9 49

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9 48.gif

9_48

10 1.gif

10_1

9 49.gif
(KnotPlot image)

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[edit 9 49 Quick Notes]
[edit 9 49 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X12,8,13,7 X5,15,6,14 X3,11,4,10 X11,3,12,2 X15,5,16,4 X17,9,18,8 X9,17,10,16 X18,14,1,13
Gauss code 1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9
Dowker-Thistlethwaite code 6 -10 -14 12 -16 -2 18 -4 -8
Conway Notation [-20:-20:-20]


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

Length is 11, width is 4,

Braid index is 4

9 49 ML.gif 9 49 AP.gif
[{2, 7}, {1, 5}, {8, 3}, {7, 9}, {6, 2}, {4, 1}, {5, 8}, {3, 6}, {9, 4}]

[edit Notes on presentations of 9 49]


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 49"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X12,8,13,7 X5,15,6,14 X3,11,4,10 X11,3,12,2 X15,5,16,4 X17,9,18,8 X9,17,10,16 X18,14,1,13
In[5]:= GaussCode[K]
Out[5]= 1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9
In[6]:= DTCode[K]
Out[6]= 6 -10 -14 12 -16 -2 18 -4 -8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [-20:-20:-20]
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,\{1,1,2,1,1,-3,2,-1,2,3,3\}) }[/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, 11, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.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 49 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{2, 7}, {1, 5}, {8, 3}, {7, 9}, {6, 2}, {4, 1}, {5, 8}, {3, 6}, {9, 4}]
In[14]:= Draw[ap]
9 49 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 2
Bridge index 3
Super bridge index [math]\displaystyle{ \{4,5\} }[/math]
Nakanishi index 2
Maximal Thurston-Bennequin number [3][-12]
Hyperbolic Volume 9.42707
A-Polynomial See Data:9 49/A-polynomial

[edit Notes for 9 49's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math]
Topological 4 genus [math]\displaystyle{ 2 }[/math]
Concordance genus [math]\displaystyle{ 2 }[/math]
Rasmussen s-Invariant -4

[edit Notes for 9 49's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t^2-6 t+7-6 t^{-1} +3 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 3 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{5,t+1\} }[/math]
Determinant and Signature { 25, 4 }
Jones polynomial [math]\displaystyle{ -2 q^9+3 q^8-4 q^7+5 q^6-4 q^5+4 q^4-2 q^3+q^2 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^{-4} +2 z^4 a^{-6} +2 z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^7 a^{-7} +z^7 a^{-9} +3 z^6 a^{-6} +4 z^6 a^{-8} +z^6 a^{-10} +2 z^5 a^{-5} +z^5 a^{-7} -z^5 a^{-9} +z^4 a^{-4} -8 z^4 a^{-6} -9 z^4 a^{-8} -3 z^3 a^{-5} -3 z^3 a^{-7} +3 z^3 a^{-9} +3 z^3 a^{-11} -2 z^2 a^{-4} +9 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +2 z a^{-7} -2 z a^{-9} -4 z a^{-11} -4 a^{-6} -3 a^{-8} }[/math]
The A2 invariant [math]\displaystyle{ q^{-6} - q^{-8} + q^{-10} + q^{-14} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} - q^{-26} -2 q^{-28} }[/math]
The G2 invariant [math]\displaystyle{ q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +8 q^{-44} -10 q^{-46} +12 q^{-48} -7 q^{-50} - q^{-52} +10 q^{-54} -16 q^{-56} +19 q^{-58} -11 q^{-60} + q^{-62} +10 q^{-64} -14 q^{-66} +13 q^{-68} -2 q^{-70} -6 q^{-72} +14 q^{-74} -12 q^{-76} +4 q^{-78} +9 q^{-80} -15 q^{-82} +21 q^{-84} -16 q^{-86} +9 q^{-88} +5 q^{-90} -13 q^{-92} +22 q^{-94} -22 q^{-96} +16 q^{-98} -4 q^{-100} -7 q^{-102} +13 q^{-104} -16 q^{-106} +9 q^{-108} - q^{-110} -10 q^{-112} +10 q^{-114} -11 q^{-116} -2 q^{-118} +10 q^{-120} -20 q^{-122} +16 q^{-124} -9 q^{-126} -4 q^{-128} +11 q^{-130} -16 q^{-132} +15 q^{-134} -7 q^{-136} + q^{-138} +3 q^{-140} -7 q^{-142} +7 q^{-144} -2 q^{-146} + q^{-148} + q^{-150} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_49.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{-3} - q^{-5} +2 q^{-7} + q^{-11} + q^{-13} - q^{-15} + q^{-17} -2 q^{-19} }[/math]
2 [math]\displaystyle{ q^{-6} - q^{-8} +5 q^{-12} - q^{-14} -5 q^{-16} +5 q^{-18} +2 q^{-20} -5 q^{-22} +4 q^{-24} +4 q^{-26} -3 q^{-28} - q^{-30} + q^{-32} + q^{-34} -5 q^{-36} + q^{-38} +5 q^{-40} -6 q^{-42} -2 q^{-44} +5 q^{-46} -3 q^{-48} -3 q^{-50} +3 q^{-52} + q^{-54} }[/math]
3 [math]\displaystyle{ q^{-9} - q^{-11} +3 q^{-15} +3 q^{-17} -3 q^{-19} -7 q^{-21} +3 q^{-23} +14 q^{-25} +4 q^{-27} -14 q^{-29} -13 q^{-31} +14 q^{-33} +18 q^{-35} -9 q^{-37} -21 q^{-39} +4 q^{-41} +23 q^{-43} + q^{-45} -19 q^{-47} -3 q^{-49} +14 q^{-51} +6 q^{-53} -9 q^{-55} -9 q^{-57} +2 q^{-59} +8 q^{-61} + q^{-63} -14 q^{-65} -7 q^{-67} +14 q^{-69} +14 q^{-71} -16 q^{-73} -17 q^{-75} +12 q^{-77} +21 q^{-79} -6 q^{-81} -21 q^{-83} - q^{-85} +18 q^{-87} +5 q^{-89} -10 q^{-91} -7 q^{-93} +5 q^{-95} +8 q^{-97} - q^{-99} -2 q^{-101} -2 q^{-103} }[/math]
4 [math]\displaystyle{ q^{-12} - q^{-14} +3 q^{-18} + q^{-20} + q^{-22} -6 q^{-24} -4 q^{-26} +8 q^{-28} +10 q^{-30} +14 q^{-32} -12 q^{-34} -28 q^{-36} -13 q^{-38} +12 q^{-40} +51 q^{-42} +23 q^{-44} -29 q^{-46} -58 q^{-48} -37 q^{-50} +56 q^{-52} +76 q^{-54} +19 q^{-56} -66 q^{-58} -93 q^{-60} +10 q^{-62} +84 q^{-64} +69 q^{-66} -28 q^{-68} -97 q^{-70} -28 q^{-72} +51 q^{-74} +71 q^{-76} +4 q^{-78} -62 q^{-80} -38 q^{-82} +14 q^{-84} +45 q^{-86} +17 q^{-88} -25 q^{-90} -34 q^{-92} -9 q^{-94} +25 q^{-96} +32 q^{-98} +7 q^{-100} -43 q^{-102} -38 q^{-104} +13 q^{-106} +56 q^{-108} +43 q^{-110} -49 q^{-112} -73 q^{-114} -13 q^{-116} +71 q^{-118} +87 q^{-120} -25 q^{-122} -88 q^{-124} -57 q^{-126} +39 q^{-128} +99 q^{-130} +24 q^{-132} -50 q^{-134} -75 q^{-136} -17 q^{-138} +58 q^{-140} +45 q^{-142} +7 q^{-144} -37 q^{-146} -35 q^{-148} +5 q^{-150} +19 q^{-152} +20 q^{-154} - q^{-156} -13 q^{-158} -7 q^{-160} -3 q^{-162} +3 q^{-164} +3 q^{-166} + q^{-168} }[/math]
5 [math]\displaystyle{ q^{-15} - q^{-17} +3 q^{-21} + q^{-23} - q^{-25} -2 q^{-27} -3 q^{-29} +9 q^{-33} +14 q^{-35} +3 q^{-37} -10 q^{-39} -24 q^{-41} -23 q^{-43} +40 q^{-47} +59 q^{-49} +34 q^{-51} -28 q^{-53} -92 q^{-55} -97 q^{-57} -27 q^{-59} +96 q^{-61} +171 q^{-63} +117 q^{-65} -44 q^{-67} -206 q^{-69} -231 q^{-71} -64 q^{-73} +197 q^{-75} +325 q^{-77} +194 q^{-79} -114 q^{-81} -363 q^{-83} -326 q^{-85} -8 q^{-87} +339 q^{-89} +402 q^{-91} +127 q^{-93} -257 q^{-95} -424 q^{-97} -219 q^{-99} +159 q^{-101} +387 q^{-103} +263 q^{-105} -72 q^{-107} -310 q^{-109} -263 q^{-111} +5 q^{-113} +228 q^{-115} +229 q^{-117} +33 q^{-119} -156 q^{-121} -183 q^{-123} -56 q^{-125} +92 q^{-127} +141 q^{-129} +66 q^{-131} -52 q^{-133} -117 q^{-135} -81 q^{-137} +20 q^{-139} +108 q^{-141} +112 q^{-143} +16 q^{-145} -116 q^{-147} -156 q^{-149} -50 q^{-151} +121 q^{-153} +216 q^{-155} +116 q^{-157} -128 q^{-159} -281 q^{-161} -184 q^{-163} +100 q^{-165} +334 q^{-167} +279 q^{-169} -42 q^{-171} -353 q^{-173} -359 q^{-175} -46 q^{-177} +317 q^{-179} +417 q^{-181} +151 q^{-183} -229 q^{-185} -414 q^{-187} -254 q^{-189} +106 q^{-191} +349 q^{-193} +299 q^{-195} +25 q^{-197} -232 q^{-199} -292 q^{-201} -124 q^{-203} +107 q^{-205} +219 q^{-207} +160 q^{-209} +3 q^{-211} -120 q^{-213} -141 q^{-215} -61 q^{-217} +42 q^{-219} +82 q^{-221} +63 q^{-223} +12 q^{-225} -31 q^{-227} -46 q^{-229} -21 q^{-231} +3 q^{-233} +12 q^{-235} +17 q^{-237} +8 q^{-239} - q^{-241} -4 q^{-243} -2 q^{-245} -2 q^{-247} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-6} - q^{-8} + q^{-10} + q^{-14} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} - q^{-26} -2 q^{-28} }[/math]
1,1 [math]\displaystyle{ q^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +19 q^{-20} -22 q^{-22} +36 q^{-24} -42 q^{-26} +48 q^{-28} -40 q^{-30} +34 q^{-32} -10 q^{-34} -6 q^{-36} +38 q^{-38} -56 q^{-40} +70 q^{-42} -87 q^{-44} +78 q^{-46} -84 q^{-48} +58 q^{-50} -45 q^{-52} +18 q^{-54} +10 q^{-56} -22 q^{-58} +45 q^{-60} -48 q^{-62} +48 q^{-64} -42 q^{-66} +25 q^{-68} -22 q^{-70} +10 q^{-72} -2 q^{-74} +2 q^{-76} +2 q^{-78} }[/math]
2,0 [math]\displaystyle{ q^{-12} - q^{-14} - q^{-16} +3 q^{-18} +3 q^{-20} - q^{-22} - q^{-24} +2 q^{-26} +2 q^{-28} -2 q^{-30} + q^{-32} +4 q^{-34} +2 q^{-36} +3 q^{-38} +5 q^{-40} +2 q^{-42} - q^{-44} - q^{-46} -2 q^{-48} -6 q^{-50} -4 q^{-52} - q^{-54} - q^{-56} -5 q^{-58} - q^{-60} + q^{-62} +3 q^{-70} + q^{-72} + q^{-74} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{-12} - q^{-14} +2 q^{-18} -3 q^{-20} +2 q^{-22} +6 q^{-24} -2 q^{-26} +5 q^{-28} +6 q^{-30} + q^{-34} +2 q^{-36} - q^{-38} -2 q^{-40} -3 q^{-42} -3 q^{-46} -6 q^{-48} +2 q^{-50} -2 q^{-52} -5 q^{-54} +4 q^{-56} + q^{-58} - q^{-60} +3 q^{-62} }[/math]
1,0,0 [math]\displaystyle{ q^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17} + q^{-19} +3 q^{-21} +3 q^{-23} +2 q^{-25} +2 q^{-27} - q^{-29} - q^{-31} -3 q^{-33} - q^{-35} -2 q^{-37} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{-18} - q^{-20} + q^{-24} - q^{-26} - q^{-28} +4 q^{-30} +2 q^{-32} - q^{-34} +4 q^{-36} +7 q^{-38} +3 q^{-40} +2 q^{-42} +8 q^{-44} +8 q^{-46} +3 q^{-50} +5 q^{-52} -3 q^{-54} -6 q^{-56} -2 q^{-58} -8 q^{-60} -13 q^{-62} -8 q^{-64} -5 q^{-66} -6 q^{-68} -3 q^{-70} +6 q^{-72} +6 q^{-74} +2 q^{-76} +3 q^{-78} +4 q^{-80} - q^{-84} }[/math]
1,0,0,0 [math]\displaystyle{ q^{-12} - q^{-14} + q^{-16} - q^{-18} + q^{-22} + q^{-24} +3 q^{-26} +3 q^{-28} +4 q^{-30} +2 q^{-32} +2 q^{-34} - q^{-36} - q^{-38} -3 q^{-40} -3 q^{-42} - q^{-44} -2 q^{-46} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{-12} - q^{-14} +2 q^{-16} -4 q^{-18} +5 q^{-20} -4 q^{-22} +4 q^{-24} +3 q^{-28} +2 q^{-30} -2 q^{-32} +7 q^{-34} -8 q^{-36} +7 q^{-38} -8 q^{-40} +5 q^{-42} -6 q^{-44} +3 q^{-46} -2 q^{-50} +4 q^{-52} -3 q^{-54} +4 q^{-56} -5 q^{-58} +3 q^{-60} -3 q^{-62} }[/math]
1,0 [math]\displaystyle{ q^{-18} - q^{-22} - q^{-24} + q^{-26} +3 q^{-28} -4 q^{-32} - q^{-34} +5 q^{-36} +5 q^{-38} -2 q^{-40} -2 q^{-42} + q^{-44} +6 q^{-46} +2 q^{-48} -2 q^{-50} - q^{-52} +4 q^{-54} +2 q^{-56} -2 q^{-58} -2 q^{-60} + q^{-62} +3 q^{-64} - q^{-66} -4 q^{-68} -2 q^{-70} +2 q^{-72} -5 q^{-76} -4 q^{-78} +2 q^{-80} +3 q^{-82} -3 q^{-84} -5 q^{-86} - q^{-88} +4 q^{-90} +3 q^{-92} -2 q^{-94} -2 q^{-96} + q^{-98} +3 q^{-100} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{-18} - q^{-20} + q^{-22} -2 q^{-24} +3 q^{-26} -4 q^{-28} +4 q^{-30} -2 q^{-32} +5 q^{-34} + q^{-36} +4 q^{-38} +4 q^{-40} +5 q^{-42} +6 q^{-44} -2 q^{-46} +6 q^{-48} -5 q^{-50} +5 q^{-52} -8 q^{-54} +2 q^{-56} -8 q^{-58} +3 q^{-60} -5 q^{-62} -4 q^{-66} -2 q^{-68} + q^{-70} -4 q^{-72} + q^{-74} -4 q^{-76} +4 q^{-78} -2 q^{-80} +4 q^{-82} -2 q^{-84} +3 q^{-86} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +8 q^{-44} -10 q^{-46} +12 q^{-48} -7 q^{-50} - q^{-52} +10 q^{-54} -16 q^{-56} +19 q^{-58} -11 q^{-60} + q^{-62} +10 q^{-64} -14 q^{-66} +13 q^{-68} -2 q^{-70} -6 q^{-72} +14 q^{-74} -12 q^{-76} +4 q^{-78} +9 q^{-80} -15 q^{-82} +21 q^{-84} -16 q^{-86} +9 q^{-88} +5 q^{-90} -13 q^{-92} +22 q^{-94} -22 q^{-96} +16 q^{-98} -4 q^{-100} -7 q^{-102} +13 q^{-104} -16 q^{-106} +9 q^{-108} - q^{-110} -10 q^{-112} +10 q^{-114} -11 q^{-116} -2 q^{-118} +10 q^{-120} -20 q^{-122} +16 q^{-124} -9 q^{-126} -4 q^{-128} +11 q^{-130} -16 q^{-132} +15 q^{-134} -7 q^{-136} + q^{-138} +3 q^{-140} -7 q^{-142} +7 q^{-144} -2 q^{-146} + q^{-148} + q^{-150} }[/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 49"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t^2-6 t+7-6 t^{-1} +3 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 z^4+6 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{ \{5,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 25, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -2 q^9+3 q^8-4 q^7+5 q^6-4 q^5+4 q^4-2 q^3+q^2 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^4 a^{-4} +2 z^4 a^{-6} +2 z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^7 a^{-7} +z^7 a^{-9} +3 z^6 a^{-6} +4 z^6 a^{-8} +z^6 a^{-10} +2 z^5 a^{-5} +z^5 a^{-7} -z^5 a^{-9} +z^4 a^{-4} -8 z^4 a^{-6} -9 z^4 a^{-8} -3 z^3 a^{-5} -3 z^3 a^{-7} +3 z^3 a^{-9} +3 z^3 a^{-11} -2 z^2 a^{-4} +9 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +2 z a^{-7} -2 z a^{-9} -4 z a^{-11} -4 a^{-6} -3 a^{-8} }[/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["9 49"];
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{ 3 t^2-6 t+7-6 t^{-1} +3 t^{-2} }[/math], [math]\displaystyle{ -2 q^9+3 q^8-4 q^7+5 q^6-4 q^5+4 q^4-2 q^3+q^2 }[/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: (6, 14)
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{ 24 }[/math] [math]\displaystyle{ 112 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 700 }[/math] [math]\displaystyle{ 116 }[/math] [math]\displaystyle{ 2688 }[/math] [math]\displaystyle{ \frac{14368}{3} }[/math] [math]\displaystyle{ \frac{2560}{3} }[/math] [math]\displaystyle{ 688 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 6272 }[/math] [math]\displaystyle{ 16800 }[/math] [math]\displaystyle{ 2784 }[/math] [math]\displaystyle{ \frac{166191}{5} }[/math] [math]\displaystyle{ \frac{1876}{5} }[/math] [math]\displaystyle{ \frac{207244}{15} }[/math] [math]\displaystyle{ \frac{689}{3} }[/math] [math]\displaystyle{ \frac{9391}{5} }[/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]4 is the signature of 9 49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 01234567χ
19       2-2
17      1 1
15     32 -1
13    21  1
11   23   1
9  22    0
7  2     2
512      -1
31       1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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^{26}+2 q^{25}-6 q^{24}+q^{23}+10 q^{22}-13 q^{21}-3 q^{20}+21 q^{19}-17 q^{18}-9 q^{17}+27 q^{16}-17 q^{15}-11 q^{14}+25 q^{13}-10 q^{12}-11 q^{11}+16 q^{10}-3 q^9-8 q^8+6 q^7+q^6-2 q^5+q^4 }[/math]
3 [math]\displaystyle{ -2 q^{50}+q^{48}+9 q^{47}-5 q^{46}-12 q^{45}-2 q^{44}+24 q^{43}+8 q^{42}-31 q^{41}-22 q^{40}+39 q^{39}+35 q^{38}-40 q^{37}-51 q^{36}+40 q^{35}+65 q^{34}-40 q^{33}-72 q^{32}+33 q^{31}+80 q^{30}-33 q^{29}-78 q^{28}+22 q^{27}+80 q^{26}-18 q^{25}-70 q^{24}+5 q^{23}+64 q^{22}+2 q^{21}-48 q^{20}-14 q^{19}+39 q^{18}+14 q^{17}-21 q^{16}-18 q^{15}+12 q^{14}+13 q^{13}-3 q^{12}-8 q^{11}+q^{10}+3 q^9+q^8-2 q^7+q^6 }[/math]
4 [math]\displaystyle{ q^{82}+2 q^{81}-6 q^{79}-4 q^{78}-5 q^{77}+14 q^{76}+21 q^{75}-7 q^{74}-18 q^{73}-45 q^{72}+12 q^{71}+65 q^{70}+31 q^{69}-5 q^{68}-120 q^{67}-46 q^{66}+90 q^{65}+105 q^{64}+70 q^{63}-180 q^{62}-142 q^{61}+59 q^{60}+168 q^{59}+182 q^{58}-196 q^{57}-226 q^{56}-q^{55}+192 q^{54}+274 q^{53}-183 q^{52}-269 q^{51}-52 q^{50}+187 q^{49}+324 q^{48}-158 q^{47}-276 q^{46}-86 q^{45}+162 q^{44}+333 q^{43}-116 q^{42}-248 q^{41}-117 q^{40}+110 q^{39}+309 q^{38}-50 q^{37}-181 q^{36}-137 q^{35}+31 q^{34}+240 q^{33}+19 q^{32}-84 q^{31}-122 q^{30}-43 q^{29}+137 q^{28}+46 q^{27}+q^{26}-65 q^{25}-63 q^{24}+44 q^{23}+25 q^{22}+30 q^{21}-13 q^{20}-35 q^{19}+5 q^{18}+15 q^{16}+3 q^{15}-9 q^{14}+q^{13}-2 q^{12}+3 q^{11}+q^{10}-2 q^9+q^8 }[/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 48.gif

9_48

10 1.gif

10_1

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