9 3

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9 2.gif

9_2

9 4.gif

9_4

9 3.gif
(KnotPlot image)

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[edit 9 3 Quick Notes]
[edit 9 3 Further Notes and Views]


Knot presentations

Planar diagram presentation X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X14,6,15,5 X16,8,17,7 X2,12,3,11 X4,14,5,13 X6,16,7,15
Gauss code 1, -7, 2, -8, 5, -9, 6, -1, 3, -4, 7, -2, 8, -5, 9, -6, 4, -3
Dowker-Thistlethwaite code 8 12 14 16 18 2 4 6 10
Conway Notation [63]


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

Length is 10, width is 3,

Braid index is 3

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

[edit Notes on presentations of 9 3]


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 3"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X14,6,15,5 X16,8,17,7 X2,12,3,11 X4,14,5,13 X6,16,7,15
In[5]:= GaussCode[K]
Out[5]= 1, -7, 2, -8, 5, -9, 6, -1, 3, -4, 7, -2, 8, -5, 9, -6, 4, -3
In[6]:= DTCode[K]
Out[6]= 8 12 14 16 18 2 4 6 10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [63]
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}(3,\{1,1,1,1,1,1,1,2,-1,2\}) }[/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]= { 3, 10, 3 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.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 3 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{4, 11}, {3, 5}, {6, 4}, {5, 10}, {2, 6}, {11, 9}, {10, 8}, {9, 7}, {1, 3}, {8, 2}, {7, 1}]
In[14]:= Draw[ap]
9 3 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [5][-16]
Hyperbolic Volume 4.99486
A-Polynomial See Data:9 3/A-polynomial

[edit Notes for 9 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 3 }[/math]
Topological 4 genus [math]\displaystyle{ 3 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -6

[edit Notes for 9 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-3 t^2+3 t-3+3 t^{-1} -3 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+9 z^4+9 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 19, 6 }
Jones polynomial [math]\displaystyle{ -q^{12}+q^{11}-2 q^{10}+3 q^9-3 q^8+3 q^7-2 q^6+2 q^5-q^4+q^3 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} +6 z^2 a^{-6} +7 z^2 a^{-8} -4 z^2 a^{-10} + a^{-6} +3 a^{-8} -3 a^{-10} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -5 z^6 a^{-10} +z^6 a^{-12} -4 z^5 a^{-7} -8 z^5 a^{-9} -3 z^5 a^{-11} +z^5 a^{-13} -5 z^4 a^{-6} +9 z^4 a^{-8} +11 z^4 a^{-10} -2 z^4 a^{-12} +z^4 a^{-14} +3 z^3 a^{-7} +9 z^3 a^{-9} +4 z^3 a^{-11} -z^3 a^{-13} +z^3 a^{-15} +6 z^2 a^{-6} -9 z^2 a^{-8} -11 z^2 a^{-10} +3 z^2 a^{-12} -z^2 a^{-14} -4 z a^{-9} -z a^{-11} +z a^{-13} -2 z a^{-15} - a^{-6} +3 a^{-8} +3 a^{-10} }[/math]
The A2 invariant [math]\displaystyle{ q^{-10} + q^{-14} + q^{-18} + q^{-20} + q^{-22} +2 q^{-24} - q^{-30} - q^{-32} - q^{-34} - q^{-36} }[/math]
The G2 invariant [math]\displaystyle{ q^{-50} + q^{-54} - q^{-56} + q^{-58} +3 q^{-64} -2 q^{-66} +3 q^{-68} - q^{-70} + q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} + q^{-84} - q^{-86} +2 q^{-88} + q^{-90} +2 q^{-94} - q^{-96} + q^{-98} +2 q^{-100} -2 q^{-102} +3 q^{-104} - q^{-106} +2 q^{-108} - q^{-112} +2 q^{-114} -2 q^{-116} +3 q^{-118} -2 q^{-120} + q^{-124} -2 q^{-126} + q^{-128} - q^{-130} - q^{-132} + q^{-134} - q^{-136} - q^{-138} - q^{-142} - q^{-146} -2 q^{-148} - q^{-152} - q^{-156} - q^{-160} - q^{-162} - q^{-164} - q^{-166} +2 q^{-168} -2 q^{-170} + q^{-172} + q^{-178} - q^{-180} + q^{-182} - q^{-184} + q^{-186} - q^{-190} + q^{-192} + q^{-196} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_3.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{-5} + q^{-9} + q^{-13} + q^{-19} - q^{-21} - q^{-25} }[/math]
2 [math]\displaystyle{ q^{-10} +2 q^{-16} + q^{-18} - q^{-20} + q^{-22} + q^{-24} - q^{-26} + q^{-30} + q^{-36} - q^{-40} - q^{-48} + q^{-50} - q^{-52} -2 q^{-54} - q^{-58} + q^{-62} + q^{-68} }[/math]
3 [math]\displaystyle{ q^{-15} + q^{-21} +2 q^{-23} + q^{-25} - q^{-27} - q^{-29} +2 q^{-31} +2 q^{-33} -2 q^{-37} +2 q^{-41} + q^{-43} - q^{-45} - q^{-47} + q^{-51} + q^{-53} - q^{-57} +2 q^{-61} -2 q^{-65} - q^{-67} + q^{-69} - q^{-71} -2 q^{-73} + q^{-77} - q^{-81} - q^{-83} - q^{-85} + q^{-87} + q^{-89} -2 q^{-91} - q^{-93} - q^{-95} +2 q^{-97} - q^{-101} - q^{-103} + q^{-105} +2 q^{-107} + q^{-109} - q^{-111} +2 q^{-115} + q^{-117} - q^{-121} - q^{-129} }[/math]
4 [math]\displaystyle{ q^{-20} + q^{-26} + q^{-28} +2 q^{-30} - q^{-34} +4 q^{-40} +2 q^{-42} - q^{-44} -2 q^{-46} -3 q^{-48} +2 q^{-50} +3 q^{-52} +2 q^{-54} -4 q^{-58} - q^{-60} + q^{-62} +2 q^{-64} +2 q^{-66} - q^{-68} - q^{-72} - q^{-74} + q^{-76} + q^{-78} +3 q^{-80} -4 q^{-84} -3 q^{-86} +4 q^{-90} +2 q^{-92} -4 q^{-94} -4 q^{-96} - q^{-98} +4 q^{-100} +3 q^{-102} -3 q^{-104} -4 q^{-106} - q^{-108} +2 q^{-110} +2 q^{-112} -2 q^{-114} -2 q^{-116} + q^{-120} + q^{-122} -2 q^{-124} -2 q^{-126} + q^{-130} + q^{-132} + q^{-134} - q^{-136} -3 q^{-138} -2 q^{-140} +2 q^{-142} +6 q^{-144} +3 q^{-146} - q^{-148} -7 q^{-150} -3 q^{-152} +6 q^{-154} +6 q^{-156} +2 q^{-158} -7 q^{-160} -4 q^{-162} +3 q^{-164} +5 q^{-166} +4 q^{-168} -3 q^{-170} -4 q^{-172} + q^{-174} +2 q^{-176} +3 q^{-178} -2 q^{-180} -3 q^{-182} - q^{-184} +2 q^{-188} - q^{-190} - q^{-192} - q^{-194} - q^{-196} + q^{-198} + q^{-208} }[/math]
5 [math]\displaystyle{ q^{-25} + q^{-31} + q^{-33} + q^{-35} + q^{-37} - q^{-41} +2 q^{-45} +2 q^{-47} +3 q^{-49} + q^{-51} -2 q^{-53} -4 q^{-55} -2 q^{-57} + q^{-59} +4 q^{-61} +5 q^{-63} +2 q^{-65} -2 q^{-67} -5 q^{-69} -4 q^{-71} +3 q^{-75} +5 q^{-77} +3 q^{-79} - q^{-81} -4 q^{-83} -3 q^{-85} + q^{-89} +2 q^{-91} +2 q^{-93} + q^{-95} + q^{-97} + q^{-99} -2 q^{-101} -4 q^{-103} -4 q^{-105} +5 q^{-109} +6 q^{-111} +2 q^{-113} -5 q^{-115} -10 q^{-117} -6 q^{-119} +3 q^{-121} +9 q^{-123} +7 q^{-125} - q^{-127} -9 q^{-129} -10 q^{-131} -3 q^{-133} +7 q^{-135} +10 q^{-137} +4 q^{-139} -4 q^{-141} -9 q^{-143} -6 q^{-145} +2 q^{-147} +8 q^{-149} +5 q^{-151} -3 q^{-153} -7 q^{-155} -7 q^{-157} +6 q^{-161} +5 q^{-163} -4 q^{-167} -3 q^{-169} + q^{-171} +3 q^{-173} +3 q^{-175} -2 q^{-177} -3 q^{-179} +2 q^{-183} +2 q^{-185} + q^{-187} - q^{-189} -2 q^{-191} -2 q^{-193} +2 q^{-197} +4 q^{-199} +6 q^{-201} +2 q^{-203} -5 q^{-205} -8 q^{-207} -4 q^{-209} +3 q^{-211} +11 q^{-213} +13 q^{-215} - q^{-217} -12 q^{-219} -13 q^{-221} -3 q^{-223} +9 q^{-225} +15 q^{-227} +5 q^{-229} -8 q^{-231} -11 q^{-233} -6 q^{-235} +5 q^{-237} +8 q^{-239} +3 q^{-241} -4 q^{-243} -6 q^{-245} -3 q^{-247} +3 q^{-249} +4 q^{-251} + q^{-253} -4 q^{-255} -3 q^{-257} - q^{-259} +2 q^{-261} +2 q^{-263} + q^{-265} -2 q^{-267} -3 q^{-269} - q^{-271} + q^{-273} + q^{-275} +2 q^{-277} + q^{-279} - q^{-281} + q^{-289} + q^{-291} - q^{-305} }[/math]
6 [math]\displaystyle{ q^{-30} + q^{-36} + q^{-38} + q^{-40} + q^{-44} - q^{-48} + q^{-50} +2 q^{-52} +3 q^{-54} + q^{-56} +2 q^{-58} - q^{-60} -4 q^{-62} -3 q^{-64} - q^{-66} +3 q^{-68} +3 q^{-70} +7 q^{-72} +4 q^{-74} -2 q^{-76} -5 q^{-78} -7 q^{-80} -4 q^{-82} -2 q^{-84} +6 q^{-86} +8 q^{-88} +6 q^{-90} +2 q^{-92} -3 q^{-94} -6 q^{-96} -9 q^{-98} -2 q^{-100} +2 q^{-102} +6 q^{-104} +6 q^{-106} +3 q^{-108} + q^{-110} -4 q^{-112} -2 q^{-114} -2 q^{-116} - q^{-120} -2 q^{-122} +5 q^{-128} +6 q^{-130} +5 q^{-132} -3 q^{-134} -11 q^{-136} -11 q^{-138} -9 q^{-140} +2 q^{-142} +12 q^{-144} +17 q^{-146} +9 q^{-148} -5 q^{-150} -15 q^{-152} -20 q^{-154} -11 q^{-156} +4 q^{-158} +18 q^{-160} +19 q^{-162} +9 q^{-164} -3 q^{-166} -18 q^{-168} -20 q^{-170} -11 q^{-172} +5 q^{-174} +16 q^{-176} +17 q^{-178} +12 q^{-180} -5 q^{-182} -18 q^{-184} -19 q^{-186} -9 q^{-188} +5 q^{-190} +16 q^{-192} +20 q^{-194} +7 q^{-196} -10 q^{-198} -19 q^{-200} -15 q^{-202} -3 q^{-204} +10 q^{-206} +20 q^{-208} +11 q^{-210} -5 q^{-212} -14 q^{-214} -12 q^{-216} -3 q^{-218} +7 q^{-220} +16 q^{-222} +10 q^{-224} -4 q^{-226} -10 q^{-228} -8 q^{-230} -2 q^{-232} +4 q^{-234} +8 q^{-236} +4 q^{-238} -4 q^{-240} -4 q^{-242} +2 q^{-246} +3 q^{-248} +3 q^{-250} -3 q^{-254} -3 q^{-256} +3 q^{-260} +4 q^{-262} +4 q^{-264} +2 q^{-266} + q^{-268} -4 q^{-270} -8 q^{-272} -7 q^{-274} -2 q^{-276} +4 q^{-278} +12 q^{-280} +15 q^{-282} +6 q^{-284} -9 q^{-286} -18 q^{-288} -17 q^{-290} -10 q^{-292} +12 q^{-294} +25 q^{-296} +22 q^{-298} +5 q^{-300} -15 q^{-302} -24 q^{-304} -25 q^{-306} -3 q^{-308} +15 q^{-310} +22 q^{-312} +13 q^{-314} - q^{-316} -12 q^{-318} -18 q^{-320} -7 q^{-322} +7 q^{-326} +7 q^{-328} +4 q^{-330} +3 q^{-332} -3 q^{-334} -3 q^{-338} -4 q^{-340} -4 q^{-342} +6 q^{-346} +3 q^{-348} +6 q^{-350} -4 q^{-354} -6 q^{-356} -3 q^{-358} +2 q^{-360} +2 q^{-362} +6 q^{-364} +3 q^{-366} -3 q^{-370} -3 q^{-372} - q^{-374} - q^{-376} +3 q^{-378} +2 q^{-380} +2 q^{-382} - q^{-388} -2 q^{-390} - q^{-394} + q^{-396} - q^{-404} - q^{-408} + q^{-420} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-10} + q^{-14} + q^{-18} + q^{-20} + q^{-22} +2 q^{-24} - q^{-30} - q^{-32} - q^{-34} - q^{-36} }[/math]
1,1 [math]\displaystyle{ q^{-20} +2 q^{-24} -2 q^{-26} +6 q^{-28} -2 q^{-30} +10 q^{-32} -4 q^{-34} +9 q^{-36} -2 q^{-38} +4 q^{-40} -4 q^{-44} +6 q^{-46} -8 q^{-48} +8 q^{-50} -11 q^{-52} +8 q^{-54} -10 q^{-56} +4 q^{-58} -8 q^{-60} -2 q^{-64} -2 q^{-66} +2 q^{-68} +2 q^{-72} +2 q^{-74} + q^{-76} -2 q^{-78} -2 q^{-82} +3 q^{-84} -4 q^{-86} +2 q^{-88} -2 q^{-90} +2 q^{-92} -2 q^{-94} +2 q^{-96} + q^{-100} }[/math]
2,0 [math]\displaystyle{ q^{-20} + q^{-26} +2 q^{-28} + q^{-30} + q^{-32} +2 q^{-34} +2 q^{-36} + q^{-42} + q^{-44} + q^{-46} +2 q^{-48} + q^{-50} - q^{-58} -3 q^{-66} -3 q^{-68} -3 q^{-70} -3 q^{-72} -3 q^{-74} -2 q^{-76} + q^{-80} +2 q^{-82} + q^{-84} +2 q^{-86} + q^{-88} + q^{-90} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{-20} + q^{-24} + q^{-26} + q^{-30} +2 q^{-32} +2 q^{-34} +4 q^{-36} +3 q^{-38} +2 q^{-40} +2 q^{-42} + q^{-44} -2 q^{-46} - q^{-48} -2 q^{-50} -3 q^{-52} -3 q^{-54} -2 q^{-56} - q^{-58} - q^{-60} + q^{-64} - q^{-66} - q^{-68} + q^{-70} - q^{-72} - q^{-74} + q^{-76} + q^{-78} + q^{-82} }[/math]
1,0,0 [math]\displaystyle{ q^{-15} + q^{-19} + q^{-23} +2 q^{-27} +2 q^{-29} +2 q^{-31} +2 q^{-33} -2 q^{-39} - q^{-41} -2 q^{-43} - q^{-45} - q^{-47} }[/math]
1,0,1 [math]\displaystyle{ q^{-30} +2 q^{-34} + q^{-38} +3 q^{-40} +2 q^{-42} +9 q^{-44} +5 q^{-46} +8 q^{-48} +7 q^{-50} +5 q^{-52} +7 q^{-54} +2 q^{-56} +5 q^{-58} - q^{-60} +4 q^{-62} -4 q^{-64} -2 q^{-66} -7 q^{-68} -12 q^{-70} -7 q^{-72} -14 q^{-74} -4 q^{-76} -9 q^{-78} + q^{-80} -3 q^{-82} +2 q^{-84} +3 q^{-86} -2 q^{-88} +6 q^{-90} -2 q^{-92} +6 q^{-94} + q^{-96} +3 q^{-98} +2 q^{-100} - q^{-102} -2 q^{-104} - q^{-106} + q^{-108} -2 q^{-110} +3 q^{-112} -2 q^{-114} - q^{-116} + q^{-118} -2 q^{-120} + q^{-122} - q^{-124} +2 q^{-128} + q^{-132} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{-30} + q^{-34} + q^{-36} + q^{-38} +2 q^{-42} +2 q^{-44} +3 q^{-46} +4 q^{-48} +4 q^{-50} +5 q^{-52} +5 q^{-54} +4 q^{-56} +3 q^{-58} +3 q^{-60} +2 q^{-62} - q^{-64} -3 q^{-66} -4 q^{-68} -5 q^{-70} -7 q^{-72} -5 q^{-74} -4 q^{-76} -4 q^{-78} -2 q^{-80} - q^{-82} -2 q^{-84} -2 q^{-86} + q^{-94} +3 q^{-96} +2 q^{-98} + q^{-100} + q^{-102} + q^{-104} }[/math]
1,0,0,0 [math]\displaystyle{ q^{-20} + q^{-24} + q^{-28} + q^{-32} +2 q^{-34} +2 q^{-36} +3 q^{-38} +2 q^{-40} +2 q^{-42} -2 q^{-48} -2 q^{-50} -2 q^{-52} -2 q^{-54} - q^{-56} - q^{-58} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{-20} + q^{-24} - q^{-26} +2 q^{-28} - q^{-30} +2 q^{-32} +2 q^{-36} + q^{-38} +2 q^{-42} - q^{-44} +2 q^{-46} -3 q^{-48} +2 q^{-50} -3 q^{-52} + q^{-54} -2 q^{-56} + q^{-58} - q^{-60} + q^{-64} - q^{-66} + q^{-68} - q^{-70} + q^{-72} - q^{-74} + q^{-76} - q^{-78} - q^{-82} }[/math]
1,0 [math]\displaystyle{ q^{-30} + q^{-38} + q^{-40} - q^{-44} + q^{-46} +2 q^{-48} + q^{-50} - q^{-52} + q^{-54} +2 q^{-56} +3 q^{-58} + q^{-60} +2 q^{-66} + q^{-68} - q^{-72} - q^{-78} - q^{-80} - q^{-82} - q^{-86} -2 q^{-88} -2 q^{-90} - q^{-96} - q^{-98} + q^{-102} - q^{-106} - q^{-108} + q^{-112} - q^{-116} - q^{-118} + q^{-122} + q^{-124} + q^{-132} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{-30} + q^{-34} +2 q^{-38} - q^{-40} +2 q^{-42} +3 q^{-46} +2 q^{-48} +3 q^{-50} +3 q^{-52} +4 q^{-54} +4 q^{-56} +2 q^{-58} +3 q^{-60} - q^{-62} + q^{-64} -4 q^{-66} - q^{-68} -5 q^{-70} -2 q^{-72} -5 q^{-74} - q^{-76} -3 q^{-78} - q^{-82} + q^{-90} - q^{-92} - q^{-96} + q^{-98} - q^{-100} - q^{-104} + q^{-106} + q^{-110} + q^{-114} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-50} + q^{-54} - q^{-56} + q^{-58} +3 q^{-64} -2 q^{-66} +3 q^{-68} - q^{-70} + q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} + q^{-84} - q^{-86} +2 q^{-88} + q^{-90} +2 q^{-94} - q^{-96} + q^{-98} +2 q^{-100} -2 q^{-102} +3 q^{-104} - q^{-106} +2 q^{-108} - q^{-112} +2 q^{-114} -2 q^{-116} +3 q^{-118} -2 q^{-120} + q^{-124} -2 q^{-126} + q^{-128} - q^{-130} - q^{-132} + q^{-134} - q^{-136} - q^{-138} - q^{-142} - q^{-146} -2 q^{-148} - q^{-152} - q^{-156} - q^{-160} - q^{-162} - q^{-164} - q^{-166} +2 q^{-168} -2 q^{-170} + q^{-172} + q^{-178} - q^{-180} + q^{-182} - q^{-184} + q^{-186} - q^{-190} + q^{-192} + q^{-196} }[/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 3"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-3 t^2+3 t-3+3 t^{-1} -3 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+9 z^4+9 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]= { 19, 6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^{12}+q^{11}-2 q^{10}+3 q^9-3 q^8+3 q^7-2 q^6+2 q^5-q^4+q^3 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} +6 z^2 a^{-6} +7 z^2 a^{-8} -4 z^2 a^{-10} + a^{-6} +3 a^{-8} -3 a^{-10} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -5 z^6 a^{-10} +z^6 a^{-12} -4 z^5 a^{-7} -8 z^5 a^{-9} -3 z^5 a^{-11} +z^5 a^{-13} -5 z^4 a^{-6} +9 z^4 a^{-8} +11 z^4 a^{-10} -2 z^4 a^{-12} +z^4 a^{-14} +3 z^3 a^{-7} +9 z^3 a^{-9} +4 z^3 a^{-11} -z^3 a^{-13} +z^3 a^{-15} +6 z^2 a^{-6} -9 z^2 a^{-8} -11 z^2 a^{-10} +3 z^2 a^{-12} -z^2 a^{-14} -4 z a^{-9} -z a^{-11} +z a^{-13} -2 z a^{-15} - a^{-6} +3 a^{-8} +3 a^{-10} }[/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 3"];
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^3-3 t^2+3 t-3+3 t^{-1} -3 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ -q^{12}+q^{11}-2 q^{10}+3 q^9-3 q^8+3 q^7-2 q^6+2 q^5-q^4+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: (9, 26)
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{ 36 }[/math] [math]\displaystyle{ 208 }[/math] [math]\displaystyle{ 648 }[/math] [math]\displaystyle{ 1578 }[/math] [math]\displaystyle{ 246 }[/math] [math]\displaystyle{ 7488 }[/math] [math]\displaystyle{ \frac{39616}{3} }[/math] [math]\displaystyle{ \frac{7072}{3} }[/math] [math]\displaystyle{ 1744 }[/math] [math]\displaystyle{ 7776 }[/math] [math]\displaystyle{ 21632 }[/math] [math]\displaystyle{ 56808 }[/math] [math]\displaystyle{ 8856 }[/math] [math]\displaystyle{ \frac{1125053}{10} }[/math] [math]\displaystyle{ \frac{16214}{5} }[/math] [math]\displaystyle{ \frac{218942}{5} }[/math] [math]\displaystyle{ \frac{1313}{2} }[/math] [math]\displaystyle{ \frac{57373}{10} }[/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]6 is the signature of 9 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 0123456789χ
25         1-1
23          0
21       21 -1
19      1   1
17     22   0
15    11    0
13   12     1
11  11      0
9  1       1
711        0
51         1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ i=7 }[/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} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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]
[math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=9 }[/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^{33}-q^{32}+2 q^{30}-2 q^{29}-q^{28}+3 q^{27}-4 q^{26}+5 q^{24}-6 q^{23}+q^{22}+5 q^{21}-6 q^{20}+6 q^{18}-5 q^{17}-q^{16}+6 q^{15}-4 q^{14}-2 q^{13}+5 q^{12}-2 q^{11}-2 q^{10}+3 q^9-q^7+q^6 }[/math]
3 [math]\displaystyle{ -q^{63}+q^{62}-2 q^{59}+2 q^{58}+q^{57}+q^{56}-4 q^{55}+q^{54}+3 q^{53}+2 q^{52}-5 q^{51}-q^{50}+3 q^{49}+3 q^{48}-3 q^{47}-4 q^{46}+3 q^{45}+2 q^{44}-4 q^{42}+q^{41}+2 q^{40}-3 q^{38}+2 q^{37}+q^{36}-2 q^{35}-2 q^{34}+4 q^{33}-q^{32}-3 q^{31}+6 q^{29}-3 q^{28}-4 q^{27}+q^{26}+7 q^{25}-3 q^{24}-5 q^{23}+7 q^{21}-q^{20}-4 q^{19}-2 q^{18}+5 q^{17}+q^{16}-2 q^{15}-2 q^{14}+2 q^{13}+q^{12}-q^{10}+q^9 }[/math]
4 [math]\displaystyle{ q^{102}-q^{101}+2 q^{97}-3 q^{96}+5 q^{92}-5 q^{91}-q^{90}-2 q^{89}+q^{88}+10 q^{87}-6 q^{86}-2 q^{85}-7 q^{84}+2 q^{83}+17 q^{82}-5 q^{81}-4 q^{80}-14 q^{79}-q^{78}+26 q^{77}-q^{76}-4 q^{75}-23 q^{74}-5 q^{73}+32 q^{72}+3 q^{71}-q^{70}-27 q^{69}-9 q^{68}+31 q^{67}+5 q^{66}+q^{65}-27 q^{64}-9 q^{63}+30 q^{62}+3 q^{61}+q^{60}-24 q^{59}-9 q^{58}+29 q^{57}+q^{56}+q^{55}-20 q^{54}-9 q^{53}+26 q^{52}-2 q^{51}+2 q^{50}-14 q^{49}-8 q^{48}+21 q^{47}-5 q^{46}+2 q^{45}-8 q^{44}-6 q^{43}+17 q^{42}-8 q^{41}+q^{40}-4 q^{39}-3 q^{38}+15 q^{37}-8 q^{36}-q^{35}-4 q^{34}-2 q^{33}+14 q^{32}-5 q^{31}-q^{30}-5 q^{29}-4 q^{28}+11 q^{27}-q^{26}+q^{25}-4 q^{24}-5 q^{23}+6 q^{22}+2 q^{20}-q^{19}-3 q^{18}+2 q^{17}+q^{15}-q^{13}+q^{12} }[/math]
5 [math]\displaystyle{ -q^{150}+q^{149}-q^{144}+2 q^{143}-q^{141}-2 q^{138}+4 q^{137}+q^{136}-2 q^{135}-2 q^{133}-4 q^{132}+5 q^{131}+4 q^{130}-q^{129}-5 q^{127}-6 q^{126}+4 q^{125}+9 q^{124}+2 q^{123}-q^{122}-11 q^{121}-9 q^{120}+6 q^{119}+16 q^{118}+7 q^{117}-4 q^{116}-22 q^{115}-14 q^{114}+9 q^{113}+29 q^{112}+17 q^{111}-10 q^{110}-34 q^{109}-24 q^{108}+10 q^{107}+40 q^{106}+31 q^{105}-12 q^{104}-42 q^{103}-31 q^{102}+6 q^{101}+43 q^{100}+38 q^{99}-8 q^{98}-44 q^{97}-33 q^{96}+4 q^{95}+41 q^{94}+38 q^{93}-7 q^{92}-42 q^{91}-32 q^{90}+4 q^{89}+39 q^{88}+35 q^{87}-6 q^{86}-37 q^{85}-32 q^{84}+2 q^{83}+35 q^{82}+34 q^{81}-2 q^{80}-32 q^{79}-31 q^{78}-4 q^{77}+28 q^{76}+34 q^{75}+2 q^{74}-24 q^{73}-28 q^{72}-10 q^{71}+20 q^{70}+31 q^{69}+7 q^{68}-16 q^{67}-22 q^{66}-13 q^{65}+10 q^{64}+24 q^{63}+8 q^{62}-8 q^{61}-14 q^{60}-11 q^{59}+4 q^{58}+15 q^{57}+4 q^{56}-3 q^{55}-7 q^{54}-7 q^{53}+3 q^{52}+10 q^{51}-3 q^{49}-5 q^{48}-4 q^{47}+3 q^{46}+10 q^{45}+q^{44}-3 q^{43}-6 q^{42}-5 q^{41}+9 q^{39}+4 q^{38}+q^{37}-4 q^{36}-7 q^{35}-3 q^{34}+5 q^{33}+3 q^{32}+4 q^{31}-4 q^{29}-4 q^{28}+2 q^{27}+2 q^{25}+2 q^{24}-q^{23}-2 q^{22}+q^{21}+q^{18}-q^{16}+q^{15} }[/math]
6 [math]\displaystyle{ q^{207}-q^{206}-q^{201}+2 q^{200}-2 q^{199}+q^{198}+q^{195}-3 q^{194}+3 q^{193}-4 q^{192}+2 q^{191}+q^{190}+3 q^{188}-3 q^{187}+4 q^{186}-8 q^{185}+2 q^{184}-q^{183}+6 q^{181}+7 q^{179}-12 q^{178}+2 q^{177}-6 q^{176}-3 q^{175}+8 q^{174}+4 q^{173}+13 q^{172}-15 q^{171}+5 q^{170}-12 q^{169}-7 q^{168}+8 q^{167}+5 q^{166}+16 q^{165}-18 q^{164}+11 q^{163}-11 q^{162}-4 q^{161}+8 q^{160}-2 q^{159}+9 q^{158}-29 q^{157}+17 q^{156}+10 q^{154}+17 q^{153}-9 q^{152}-9 q^{151}-51 q^{150}+18 q^{149}+9 q^{148}+30 q^{147}+34 q^{146}-6 q^{145}-22 q^{144}-73 q^{143}+11 q^{142}+8 q^{141}+39 q^{140}+49 q^{139}+3 q^{138}-25 q^{137}-81 q^{136}+5 q^{135}+3 q^{134}+38 q^{133}+53 q^{132}+8 q^{131}-24 q^{130}-79 q^{129}+5 q^{128}+2 q^{127}+35 q^{126}+50 q^{125}+8 q^{124}-21 q^{123}-76 q^{122}+5 q^{121}+q^{120}+33 q^{119}+46 q^{118}+8 q^{117}-13 q^{116}-72 q^{115}+q^{114}-5 q^{113}+27 q^{112}+44 q^{111}+14 q^{110}+q^{109}-66 q^{108}-8 q^{107}-15 q^{106}+18 q^{105}+42 q^{104}+23 q^{103}+17 q^{102}-57 q^{101}-18 q^{100}-28 q^{99}+6 q^{98}+38 q^{97}+32 q^{96}+34 q^{95}-44 q^{94}-22 q^{93}-39 q^{92}-8 q^{91}+28 q^{90}+33 q^{89}+47 q^{88}-27 q^{87}-17 q^{86}-40 q^{85}-19 q^{84}+12 q^{83}+24 q^{82}+49 q^{81}-12 q^{80}-5 q^{79}-30 q^{78}-20 q^{77}-2 q^{76}+9 q^{75}+40 q^{74}-7 q^{73}+5 q^{72}-16 q^{71}-12 q^{70}-7 q^{69}-q^{68}+29 q^{67}-9 q^{66}+5 q^{65}-8 q^{64}-4 q^{63}-6 q^{62}-2 q^{61}+24 q^{60}-9 q^{59}+3 q^{58}-7 q^{57}-3 q^{56}-8 q^{55}-2 q^{54}+22 q^{53}-4 q^{52}+5 q^{51}-4 q^{50}-3 q^{49}-12 q^{48}-6 q^{47}+15 q^{46}-q^{45}+8 q^{44}+q^{43}+q^{42}-10 q^{41}-8 q^{40}+7 q^{39}-3 q^{38}+5 q^{37}+3 q^{36}+4 q^{35}-4 q^{34}-5 q^{33}+3 q^{32}-3 q^{31}+q^{30}+q^{29}+3 q^{28}-q^{27}-2 q^{26}+2 q^{25}-q^{24}+q^{21}-q^{19}+q^{18} }[/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 2.gif

9_2

9 4.gif

9_4

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