8 13

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8 12.gif

8_12

8 14.gif

8_14

8 13.gif
(KnotPlot image)

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[edit 8 13 Quick Notes]
[edit 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]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 9, width is 4,

Braid index is 4

8 13 ML.gif 8 13 AP.gif
[{10, 5}, {1, 8}, {6, 9}, {8, 10}, {9, 4}, {5, 2}, {3, 1}, {4, 7}, {2, 6}, {7, 3}]

[edit Notes on presentations of 8 13]

Knot 8_13.
A graph, knot 8_13.
Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["8 13"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 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
In[5]:= GaussCode[K]
Out[5]= -1, 8, -2, 1, -4, 5, -6, 7, -8, 2, -3, 4, -7, 6, -5, 3
In[6]:= DTCode[K]
Out[6]= 4 10 12 14 2 16 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [31112]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= [math]\displaystyle{ \textrm{BR}(4,\{-1,-1,2,-1,2,2,3,-2,3\}) }[/math]
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 4, 9, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
8 13 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{10, 5}, {1, 8}, {6, 9}, {8, 10}, {9, 4}, {5, 2}, {3, 1}, {4, 7}, {2, 6}, {7, 3}]
In[14]:= Draw[ap]
8 13 AP.gif
Out[14]= -Graphics-

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]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["8 13"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { [math]\displaystyle{ 2 t^2-7 t+11-7 t^{-1} +2 t^{-2} }[/math], [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[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

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]

V2,1 through V6,9 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 \
 -3-2-1012345χ
11        1-1
9       1 1
7      21 -1
5     31  2
3    22   0
1   33    0
-1  23     1
-3 12      -1
-5 2       2
-71        -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]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^{15}-2 q^{14}-q^{13}+6 q^{12}-5 q^{11}-6 q^{10}+15 q^9-6 q^8-15 q^7+24 q^6-5 q^5-24 q^4+28 q^3-q^2-27 q+26+2 q^{-1} -22 q^{-2} +17 q^{-3} +2 q^{-4} -12 q^{-5} +7 q^{-6} + q^{-7} -3 q^{-8} + q^{-9} }[/math]
3 [math]\displaystyle{ -q^{30}+2 q^{29}+q^{28}-2 q^{27}-5 q^{26}+4 q^{25}+9 q^{24}-3 q^{23}-17 q^{22}+q^{21}+24 q^{20}+6 q^{19}-32 q^{18}-15 q^{17}+39 q^{16}+25 q^{15}-41 q^{14}-40 q^{13}+46 q^{12}+48 q^{11}-41 q^{10}-64 q^9+42 q^8+71 q^7-35 q^6-81 q^5+33 q^4+82 q^3-24 q^2-84 q+20+77 q^{-1} -12 q^{-2} -69 q^{-3} +9 q^{-4} +54 q^{-5} -2 q^{-6} -42 q^{-7} +2 q^{-8} +27 q^{-9} + q^{-10} -19 q^{-11} + q^{-12} +9 q^{-13} -4 q^{-15} - q^{-16} +3 q^{-17} - q^{-18} }[/math]
4 [math]\displaystyle{ q^{50}-2 q^{49}-q^{48}+2 q^{47}+q^{46}+6 q^{45}-8 q^{44}-7 q^{43}+2 q^{42}+3 q^{41}+26 q^{40}-12 q^{39}-23 q^{38}-12 q^{37}-4 q^{36}+65 q^{35}+3 q^{34}-31 q^{33}-45 q^{32}-43 q^{31}+104 q^{30}+41 q^{29}-7 q^{28}-77 q^{27}-115 q^{26}+116 q^{25}+79 q^{24}+53 q^{23}-82 q^{22}-198 q^{21}+96 q^{20}+97 q^{19}+128 q^{18}-59 q^{17}-269 q^{16}+56 q^{15}+98 q^{14}+200 q^{13}-26 q^{12}-319 q^{11}+12 q^{10}+89 q^9+253 q^8+8 q^7-338 q^6-31 q^5+69 q^4+279 q^3+40 q^2-318 q-59+36 q^{-1} +258 q^{-2} +65 q^{-3} -251 q^{-4} -62 q^{-5} -4 q^{-6} +192 q^{-7} +70 q^{-8} -161 q^{-9} -37 q^{-10} -28 q^{-11} +109 q^{-12} +50 q^{-13} -83 q^{-14} -8 q^{-15} -25 q^{-16} +47 q^{-17} +22 q^{-18} -36 q^{-19} +5 q^{-20} -12 q^{-21} +15 q^{-22} +7 q^{-23} -12 q^{-24} +3 q^{-25} -3 q^{-26} +4 q^{-27} + q^{-28} -3 q^{-29} + q^{-30} }[/math]
5 [math]\displaystyle{ -q^{75}+2 q^{74}+q^{73}-2 q^{72}-q^{71}-2 q^{70}-2 q^{69}+6 q^{68}+9 q^{67}-2 q^{66}-7 q^{65}-11 q^{64}-12 q^{63}+9 q^{62}+29 q^{61}+20 q^{60}-5 q^{59}-34 q^{58}-48 q^{57}-15 q^{56}+47 q^{55}+73 q^{54}+46 q^{53}-33 q^{52}-105 q^{51}-94 q^{50}+6 q^{49}+118 q^{48}+150 q^{47}+52 q^{46}-115 q^{45}-203 q^{44}-123 q^{43}+77 q^{42}+239 q^{41}+211 q^{40}-11 q^{39}-254 q^{38}-298 q^{37}-72 q^{36}+233 q^{35}+365 q^{34}+190 q^{33}-192 q^{32}-434 q^{31}-281 q^{30}+121 q^{29}+454 q^{28}+409 q^{27}-44 q^{26}-493 q^{25}-488 q^{24}-38 q^{23}+478 q^{22}+596 q^{21}+122 q^{20}-499 q^{19}-653 q^{18}-198 q^{17}+475 q^{16}+735 q^{15}+270 q^{14}-481 q^{13}-774 q^{12}-336 q^{11}+450 q^{10}+832 q^9+391 q^8-435 q^7-837 q^6-449 q^5+383 q^4+849 q^3+490 q^2-336 q-805-518 q^{-1} +249 q^{-2} +754 q^{-3} +525 q^{-4} -178 q^{-5} -649 q^{-6} -506 q^{-7} +88 q^{-8} +545 q^{-9} +455 q^{-10} -28 q^{-11} -408 q^{-12} -386 q^{-13} -29 q^{-14} +302 q^{-15} +298 q^{-16} +40 q^{-17} -184 q^{-18} -216 q^{-19} -56 q^{-20} +123 q^{-21} +139 q^{-22} +33 q^{-23} -59 q^{-24} -80 q^{-25} -32 q^{-26} +37 q^{-27} +45 q^{-28} +11 q^{-29} -17 q^{-30} -20 q^{-31} -4 q^{-32} +5 q^{-33} +11 q^{-34} +5 q^{-35} -10 q^{-36} -3 q^{-37} +4 q^{-38} +3 q^{-41} -4 q^{-42} - q^{-43} +3 q^{-44} - q^{-45} }[/math]
6 [math]\displaystyle{ q^{105}-2 q^{104}-q^{103}+2 q^{102}+q^{101}+2 q^{100}-2 q^{99}+4 q^{98}-8 q^{97}-9 q^{96}+5 q^{95}+6 q^{94}+12 q^{93}+16 q^{91}-22 q^{90}-35 q^{89}-9 q^{88}+5 q^{87}+35 q^{86}+22 q^{85}+72 q^{84}-21 q^{83}-80 q^{82}-71 q^{81}-50 q^{80}+25 q^{79}+48 q^{78}+207 q^{77}+70 q^{76}-60 q^{75}-148 q^{74}-188 q^{73}-118 q^{72}-42 q^{71}+346 q^{70}+265 q^{69}+146 q^{68}-77 q^{67}-283 q^{66}-387 q^{65}-373 q^{64}+283 q^{63}+390 q^{62}+491 q^{61}+264 q^{60}-98 q^{59}-565 q^{58}-852 q^{57}-94 q^{56}+195 q^{55}+718 q^{54}+750 q^{53}+454 q^{52}-405 q^{51}-1201 q^{50}-632 q^{49}-372 q^{48}+593 q^{47}+1106 q^{46}+1193 q^{45}+118 q^{44}-1213 q^{43}-1073 q^{42}-1118 q^{41}+124 q^{40}+1165 q^{39}+1866 q^{38}+808 q^{37}-921 q^{36}-1286 q^{35}-1807 q^{34}-489 q^{33}+981 q^{32}+2349 q^{31}+1453 q^{30}-504 q^{29}-1325 q^{28}-2337 q^{27}-1057 q^{26}+720 q^{25}+2671 q^{24}+1961 q^{23}-114 q^{22}-1313 q^{21}-2725 q^{20}-1506 q^{19}+492 q^{18}+2897 q^{17}+2343 q^{16}+215 q^{15}-1294 q^{14}-3006 q^{13}-1861 q^{12}+281 q^{11}+3008 q^{10}+2627 q^9+542 q^8-1193 q^7-3132 q^6-2158 q^5-11 q^4+2891 q^3+2759 q^2+906 q-888-2973 q^{-1} -2331 q^{-2} -420 q^{-3} +2429 q^{-4} +2600 q^{-5} +1207 q^{-6} -376 q^{-7} -2423 q^{-8} -2213 q^{-9} -802 q^{-10} +1667 q^{-11} +2056 q^{-12} +1246 q^{-13} +147 q^{-14} -1585 q^{-15} -1725 q^{-16} -928 q^{-17} +876 q^{-18} +1271 q^{-19} +950 q^{-20} +421 q^{-21} -775 q^{-22} -1040 q^{-23} -739 q^{-24} +343 q^{-25} +570 q^{-26} +506 q^{-27} +391 q^{-28} -266 q^{-29} -463 q^{-30} -418 q^{-31} +119 q^{-32} +167 q^{-33} +168 q^{-34} +228 q^{-35} -63 q^{-36} -148 q^{-37} -172 q^{-38} +57 q^{-39} +25 q^{-40} +19 q^{-41} +94 q^{-42} -12 q^{-43} -35 q^{-44} -54 q^{-45} +34 q^{-46} -2 q^{-47} -10 q^{-48} +28 q^{-49} -5 q^{-50} -5 q^{-51} -15 q^{-52} +17 q^{-53} -2 q^{-54} -8 q^{-55} +8 q^{-56} -3 q^{-57} -3 q^{-59} +4 q^{-60} + q^{-61} -3 q^{-62} + q^{-63} }[/math]
7 [math]\displaystyle{ -q^{140}+2 q^{139}+q^{138}-2 q^{137}-q^{136}-2 q^{135}+2 q^{134}-2 q^{132}+8 q^{131}+6 q^{130}-4 q^{129}-6 q^{128}-14 q^{127}-2 q^{126}+3 q^{125}-6 q^{124}+25 q^{123}+28 q^{122}+10 q^{121}-5 q^{120}-48 q^{119}-35 q^{118}-22 q^{117}-32 q^{116}+45 q^{115}+85 q^{114}+82 q^{113}+69 q^{112}-57 q^{111}-101 q^{110}-121 q^{109}-170 q^{108}-25 q^{107}+102 q^{106}+211 q^{105}+300 q^{104}+122 q^{103}-33 q^{102}-201 q^{101}-442 q^{100}-335 q^{99}-149 q^{98}+141 q^{97}+551 q^{96}+538 q^{95}+423 q^{94}+101 q^{93}-521 q^{92}-731 q^{91}-781 q^{90}-482 q^{89}+315 q^{88}+768 q^{87}+1112 q^{86}+1011 q^{85}+115 q^{84}-594 q^{83}-1312 q^{82}-1559 q^{81}-751 q^{80}+118 q^{79}+1284 q^{78}+2043 q^{77}+1489 q^{76}+608 q^{75}-924 q^{74}-2293 q^{73}-2229 q^{72}-1563 q^{71}+231 q^{70}+2253 q^{69}+2838 q^{68}+2603 q^{67}+735 q^{66}-1864 q^{65}-3166 q^{64}-3609 q^{63}-1943 q^{62}+1125 q^{61}+3246 q^{60}+4506 q^{59}+3170 q^{58}-168 q^{57}-2947 q^{56}-5125 q^{55}-4448 q^{54}-1027 q^{53}+2453 q^{52}+5588 q^{51}+5550 q^{50}+2183 q^{49}-1703 q^{48}-5715 q^{47}-6547 q^{46}-3436 q^{45}+894 q^{44}+5790 q^{43}+7334 q^{42}+4473 q^{41}-40 q^{40}-5607 q^{39}-7992 q^{38}-5517 q^{37}-752 q^{36}+5502 q^{35}+8509 q^{34}+6296 q^{33}+1474 q^{32}-5273 q^{31}-8946 q^{30}-7071 q^{29}-2070 q^{28}+5177 q^{27}+9302 q^{26}+7641 q^{25}+2595 q^{24}-5016 q^{23}-9657 q^{22}-8219 q^{21}-3010 q^{20}+4948 q^{19}+9926 q^{18}+8672 q^{17}+3456 q^{16}-4798 q^{15}-10186 q^{14}-9151 q^{13}-3854 q^{12}+4617 q^{11}+10293 q^{10}+9531 q^9+4354 q^8-4240 q^7-10292 q^6-9891 q^5-4866 q^4+3744 q^3+10026 q^2+10041 q+5442-2978 q^{-1} -9500 q^{-2} -10054 q^{-3} -5946 q^{-4} +2107 q^{-5} +8639 q^{-6} +9702 q^{-7} +6341 q^{-8} -1071 q^{-9} -7507 q^{-10} -9069 q^{-11} -6480 q^{-12} +92 q^{-13} +6140 q^{-14} +8055 q^{-15} +6347 q^{-16} +783 q^{-17} -4706 q^{-18} -6812 q^{-19} -5852 q^{-20} -1392 q^{-21} +3288 q^{-22} +5393 q^{-23} +5128 q^{-24} +1742 q^{-25} -2107 q^{-26} -4026 q^{-27} -4165 q^{-28} -1758 q^{-29} +1145 q^{-30} +2734 q^{-31} +3215 q^{-32} +1621 q^{-33} -534 q^{-34} -1768 q^{-35} -2280 q^{-36} -1257 q^{-37} +140 q^{-38} +973 q^{-39} +1532 q^{-40} +966 q^{-41} +22 q^{-42} -532 q^{-43} -954 q^{-44} -606 q^{-45} -73 q^{-46} +203 q^{-47} +554 q^{-48} +390 q^{-49} +75 q^{-50} -72 q^{-51} -317 q^{-52} -209 q^{-53} -37 q^{-54} +8 q^{-55} +153 q^{-56} +98 q^{-57} +27 q^{-58} +30 q^{-59} -88 q^{-60} -61 q^{-61} +4 q^{-62} -9 q^{-63} +35 q^{-64} +5 q^{-65} +29 q^{-67} -22 q^{-68} -16 q^{-69} +6 q^{-70} - q^{-71} +9 q^{-72} -7 q^{-73} -5 q^{-74} +13 q^{-75} -4 q^{-76} -5 q^{-77} +3 q^{-78} +3 q^{-80} -4 q^{-81} - q^{-82} +3 q^{-83} - q^{-84} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

8 12.gif

8_12

8 14.gif

8_14

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