q^{14}-4 q^{12}-3 q^{10}+7 q^8-3 q^6-3 q^4+6 q^2+1-2 q^{-2} +3 q^{-6} - q^{-8} -5 q^{-10} +4 q^{-12} +3 q^{-14} -6 q^{-16} +3 q^{-18} +4 q^{-20} -5 q^{-22} +3 q^{-26} -2 q^{-28} - q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{39}+2 q^{37}+q^{35}-3 q^{33}-2 q^{31}+4 q^{29}+6 q^{27}-9 q^{25}-8 q^{23}+10 q^{21}+11 q^{19}-12 q^{17}-15 q^{15}+12 q^{13}+19 q^{11}-8 q^9-18 q^7+5 q^5+16 q^3+q-11 q^{-1} -6 q^{-3} +7 q^{-5} +10 q^{-7} - q^{-9} -12 q^{-11} -3 q^{-13} +14 q^{-15} +8 q^{-17} -15 q^{-19} -11 q^{-21} +13 q^{-23} +13 q^{-25} -10 q^{-27} -17 q^{-29} +8 q^{-31} +17 q^{-33} -2 q^{-35} -17 q^{-37} - q^{-39} +14 q^{-41} +5 q^{-43} -10 q^{-45} -7 q^{-47} +5 q^{-49} +6 q^{-51} -2 q^{-53} -4 q^{-55} +2 q^{-59} + q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{64}-2 q^{62}-q^{60}+3 q^{58}+2 q^{54}-7 q^{52}-q^{50}+10 q^{48}+q^{46}+3 q^{44}-21 q^{42}-6 q^{40}+26 q^{38}+13 q^{36}-47 q^{32}-19 q^{30}+43 q^{28}+40 q^{26}+11 q^{24}-67 q^{22}-47 q^{20}+36 q^{18}+60 q^{16}+35 q^{14}-55 q^{12}-60 q^{10}+6 q^8+46 q^6+49 q^4-18 q^2-43-22 q^{-2} +11 q^{-4} +39 q^{-6} +19 q^{-8} -13 q^{-10} -39 q^{-12} -19 q^{-14} +24 q^{-16} +43 q^{-18} +9 q^{-20} -44 q^{-22} -35 q^{-24} +9 q^{-26} +59 q^{-28} +26 q^{-30} -46 q^{-32} -47 q^{-34} -7 q^{-36} +64 q^{-38} +41 q^{-40} -34 q^{-42} -52 q^{-44} -32 q^{-46} +51 q^{-48} +56 q^{-50} -4 q^{-52} -42 q^{-54} -54 q^{-56} +18 q^{-58} +50 q^{-60} +26 q^{-62} -12 q^{-64} -51 q^{-66} -12 q^{-68} +21 q^{-70} +29 q^{-72} +14 q^{-74} -25 q^{-76} -18 q^{-78} -4 q^{-80} +12 q^{-82} +16 q^{-84} -4 q^{-86} -6 q^{-88} -6 q^{-90} +6 q^{-94} + q^{-96} -2 q^{-100} - q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{95}+2 q^{93}+q^{91}-3 q^{89}+q^{83}+2 q^{81}-6 q^{77}-4 q^{75}+7 q^{73}+12 q^{71}+4 q^{69}-13 q^{67}-20 q^{65}-14 q^{63}+20 q^{61}+52 q^{59}+24 q^{57}-36 q^{55}-78 q^{53}-56 q^{51}+38 q^{49}+124 q^{47}+100 q^{45}-36 q^{43}-161 q^{41}-154 q^{39}+5 q^{37}+184 q^{35}+211 q^{33}+41 q^{31}-183 q^{29}-251 q^{27}-94 q^{25}+149 q^{23}+266 q^{21}+146 q^{19}-95 q^{17}-245 q^{15}-175 q^{13}+34 q^{11}+195 q^9+183 q^7+27 q^5-131 q^3-166 q-71 q^{-1} +63 q^{-3} +132 q^{-5} +100 q^{-7} + q^{-9} -98 q^{-11} -115 q^{-13} -48 q^{-15} +65 q^{-17} +128 q^{-19} +82 q^{-21} -39 q^{-23} -136 q^{-25} -111 q^{-27} +27 q^{-29} +148 q^{-31} +130 q^{-33} -18 q^{-35} -157 q^{-37} -154 q^{-39} +6 q^{-41} +171 q^{-43} +177 q^{-45} +11 q^{-47} -176 q^{-49} -200 q^{-51} -41 q^{-53} +166 q^{-55} +225 q^{-57} +77 q^{-59} -142 q^{-61} -231 q^{-63} -119 q^{-65} +90 q^{-67} +226 q^{-69} +164 q^{-71} -37 q^{-73} -191 q^{-75} -185 q^{-77} -36 q^{-79} +139 q^{-81} +190 q^{-83} +86 q^{-85} -73 q^{-87} -162 q^{-89} -121 q^{-91} +8 q^{-93} +117 q^{-95} +127 q^{-97} +42 q^{-99} -62 q^{-101} -107 q^{-103} -66 q^{-105} +13 q^{-107} +70 q^{-109} +69 q^{-111} +18 q^{-113} -35 q^{-115} -53 q^{-117} -29 q^{-119} +7 q^{-121} +30 q^{-123} +28 q^{-125} +6 q^{-127} -14 q^{-129} -17 q^{-131} -7 q^{-133} +2 q^{-135} +8 q^{-137} +8 q^{-139} -4 q^{-143} -3 q^{-145} - q^{-147} +2 q^{-151} + q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{10}+q^8+q^6-q^4+q^2-1+ q^{-2} + q^{-4} + q^{-6} +2 q^{-8} - q^{-10} - q^{-16} }[/math]
1,1	[math]\displaystyle{ q^{28}-4 q^{26}+8 q^{24}-12 q^{22}+20 q^{20}-32 q^{18}+38 q^{16}-44 q^{14}+51 q^{12}-52 q^{10}+44 q^8-30 q^6+13 q^4+12 q^2-36+66 q^{-2} -82 q^{-4} +100 q^{-6} -102 q^{-8} +98 q^{-10} -86 q^{-12} +66 q^{-14} -42 q^{-16} +12 q^{-18} +10 q^{-20} -28 q^{-22} +46 q^{-24} -54 q^{-26} +55 q^{-28} -48 q^{-30} +40 q^{-32} -32 q^{-34} +19 q^{-36} -12 q^{-38} +6 q^{-40} -2 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{26}-q^{24}-2 q^{22}+q^{20}+3 q^{18}-5 q^{14}+3 q^{10}-2 q^8-2 q^6+4 q^4+4 q^2+1+ q^{-2} +2 q^{-4} - q^{-6} -2 q^{-8} + q^{-10} - q^{-12} -3 q^{-14} +3 q^{-16} +4 q^{-18} - q^{-20} -2 q^{-22} +2 q^{-24} +2 q^{-26} -2 q^{-28} -3 q^{-30} + q^{-32} + q^{-34} - q^{-36} - q^{-38} + q^{-42} }[/math]

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{28}-q^{26}-2 q^{24}+q^{22}+2 q^{20}-q^{18}-2 q^{16}+3 q^{14}+3 q^{12}-3 q^{10}-3 q^8+4 q^6-4 q^2+1+2 q^{-2} -2 q^{-4} - q^{-6} +4 q^{-8} +3 q^{-10} + q^{-12} +7 q^{-14} +8 q^{-16} - q^{-20} +4 q^{-22} -2 q^{-24} -7 q^{-26} -4 q^{-28} -2 q^{-32} -3 q^{-34} + q^{-36} +2 q^{-38} + q^{-44} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{16}+q^{14}+q^8-q^6+q^4-q^2+ q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} +2 q^{-14} - q^{-16} - q^{-20} - q^{-22} - q^{-26} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{22}+2 q^{20}-3 q^{18}+4 q^{16}-5 q^{14}+5 q^{12}-4 q^{10}+3 q^8-q^6+4 q^2-6+8 q^{-2} -8 q^{-4} +9 q^{-6} -8 q^{-8} +7 q^{-10} -4 q^{-12} +2 q^{-14} + q^{-16} -2 q^{-18} +4 q^{-20} -4 q^{-22} +5 q^{-24} -4 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{36}-2 q^{32}-2 q^{30}+q^{28}+4 q^{26}+q^{24}-4 q^{22}-3 q^{20}+3 q^{18}+5 q^{16}-5 q^{12}-2 q^{10}+3 q^8+3 q^6-3 q^4-3 q^2+2+4 q^{-2} -3 q^{-6} +4 q^{-10} +2 q^{-12} -2 q^{-14} - q^{-16} +4 q^{-18} +3 q^{-20} -2 q^{-22} -4 q^{-24} +2 q^{-26} +5 q^{-28} + q^{-30} -5 q^{-32} -4 q^{-34} +2 q^{-36} +4 q^{-38} - q^{-40} -4 q^{-42} -2 q^{-44} +2 q^{-46} +2 q^{-48} - q^{-50} - q^{-52} + q^{-56} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{30}-2 q^{28}+q^{26}-2 q^{24}+4 q^{22}-4 q^{20}+3 q^{18}-3 q^{16}+5 q^{14}-2 q^{12}+q^{10}-2 q^8+2 q^4-4 q^2+3-6 q^{-2} +7 q^{-4} -5 q^{-6} +8 q^{-8} -5 q^{-10} +9 q^{-12} - q^{-14} +7 q^{-16} - q^{-18} +2 q^{-20} + q^{-22} -2 q^{-24} -5 q^{-28} +2 q^{-30} -5 q^{-32} +2 q^{-34} -4 q^{-36} +3 q^{-38} -2 q^{-40} +2 q^{-42} - q^{-44} + q^{-46} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+9 q^{38}-11 q^{36}+12 q^{34}-8 q^{32}+6 q^{28}-13 q^{26}+18 q^{24}-16 q^{22}+10 q^{20}-8 q^{16}+14 q^{14}-10 q^{12}+4 q^{10}+2 q^8-10 q^6+9 q^4-3 q^2-7+17 q^{-2} -22 q^{-4} +19 q^{-6} -6 q^{-8} -9 q^{-10} +19 q^{-12} -26 q^{-14} +26 q^{-16} -14 q^{-18} +3 q^{-20} +11 q^{-22} -16 q^{-24} +21 q^{-26} -10 q^{-28} + q^{-30} +6 q^{-32} -9 q^{-34} +8 q^{-36} -6 q^{-40} +14 q^{-42} -15 q^{-44} +9 q^{-46} + q^{-48} -13 q^{-50} +18 q^{-52} -19 q^{-54} +11 q^{-56} -4 q^{-58} -6 q^{-60} +11 q^{-62} -13 q^{-64} +10 q^{-66} -4 q^{-68} - q^{-70} +2 q^{-72} -4 q^{-74} +3 q^{-76} - 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 13"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-7 t+11-7 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]=

{ 29, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+2 q^4-3 q^3+5 q^2-5 q+5-4 q^{-1} +3 q^{-2} - q^{-3} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^4 a^{-2} +z^4-a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +z^2+2 a^{-2} - a^{-4} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +5 z^6 a^{-2} +2 z^6 a^{-4} +3 z^6+4 a z^5+4 z^5 a^{-1} +z^5 a^{-3} +z^5 a^{-5} +3 a^2 z^4-11 z^4 a^{-2} -6 z^4 a^{-4} -2 z^4+a^3 z^3-4 a z^3-9 z^3 a^{-1} -7 z^3 a^{-3} -3 z^3 a^{-5} -2 a^2 z^2+7 z^2 a^{-2} +5 z^2 a^{-4} +a z+3 z a^{-1} +4 z a^{-3} +2 z a^{-5} -2 a^{-2} - a^{-4} }[/math]

Vassiliev invariants

V2 and V3:

(1, 1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 4 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{14}{3} }[/math] [math]\displaystyle{ -\frac{38}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{80}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{56}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ \frac{1951}{30} }[/math] [math]\displaystyle{ \frac{526}{5} }[/math] [math]\displaystyle{ -\frac{6058}{45} }[/math] [math]\displaystyle{ -\frac{31}{18} }[/math] [math]\displaystyle{ -\frac{1409}{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 8 13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 χ 11                 1 -1 9               1   1 7             2 1   -1 5           3 1     2 3         2 2       0 1       3 3         0 -1     2 3           1 -3   1 2             -1 -5   2               2 -7 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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[8, 13]]
Out[2]=	8
In[3]:=	PD[Knot[8, 13]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 16], X[5, 13, 6, 12], X[15, 7, 16, 6], X[7, 15, 8, 14], X[13, 9, 14, 8], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[8, 13]]
Out[4]=	GaussCode[-1, 8, -2, 1, -4, 5, -6, 7, -8, 2, -3, 4, -7, 6, -5, 3]
In[5]:=	BR[Knot[8, 13]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[8, 13]][t]
Out[6]=	2 7 2 11 + -- - - - 7 t + 2 t 2 t t
In[7]:=	Conway[Knot[8, 13]][z]
Out[7]=	2 4 1 + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 13]}
In[9]:=	{KnotDet[Knot[8, 13]], KnotSignature[Knot[8, 13]]}
Out[9]=	{29, 0}
In[10]:=	J=Jones[Knot[8, 13]][q]
Out[10]=	-3 3 4 2 3 4 5 5 - q + -- - - - 5 q + 5 q - 3 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 13]}
In[12]:=	A2Invariant[Knot[8, 13]][q]
Out[12]=	-10 -8 -6 -4 -2 2 4 6 8 10 16 -1 - q + q + q - q + q + q + q + q + 2 q - q - q
In[13]:=	Kauffman[Knot[8, 13]][a, z]
Out[13]=	2 2 3 -4 2 2 z 4 z 3 z 5 z 7 z 2 2 3 z -a - -- + --- + --- + --- + a z + ---- + ---- - 2 a z - ---- - 2 5 3 a 4 2 5 a a a a a a 3 3 4 4 5 7 z 9 z 3 3 3 4 6 z 11 z 2 4 z ---- - ---- - 4 a z + a z - 2 z - ---- - ----- + 3 a z + -- + 3 a 4 2 5 a a a a 5 5 6 6 7 7 z 4 z 5 6 2 z 5 z z z -- + ---- + 4 a z + 3 z + ---- + ---- + -- + -- 3 a 4 2 3 a a a a a
In[14]:=	{Vassiliev[2][Knot[8, 13]], Vassiliev[3][Knot[8, 13]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[8, 13]][q, t]
Out[15]=	3 1 2 1 2 2 3 - + 3 q + ----- + ----- + ----- + ---- + --- + 3 q t + 2 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 2 q t + 3 q t + q t + 2 q t + q t + q t + q t