9 14

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9 13.gif

9_13

9 15.gif

9_15

9 14.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 14's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 14 at Knotilus!

[edit 9 14 Quick Notes]
[edit 9 14 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,18,14,1 X9,15,10,14 X7,17,8,16 X15,9,16,8 X17,7,18,6
Gauss code -1, 4, -3, 1, -2, 9, -7, 8, -6, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code 4 10 12 16 14 2 18 8 6
Conway Notation [41112]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 10, width is 5,

Braid index is 5

9 14 ML.gif 9 14 AP.gif
[{11, 3}, {2, 9}, {10, 4}, {3, 5}, {9, 11}, {4, 1}, {6, 2}, {5, 7}, {8, 6}, {7, 10}, {1, 8}]

[edit Notes on presentations of 9 14]


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["9 14"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,18,14,1 X9,15,10,14 X7,17,8,16 X15,9,16,8 X17,7,18,6
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, -2, 9, -7, 8, -6, 3, -4, 2, -5, 6, -8, 7, -9, 5
In[6]:= DTCode[K]
Out[6]= 4 10 12 16 14 2 18 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [41112]
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}(5,\{1,1,2,-1,-3,2,-3,4,-3,4\}) }[/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]= { 5, 10, 5 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.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."
9 14 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{11, 3}, {2, 9}, {10, 4}, {3, 5}, {9, 11}, {4, 1}, {6, 2}, {5, 7}, {8, 6}, {7, 10}, {1, 8}]
In[14]:= Draw[ap]
9 14 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{ \{4,7\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-4][-7]
Hyperbolic Volume 8.95499
A-Polynomial See Data:9 14/A-polynomial

[edit Notes for 9 14'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 9 14's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^2-9 t+15-9 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 { 37, 0 }
Jones polynomial [math]\displaystyle{ q^6-2 q^5+3 q^4-5 q^3+6 q^2-6 q+6-4 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+z^2 a^{-2} -2 z^2 a^{-4} +z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +5 z^7 a^{-3} +2 z^7 a^{-5} +3 z^6 a^{-2} +z^6 a^{-6} +4 z^6+4 a z^5-4 z^5 a^{-1} -16 z^5 a^{-3} -8 z^5 a^{-5} +3 a^2 z^4-12 z^4 a^{-2} -9 z^4 a^{-4} -4 z^4 a^{-6} -4 z^4+a^3 z^3-3 a z^3+2 z^3 a^{-1} +15 z^3 a^{-3} +9 z^3 a^{-5} -2 a^2 z^2+8 z^2 a^{-2} +10 z^2 a^{-4} +4 z^2 a^{-6} -2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1 }[/math]
The A2 invariant [math]\displaystyle{ -q^{10}+q^8+q^6-q^4+2 q^2+ q^{-2} + q^{-4} + q^{-8} -2 q^{-10} - q^{-12} - q^{-16} + q^{-18} + q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+8 q^{38}-10 q^{36}+11 q^{34}-8 q^{32}+2 q^{30}+4 q^{28}-10 q^{26}+16 q^{24}-16 q^{22}+14 q^{20}-8 q^{18}-q^{16}+11 q^{14}-15 q^{12}+17 q^{10}-12 q^8+3 q^6+6 q^4-10 q^2+10-2 q^{-2} -6 q^{-4} +15 q^{-6} -16 q^{-8} +9 q^{-10} +5 q^{-12} -19 q^{-14} +30 q^{-16} -28 q^{-18} +18 q^{-20} -16 q^{-24} +28 q^{-26} -30 q^{-28} +23 q^{-30} -10 q^{-32} -6 q^{-34} +16 q^{-36} -19 q^{-38} +15 q^{-40} -5 q^{-42} -7 q^{-44} +11 q^{-46} -13 q^{-48} +5 q^{-50} +5 q^{-52} -16 q^{-54} +21 q^{-56} -19 q^{-58} +6 q^{-60} +8 q^{-62} -20 q^{-64} +25 q^{-66} -21 q^{-68} +11 q^{-70} + q^{-72} -10 q^{-74} +16 q^{-76} -14 q^{-78} +10 q^{-80} -2 q^{-82} -2 q^{-84} +3 q^{-86} -4 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_14.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^7+2 q^5-q^3+2 q+ q^{-5} -2 q^{-7} + q^{-9} - q^{-11} + q^{-13} }[/math]
2 [math]\displaystyle{ q^{20}-2 q^{18}-q^{16}+4 q^{14}-4 q^{12}+7 q^8-5 q^6-q^4+7 q^2-3-3 q^{-2} +3 q^{-4} +2 q^{-6} -3 q^{-8} -2 q^{-10} +5 q^{-12} - q^{-14} -6 q^{-16} +5 q^{-18} +2 q^{-20} -6 q^{-22} +4 q^{-24} +4 q^{-26} -5 q^{-28} +3 q^{-32} -2 q^{-34} - q^{-36} + q^{-38} }[/math]
3 [math]\displaystyle{ -q^{39}+2 q^{37}+q^{35}-2 q^{33}-2 q^{31}+q^{29}+3 q^{27}-5 q^{25}+6 q^{21}-10 q^{17}+3 q^{15}+14 q^{13}-q^{11}-17 q^9+19 q^5+5 q^3-16 q-9 q^{-1} +10 q^{-3} +11 q^{-5} -2 q^{-7} -14 q^{-9} -3 q^{-11} +12 q^{-13} +10 q^{-15} -11 q^{-17} -13 q^{-19} +8 q^{-21} +16 q^{-23} -6 q^{-25} -16 q^{-27} +2 q^{-29} +18 q^{-31} +2 q^{-33} -17 q^{-35} -6 q^{-37} +16 q^{-39} +12 q^{-41} -13 q^{-43} -15 q^{-45} +5 q^{-47} +16 q^{-49} - q^{-51} -14 q^{-53} -4 q^{-55} +10 q^{-57} +7 q^{-59} -5 q^{-61} -6 q^{-63} +2 q^{-65} +4 q^{-67} -2 q^{-71} - q^{-73} + q^{-75} }[/math]
4 [math]\displaystyle{ q^{64}-2 q^{62}-q^{60}+2 q^{58}+5 q^{54}-4 q^{52}-2 q^{50}-6 q^{46}+11 q^{44}-2 q^{42}+3 q^{40}-21 q^{36}+4 q^{34}+6 q^{32}+26 q^{30}+10 q^{28}-43 q^{26}-22 q^{24}+4 q^{22}+56 q^{20}+40 q^{18}-48 q^{16}-56 q^{14}-24 q^{12}+60 q^{10}+70 q^8-16 q^6-54 q^4-55 q^2+21+63 q^{-2} +24 q^{-4} -14 q^{-6} -52 q^{-8} -26 q^{-10} +19 q^{-12} +41 q^{-14} +30 q^{-16} -25 q^{-18} -48 q^{-20} -17 q^{-22} +41 q^{-24} +47 q^{-26} -3 q^{-28} -53 q^{-30} -36 q^{-32} +39 q^{-34} +51 q^{-36} +8 q^{-38} -57 q^{-40} -48 q^{-42} +34 q^{-44} +54 q^{-46} +29 q^{-48} -47 q^{-50} -64 q^{-52} +8 q^{-54} +45 q^{-56} +56 q^{-58} -14 q^{-60} -63 q^{-62} -29 q^{-64} +8 q^{-66} +62 q^{-68} +30 q^{-70} -28 q^{-72} -41 q^{-74} -34 q^{-76} +32 q^{-78} +44 q^{-80} +15 q^{-82} -17 q^{-84} -47 q^{-86} -6 q^{-88} +21 q^{-90} +27 q^{-92} +12 q^{-94} -25 q^{-96} -17 q^{-98} -4 q^{-100} +12 q^{-102} +16 q^{-104} -4 q^{-106} -6 q^{-108} -6 q^{-110} +6 q^{-114} + q^{-116} -2 q^{-120} - q^{-122} + q^{-124} }[/math]
5 [math]\displaystyle{ -q^{95}+2 q^{93}+q^{91}-2 q^{89}-3 q^{85}-2 q^{83}+3 q^{81}+7 q^{79}+3 q^{77}-q^{75}-8 q^{73}-12 q^{71}-4 q^{69}+13 q^{67}+26 q^{65}+10 q^{63}-15 q^{61}-36 q^{59}-38 q^{57}+9 q^{55}+66 q^{53}+62 q^{51}+q^{49}-80 q^{47}-112 q^{45}-33 q^{43}+111 q^{41}+167 q^{39}+71 q^{37}-114 q^{35}-226 q^{33}-137 q^{31}+104 q^{29}+277 q^{27}+209 q^{25}-60 q^{23}-298 q^{21}-276 q^{19}-8 q^{17}+276 q^{15}+324 q^{13}+90 q^{11}-216 q^9-326 q^7-162 q^5+119 q^3+282 q+208 q^{-1} -17 q^{-3} -202 q^{-5} -216 q^{-7} -66 q^{-9} +103 q^{-11} +184 q^{-13} +132 q^{-15} -13 q^{-17} -142 q^{-19} -158 q^{-21} -53 q^{-23} +90 q^{-25} +172 q^{-27} +104 q^{-29} -65 q^{-31} -169 q^{-33} -121 q^{-35} +46 q^{-37} +177 q^{-39} +134 q^{-41} -48 q^{-43} -189 q^{-45} -148 q^{-47} +46 q^{-49} +209 q^{-51} +168 q^{-53} -43 q^{-55} -223 q^{-57} -200 q^{-59} +18 q^{-61} +233 q^{-63} +234 q^{-65} +20 q^{-67} -213 q^{-69} -260 q^{-71} -80 q^{-73} +172 q^{-75} +272 q^{-77} +137 q^{-79} -105 q^{-81} -250 q^{-83} -190 q^{-85} +19 q^{-87} +202 q^{-89} +217 q^{-91} +59 q^{-93} -122 q^{-95} -199 q^{-97} -131 q^{-99} +32 q^{-101} +157 q^{-103} +159 q^{-105} +50 q^{-107} -79 q^{-109} -149 q^{-111} -112 q^{-113} +3 q^{-115} +104 q^{-117} +127 q^{-119} +61 q^{-121} -42 q^{-123} -107 q^{-125} -95 q^{-127} -16 q^{-129} +65 q^{-131} +93 q^{-133} +51 q^{-135} -18 q^{-137} -65 q^{-139} -62 q^{-141} -15 q^{-143} +34 q^{-145} +51 q^{-147} +27 q^{-149} -7 q^{-151} -29 q^{-153} -28 q^{-155} -6 q^{-157} +14 q^{-159} +17 q^{-161} +7 q^{-163} -2 q^{-165} -8 q^{-167} -8 q^{-169} +4 q^{-173} +3 q^{-175} + q^{-177} -2 q^{-181} - q^{-183} + q^{-185} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{10}+q^8+q^6-q^4+2 q^2+ q^{-2} + q^{-4} + q^{-8} -2 q^{-10} - q^{-12} - q^{-16} + q^{-18} + q^{-20} }[/math]
1,1 [math]\displaystyle{ q^{28}-4 q^{26}+8 q^{24}-12 q^{22}+18 q^{20}-28 q^{18}+34 q^{16}-38 q^{14}+43 q^{12}-48 q^{10}+50 q^8-44 q^6+46 q^4-32 q^2+22-24 q^{-4} +50 q^{-6} -80 q^{-8} +102 q^{-10} -118 q^{-12} +126 q^{-14} -126 q^{-16} +108 q^{-18} -91 q^{-20} +62 q^{-22} -30 q^{-24} +34 q^{-28} -50 q^{-30} +70 q^{-32} -76 q^{-34} +71 q^{-36} -62 q^{-38} +48 q^{-40} -36 q^{-42} +21 q^{-44} -12 q^{-46} +6 q^{-48} -2 q^{-50} + q^{-52} }[/math]
2,0 [math]\displaystyle{ q^{26}-q^{24}-2 q^{22}+q^{20}+2 q^{18}-q^{16}-4 q^{14}+3 q^{12}+5 q^{10}-3 q^8-2 q^6+6 q^4+4 q^2-2-2 q^{-2} +2 q^{-4} - q^{-8} + q^{-10} -3 q^{-14} +2 q^{-16} + q^{-18} -5 q^{-20} -3 q^{-22} +2 q^{-24} +3 q^{-26} -2 q^{-28} +5 q^{-32} +4 q^{-34} -2 q^{-36} -2 q^{-38} + q^{-40} + q^{-42} -2 q^{-44} -3 q^{-46} + q^{-50} + q^{-52} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{22}-2 q^{20}-q^{18}+4 q^{16}-3 q^{14}-2 q^{12}+6 q^{10}-2 q^8-4 q^6+7 q^4+q^2-1+5 q^{-2} +2 q^{-4} -2 q^{-6} -3 q^{-8} -6 q^{-14} +2 q^{-16} +5 q^{-18} -4 q^{-20} +2 q^{-22} +4 q^{-24} -4 q^{-26} +2 q^{-30} -3 q^{-32} + q^{-34} + q^{-36} - q^{-38} + q^{-40} }[/math]
1,0,0 [math]\displaystyle{ -q^{13}+q^{11}+q^7-q^5+2 q^3+ q^{-1} + q^{-3} + q^{-5} + q^{-7} + q^{-11} -2 q^{-13} - q^{-15} -2 q^{-17} - q^{-21} + q^{-23} + q^{-25} + q^{-27} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{22}+2 q^{20}-3 q^{18}+4 q^{16}-5 q^{14}+6 q^{12}-6 q^{10}+6 q^8-4 q^6+3 q^4+q^2-3+7 q^{-2} -8 q^{-4} +12 q^{-6} -11 q^{-8} +12 q^{-10} -10 q^{-12} +8 q^{-14} -6 q^{-16} + q^{-18} -4 q^{-22} +4 q^{-24} -6 q^{-26} +6 q^{-28} -6 q^{-30} +5 q^{-32} -3 q^{-34} +3 q^{-36} - q^{-38} + q^{-40} }[/math]
1,0 [math]\displaystyle{ q^{36}-2 q^{32}-2 q^{30}+q^{28}+4 q^{26}+q^{24}-4 q^{22}-4 q^{20}+2 q^{18}+6 q^{16}+2 q^{14}-5 q^{12}-4 q^{10}+3 q^8+7 q^6-4 q^2+6 q^{-2} +3 q^{-4} -3 q^{-6} -4 q^{-8} +2 q^{-10} +3 q^{-12} -2 q^{-14} -5 q^{-16} +3 q^{-20} - q^{-22} -5 q^{-24} - q^{-26} +6 q^{-28} +4 q^{-30} -3 q^{-32} -6 q^{-34} +2 q^{-36} +7 q^{-38} +3 q^{-40} -5 q^{-42} -5 q^{-44} +2 q^{-46} +5 q^{-48} -4 q^{-52} -2 q^{-54} +2 q^{-56} +2 q^{-58} - q^{-60} - q^{-62} + q^{-66} }[/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}+8 q^{38}-10 q^{36}+11 q^{34}-8 q^{32}+2 q^{30}+4 q^{28}-10 q^{26}+16 q^{24}-16 q^{22}+14 q^{20}-8 q^{18}-q^{16}+11 q^{14}-15 q^{12}+17 q^{10}-12 q^8+3 q^6+6 q^4-10 q^2+10-2 q^{-2} -6 q^{-4} +15 q^{-6} -16 q^{-8} +9 q^{-10} +5 q^{-12} -19 q^{-14} +30 q^{-16} -28 q^{-18} +18 q^{-20} -16 q^{-24} +28 q^{-26} -30 q^{-28} +23 q^{-30} -10 q^{-32} -6 q^{-34} +16 q^{-36} -19 q^{-38} +15 q^{-40} -5 q^{-42} -7 q^{-44} +11 q^{-46} -13 q^{-48} +5 q^{-50} +5 q^{-52} -16 q^{-54} +21 q^{-56} -19 q^{-58} +6 q^{-60} +8 q^{-62} -20 q^{-64} +25 q^{-66} -21 q^{-68} +11 q^{-70} + q^{-72} -10 q^{-74} +16 q^{-76} -14 q^{-78} +10 q^{-80} -2 q^{-82} -2 q^{-84} +3 q^{-86} -4 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }[/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["9 14"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^2-9 t+15-9 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]= { 37, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^6-2 q^5+3 q^4-5 q^3+6 q^2-6 q+6-4 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+z^2 a^{-2} -2 z^2 a^{-4} +z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +5 z^7 a^{-3} +2 z^7 a^{-5} +3 z^6 a^{-2} +z^6 a^{-6} +4 z^6+4 a z^5-4 z^5 a^{-1} -16 z^5 a^{-3} -8 z^5 a^{-5} +3 a^2 z^4-12 z^4 a^{-2} -9 z^4 a^{-4} -4 z^4 a^{-6} -4 z^4+a^3 z^3-3 a z^3+2 z^3 a^{-1} +15 z^3 a^{-3} +9 z^3 a^{-5} -2 a^2 z^2+8 z^2 a^{-2} +10 z^2 a^{-4} +4 z^2 a^{-6} -2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n53,}

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["9 14"];
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^2-9 t+15-9 t^{-1} +2 t^{-2} }[/math], [math]\displaystyle{ q^6-2 q^5+3 q^4-5 q^3+6 q^2-6 q+6-4 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]= {K11n53,}

Vassiliev invariants

V2 and V3: (-1, -2)
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{ -16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{110}{3} }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{440}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ \frac{10529}{30} }[/math] [math]\displaystyle{ \frac{1502}{15} }[/math] [math]\displaystyle{ \frac{4978}{45} }[/math] [math]\displaystyle{ -\frac{833}{18} }[/math] [math]\displaystyle{ \frac{449}{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 9 14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-10123456χ
13         11
11        1 -1
9       21 1
7      31  -2
5     32   1
3    33    0
1   33     0
-1  24      2
-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}^{4}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=6 }[/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^{18}-2 q^{17}-q^{16}+6 q^{15}-5 q^{14}-6 q^{13}+15 q^{12}-5 q^{11}-16 q^{10}+23 q^9-2 q^8-27 q^7+28 q^6+4 q^5-34 q^4+27 q^3+9 q^2-33 q+21+9 q^{-1} -23 q^{-2} +13 q^{-3} +5 q^{-4} -11 q^{-5} +6 q^{-6} + q^{-7} -3 q^{-8} + q^{-9} }[/math]
3 [math]\displaystyle{ q^{36}-2 q^{35}-q^{34}+2 q^{33}+5 q^{32}-4 q^{31}-9 q^{30}+3 q^{29}+17 q^{28}-q^{27}-23 q^{26}-7 q^{25}+30 q^{24}+16 q^{23}-34 q^{22}-27 q^{21}+32 q^{20}+41 q^{19}-30 q^{18}-49 q^{17}+21 q^{16}+60 q^{15}-14 q^{14}-65 q^{13}+3 q^{12}+70 q^{11}+8 q^{10}-73 q^9-18 q^8+72 q^7+29 q^6-71 q^5-33 q^4+61 q^3+41 q^2-58 q-34+42 q^{-1} +34 q^{-2} -37 q^{-3} -20 q^{-4} +23 q^{-5} +17 q^{-6} -21 q^{-7} -5 q^{-8} +12 q^{-9} +4 q^{-10} -11 q^{-11} + q^{-12} +6 q^{-13} - q^{-14} -3 q^{-15} - q^{-16} +3 q^{-17} - q^{-18} }[/math]
4 [math]\displaystyle{ q^{60}-2 q^{59}-q^{58}+2 q^{57}+q^{56}+6 q^{55}-8 q^{54}-7 q^{53}+2 q^{52}+3 q^{51}+26 q^{50}-12 q^{49}-23 q^{48}-11 q^{47}-5 q^{46}+63 q^{45}+3 q^{44}-29 q^{43}-38 q^{42}-46 q^{41}+93 q^{40}+35 q^{39}-50 q^{37}-112 q^{36}+86 q^{35}+48 q^{34}+58 q^{33}-18 q^{32}-166 q^{31}+49 q^{30}+14 q^{29}+107 q^{28}+52 q^{27}-177 q^{26}+12 q^{25}-58 q^{24}+124 q^{23}+128 q^{22}-152 q^{21}-8 q^{20}-140 q^{19}+115 q^{18}+193 q^{17}-109 q^{16}-20 q^{15}-215 q^{14}+98 q^{13}+243 q^{12}-59 q^{11}-26 q^{10}-273 q^9+67 q^8+266 q^7-4 q^6-15 q^5-295 q^4+22 q^3+240 q^2+34 q+23-256 q^{-1} -20 q^{-2} +164 q^{-3} +35 q^{-4} +61 q^{-5} -170 q^{-6} -30 q^{-7} +80 q^{-8} +3 q^{-9} +69 q^{-10} -82 q^{-11} -14 q^{-12} +28 q^{-13} -23 q^{-14} +48 q^{-15} -29 q^{-16} +2 q^{-17} +8 q^{-18} -25 q^{-19} +23 q^{-20} -8 q^{-21} +5 q^{-22} +3 q^{-23} -12 q^{-24} +6 q^{-25} -2 q^{-26} +3 q^{-27} + q^{-28} -3 q^{-29} + q^{-30} }[/math]
5 [math]\displaystyle{ q^{90}-2 q^{89}-q^{88}+2 q^{87}+q^{86}+2 q^{85}+2 q^{84}-6 q^{83}-9 q^{82}+2 q^{81}+7 q^{80}+11 q^{79}+12 q^{78}-9 q^{77}-29 q^{76}-20 q^{75}+6 q^{74}+33 q^{73}+46 q^{72}+15 q^{71}-46 q^{70}-69 q^{69}-41 q^{68}+30 q^{67}+93 q^{66}+84 q^{65}-4 q^{64}-97 q^{63}-122 q^{62}-49 q^{61}+81 q^{60}+149 q^{59}+99 q^{58}-31 q^{57}-145 q^{56}-150 q^{55}-34 q^{54}+112 q^{53}+169 q^{52}+98 q^{51}-36 q^{50}-152 q^{49}-159 q^{48}-51 q^{47}+101 q^{46}+175 q^{45}+145 q^{44}+6 q^{43}-174 q^{42}-234 q^{41}-108 q^{40}+115 q^{39}+290 q^{38}+248 q^{37}-39 q^{36}-334 q^{35}-360 q^{34}-65 q^{33}+337 q^{32}+481 q^{31}+175 q^{30}-335 q^{29}-575 q^{28}-283 q^{27}+314 q^{26}+661 q^{25}+386 q^{24}-294 q^{23}-738 q^{22}-477 q^{21}+273 q^{20}+802 q^{19}+568 q^{18}-251 q^{17}-869 q^{16}-644 q^{15}+225 q^{14}+906 q^{13}+737 q^{12}-183 q^{11}-951 q^{10}-787 q^9+120 q^8+922 q^7+866 q^6-38 q^5-899 q^4-868 q^3-49 q^2+772 q+880+147 q^{-1} -674 q^{-2} -794 q^{-3} -212 q^{-4} +491 q^{-5} +716 q^{-6} +257 q^{-7} -368 q^{-8} -560 q^{-9} -260 q^{-10} +207 q^{-11} +448 q^{-12} +235 q^{-13} -130 q^{-14} -291 q^{-15} -192 q^{-16} +34 q^{-17} +207 q^{-18} +146 q^{-19} -18 q^{-20} -106 q^{-21} -96 q^{-22} -22 q^{-23} +63 q^{-24} +67 q^{-25} +14 q^{-26} -25 q^{-27} -35 q^{-28} -18 q^{-29} +6 q^{-30} +20 q^{-31} +16 q^{-32} -4 q^{-33} -10 q^{-34} -2 q^{-35} -7 q^{-36} +3 q^{-37} +8 q^{-38} -3 q^{-40} +2 q^{-41} -3 q^{-42} - q^{-43} +3 q^{-44} - q^{-45} }[/math]
6 [math]\displaystyle{ q^{126}-2 q^{125}-q^{124}+2 q^{123}+q^{122}+2 q^{121}-2 q^{120}+4 q^{119}-8 q^{118}-9 q^{117}+5 q^{116}+6 q^{115}+12 q^{114}+16 q^{112}-22 q^{111}-35 q^{110}-9 q^{109}+5 q^{108}+36 q^{107}+21 q^{106}+70 q^{105}-21 q^{104}-79 q^{103}-68 q^{102}-48 q^{101}+30 q^{100}+43 q^{99}+193 q^{98}+62 q^{97}-60 q^{96}-133 q^{95}-165 q^{94}-86 q^{93}-49 q^{92}+295 q^{91}+209 q^{90}+108 q^{89}-57 q^{88}-200 q^{87}-245 q^{86}-311 q^{85}+201 q^{84}+214 q^{83}+285 q^{82}+170 q^{81}+27 q^{80}-189 q^{79}-516 q^{78}-37 q^{77}-79 q^{76}+170 q^{75}+243 q^{74}+381 q^{73}+195 q^{72}-341 q^{71}-51 q^{70}-447 q^{69}-301 q^{68}-148 q^{67}+447 q^{66}+615 q^{65}+212 q^{64}+436 q^{63}-453 q^{62}-784 q^{61}-930 q^{60}-15 q^{59}+645 q^{58}+765 q^{57}+1296 q^{56}+92 q^{55}-879 q^{54}-1711 q^{53}-848 q^{52}+151 q^{51}+960 q^{50}+2146 q^{49}+1001 q^{48}-494 q^{47}-2172 q^{46}-1702 q^{45}-668 q^{44}+748 q^{43}+2720 q^{42}+1943 q^{41}+167 q^{40}-2283 q^{39}-2349 q^{38}-1514 q^{37}+326 q^{36}+3007 q^{35}+2713 q^{34}+832 q^{33}-2217 q^{32}-2784 q^{31}-2212 q^{30}-69 q^{29}+3160 q^{28}+3299 q^{27}+1356 q^{26}-2150 q^{25}-3128 q^{24}-2763 q^{23}-352 q^{22}+3297 q^{21}+3795 q^{20}+1786 q^{19}-2085 q^{18}-3443 q^{17}-3262 q^{16}-644 q^{15}+3332 q^{14}+4210 q^{13}+2258 q^{12}-1817 q^{11}-3570 q^{10}-3697 q^9-1116 q^8+3007 q^7+4328 q^6+2736 q^5-1174 q^4-3219 q^3-3810 q^2-1684 q+2192+3857 q^{-1} +2909 q^{-2} -334 q^{-3} -2323 q^{-4} -3321 q^{-5} -1975 q^{-6} +1165 q^{-7} +2814 q^{-8} +2515 q^{-9} +269 q^{-10} -1243 q^{-11} -2327 q^{-12} -1750 q^{-13} +391 q^{-14} +1636 q^{-15} +1707 q^{-16} +415 q^{-17} -438 q^{-18} -1290 q^{-19} -1195 q^{-20} +47 q^{-21} +765 q^{-22} +919 q^{-23} +276 q^{-24} -49 q^{-25} -574 q^{-26} -664 q^{-27} -20 q^{-28} +291 q^{-29} +409 q^{-30} +117 q^{-31} +64 q^{-32} -207 q^{-33} -323 q^{-34} -9 q^{-35} +87 q^{-36} +155 q^{-37} +33 q^{-38} +67 q^{-39} -58 q^{-40} -139 q^{-41} + q^{-42} +14 q^{-43} +50 q^{-44} +2 q^{-45} +41 q^{-46} -11 q^{-47} -49 q^{-48} +4 q^{-49} -3 q^{-50} +14 q^{-51} -5 q^{-52} +17 q^{-53} - q^{-54} -14 q^{-55} +4 q^{-56} -3 q^{-57} +3 q^{-58} -2 q^{-59} +3 q^{-60} + q^{-61} -3 q^{-62} + q^{-63} }[/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.

9 13.gif

9_13

9 15.gif

9_15

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