7 3

From Knot Atlas
Jump to navigationJump to search

7 2.gif

7_2

7 4.gif

7_4

7 3.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 3's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 7 3 at Knotilus!

[edit 7 3 Quick Notes]
[edit 7 3 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X14,8,1,7 X8,14,9,13 X12,6,13,5 X2,10,3,9 X4,12,5,11
Gauss code 1, -6, 2, -7, 5, -1, 3, -4, 6, -2, 7, -5, 4, -3
Dowker-Thistlethwaite code 6 10 12 14 2 4 8
Conway Notation [43]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 7 3]

Knot 7_3.
A graph, knot 7_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["7 3"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X10,4,11,3 X14,8,1,7 X8,14,9,13 X12,6,13,5 X2,10,3,9 X4,12,5,11
In[5]:= GaussCode[K]
Out[5]= 1, -6, 2, -7, 5, -1, 3, -4, 6, -2, 7, -5, 4, -3
In[6]:= DTCode[K]
Out[6]= 6 10 12 14 2 4 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [43]
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,\{1,1,1,1,1,2,-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, 8, 3 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.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."
7 3 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{4, 9}, {3, 5}, {6, 4}, {5, 8}, {2, 6}, {9, 7}, {1, 3}, {8, 2}, {7, 1}]
In[14]:= Draw[ap]
7 3 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index [math]\displaystyle{ \{3,4\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [math]\displaystyle{ \text{$\$$Failed} }[/math]
Hyperbolic Volume 4.59213
A-Polynomial See Data:7 3/A-polynomial

[edit Notes for 7 3'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{ \textrm{ConcordanceGenus}(\textrm{Knot}(7,3)) }[/math]
Rasmussen s-Invariant -4

[edit Notes for 7 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^2+2 t^{-2} -3 t-3 t^{-1} +3 }[/math]
Conway polynomial [math]\displaystyle{ 2 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 13, 4 }
Jones polynomial [math]\displaystyle{ -q^9+q^8-2 q^7+3 q^6-2 q^5+2 q^4-q^3+q^2 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^2 a^{-8} -2 a^{-8} +z^4 a^{-6} +3 z^2 a^{-6} +2 a^{-6} +z^4 a^{-4} +3 z^2 a^{-4} + a^{-4} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^3 a^{-11} -2 z a^{-11} +z^4 a^{-10} -z^2 a^{-10} +z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +z^6 a^{-8} -3 z^4 a^{-8} +6 z^2 a^{-8} -2 a^{-8} +2 z^5 a^{-7} -4 z^3 a^{-7} +3 z a^{-7} +z^6 a^{-6} -3 z^4 a^{-6} +4 z^2 a^{-6} -2 a^{-6} +z^5 a^{-5} -2 z^3 a^{-5} +z^4 a^{-4} -3 z^2 a^{-4} + a^{-4} }[/math]
The A2 invariant [math]\displaystyle{ q^{-6} + q^{-10} + q^{-14} +2 q^{-16} + q^{-18} + q^{-20} - q^{-22} - q^{-24} - q^{-26} - q^{-28} }[/math]
The G2 invariant [math]\displaystyle{ q^{-30} + q^{-34} - q^{-36} + q^{-38} + q^{-40} - q^{-42} +3 q^{-44} - q^{-46} +2 q^{-48} - q^{-52} +2 q^{-54} -2 q^{-56} +2 q^{-58} - q^{-62} +2 q^{-64} + q^{-68} +2 q^{-70} - q^{-72} +2 q^{-74} +3 q^{-80} -2 q^{-82} +3 q^{-84} +2 q^{-88} + q^{-90} -3 q^{-92} +3 q^{-94} -3 q^{-96} +2 q^{-98} - q^{-100} -3 q^{-102} + q^{-104} - q^{-106} - q^{-108} -3 q^{-112} - q^{-114} - q^{-116} -2 q^{-118} +2 q^{-120} -3 q^{-122} + q^{-124} - q^{-128} + q^{-130} - q^{-132} + q^{-134} - q^{-136} + q^{-138} - q^{-142} + q^{-144} + q^{-148} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 7_3.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{-3} + q^{-7} + q^{-11} + q^{-13} - q^{-15} - q^{-19} }[/math]
2 [math]\displaystyle{ q^{-6} +2 q^{-12} + q^{-14} - q^{-16} + q^{-18} + q^{-20} - q^{-22} + q^{-24} + q^{-26} - q^{-28} + q^{-34} -2 q^{-36} - q^{-38} + q^{-40} -2 q^{-42} - q^{-44} + q^{-46} + q^{-52} }[/math]
3 [math]\displaystyle{ q^{-9} + q^{-15} +2 q^{-17} + q^{-19} - q^{-21} - q^{-23} +2 q^{-25} +2 q^{-27} -2 q^{-31} +3 q^{-35} + q^{-37} -2 q^{-39} - q^{-41} +2 q^{-43} + q^{-45} -2 q^{-47} - q^{-49} + q^{-51} -2 q^{-55} - q^{-57} - q^{-59} +2 q^{-63} -2 q^{-65} -2 q^{-67} +3 q^{-71} - q^{-73} -3 q^{-75} +3 q^{-79} + q^{-81} - q^{-83} + q^{-87} + q^{-89} - q^{-99} }[/math]
4 [math]\displaystyle{ q^{-12} + q^{-18} + q^{-20} +2 q^{-22} - q^{-26} +4 q^{-32} +2 q^{-34} - q^{-36} -2 q^{-38} -3 q^{-40} +2 q^{-42} +3 q^{-44} +3 q^{-46} -5 q^{-50} - q^{-52} +2 q^{-54} +5 q^{-56} +2 q^{-58} -5 q^{-60} -4 q^{-62} - q^{-64} +4 q^{-66} +3 q^{-68} -4 q^{-70} -3 q^{-72} - q^{-74} +3 q^{-76} +2 q^{-78} -3 q^{-80} -2 q^{-82} - q^{-84} + q^{-86} -2 q^{-90} - q^{-92} - q^{-94} - q^{-96} + q^{-98} +4 q^{-100} - q^{-102} -2 q^{-104} -2 q^{-106} + q^{-108} +7 q^{-110} + q^{-112} -2 q^{-114} -5 q^{-116} - q^{-118} +7 q^{-120} +3 q^{-122} - q^{-124} -4 q^{-126} -3 q^{-128} +3 q^{-130} +2 q^{-132} +2 q^{-134} - q^{-136} -2 q^{-138} - q^{-142} + q^{-144} - q^{-148} - q^{-152} + q^{-160} }[/math]
5 [math]\displaystyle{ q^{-15} + q^{-21} + q^{-23} + q^{-25} + q^{-27} - q^{-31} +2 q^{-35} +2 q^{-37} +3 q^{-39} + q^{-41} -2 q^{-43} -4 q^{-45} -2 q^{-47} + q^{-49} +4 q^{-51} +5 q^{-53} +2 q^{-55} - q^{-57} -5 q^{-59} -5 q^{-61} +4 q^{-65} +7 q^{-67} +5 q^{-69} -2 q^{-71} -8 q^{-73} -8 q^{-75} - q^{-77} +6 q^{-79} +9 q^{-81} +4 q^{-83} -5 q^{-85} -11 q^{-87} -7 q^{-89} +3 q^{-91} +9 q^{-93} +7 q^{-95} -2 q^{-97} -9 q^{-99} -8 q^{-101} +7 q^{-105} +5 q^{-107} - q^{-109} -6 q^{-111} -5 q^{-113} +4 q^{-117} +3 q^{-119} - q^{-121} -3 q^{-123} - q^{-125} +2 q^{-129} + q^{-131} - q^{-133} - q^{-137} - q^{-139} - q^{-141} +2 q^{-143} +6 q^{-145} + q^{-147} - q^{-149} -3 q^{-151} -4 q^{-153} +2 q^{-155} +10 q^{-157} +7 q^{-159} -7 q^{-163} -11 q^{-165} -3 q^{-167} +8 q^{-169} +11 q^{-171} +4 q^{-173} -7 q^{-175} -12 q^{-177} -7 q^{-179} +3 q^{-181} +9 q^{-183} +7 q^{-185} + q^{-187} -6 q^{-189} -6 q^{-191} -3 q^{-193} +2 q^{-195} +4 q^{-197} +3 q^{-199} -2 q^{-203} -3 q^{-205} - q^{-207} + q^{-211} +2 q^{-213} - q^{-217} + q^{-225} + q^{-227} - q^{-235} }[/math]
6 [math]\displaystyle{ q^{-18} + q^{-24} + q^{-26} + q^{-28} + q^{-32} - q^{-36} + q^{-38} +2 q^{-40} +3 q^{-42} + q^{-44} +2 q^{-46} - q^{-48} -4 q^{-50} -3 q^{-52} - q^{-54} +3 q^{-56} +3 q^{-58} +7 q^{-60} +4 q^{-62} -2 q^{-64} -5 q^{-66} -6 q^{-68} -4 q^{-70} -3 q^{-72} +6 q^{-74} +9 q^{-76} +8 q^{-78} +3 q^{-80} -2 q^{-82} -8 q^{-84} -14 q^{-86} -5 q^{-88} +2 q^{-90} +10 q^{-92} +13 q^{-94} +10 q^{-96} - q^{-98} -16 q^{-100} -16 q^{-102} -12 q^{-104} + q^{-106} +13 q^{-108} +21 q^{-110} +12 q^{-112} -6 q^{-114} -16 q^{-116} -21 q^{-118} -10 q^{-120} +5 q^{-122} +21 q^{-124} +19 q^{-126} +2 q^{-128} -11 q^{-130} -21 q^{-132} -15 q^{-134} - q^{-136} +16 q^{-138} +17 q^{-140} +5 q^{-142} -5 q^{-144} -14 q^{-146} -11 q^{-148} -2 q^{-150} +11 q^{-152} +11 q^{-154} +2 q^{-156} -3 q^{-158} -8 q^{-160} -5 q^{-162} - q^{-164} +6 q^{-166} +5 q^{-168} -2 q^{-174} + q^{-178} +3 q^{-180} +2 q^{-182} -2 q^{-190} -2 q^{-192} -2 q^{-194} +2 q^{-196} +8 q^{-198} +3 q^{-200} +2 q^{-202} -3 q^{-204} -7 q^{-206} -8 q^{-208} - q^{-210} +14 q^{-212} +12 q^{-214} +9 q^{-216} -3 q^{-218} -16 q^{-220} -23 q^{-222} -11 q^{-224} +12 q^{-226} +19 q^{-228} +20 q^{-230} +4 q^{-232} -15 q^{-234} -28 q^{-236} -20 q^{-238} +4 q^{-240} +16 q^{-242} +24 q^{-244} +14 q^{-246} -2 q^{-248} -18 q^{-250} -20 q^{-252} -6 q^{-254} +3 q^{-256} +13 q^{-258} +14 q^{-260} +8 q^{-262} -4 q^{-264} -9 q^{-266} -7 q^{-268} -6 q^{-270} + q^{-272} +6 q^{-274} +6 q^{-276} + q^{-278} - q^{-280} - q^{-282} -4 q^{-284} -2 q^{-286} + q^{-288} +3 q^{-290} + q^{-292} + q^{-294} + q^{-296} -2 q^{-298} - q^{-300} + q^{-304} + q^{-310} - q^{-312} - q^{-314} - q^{-316} + q^{-324} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-6} + q^{-10} + q^{-14} +2 q^{-16} + q^{-18} + q^{-20} - q^{-22} - q^{-24} - q^{-26} - q^{-28} }[/math]
1,1 [math]\displaystyle{ q^{-12} +2 q^{-16} -2 q^{-18} +6 q^{-20} +8 q^{-24} -2 q^{-26} +5 q^{-28} +2 q^{-34} -4 q^{-36} +6 q^{-38} -8 q^{-40} +4 q^{-42} -11 q^{-44} +4 q^{-46} -8 q^{-48} +4 q^{-50} - q^{-52} +4 q^{-56} -2 q^{-58} +3 q^{-60} -6 q^{-62} +2 q^{-64} -2 q^{-66} +2 q^{-68} -2 q^{-70} +2 q^{-72} + q^{-76} }[/math]
2,0 [math]\displaystyle{ q^{-12} + q^{-18} +2 q^{-20} + q^{-22} + q^{-24} +2 q^{-26} +2 q^{-28} + q^{-30} + q^{-32} + q^{-34} +2 q^{-40} - q^{-46} - q^{-48} -3 q^{-50} -3 q^{-52} -2 q^{-54} -2 q^{-56} -2 q^{-58} - q^{-60} + q^{-62} + q^{-64} +2 q^{-66} + q^{-68} + q^{-70} }[/math]
3,0 [math]\displaystyle{ q^{-18} +2 q^{-26} +3 q^{-28} +2 q^{-30} +2 q^{-36} +4 q^{-38} +2 q^{-40} +3 q^{-46} +4 q^{-48} +2 q^{-50} + q^{-54} +2 q^{-56} +2 q^{-58} - q^{-64} - q^{-66} -4 q^{-68} -4 q^{-70} -2 q^{-72} -3 q^{-74} -3 q^{-76} -5 q^{-78} -2 q^{-80} - q^{-82} -3 q^{-86} -3 q^{-88} - q^{-90} + q^{-92} -2 q^{-96} - q^{-98} +2 q^{-100} +4 q^{-102} +4 q^{-104} +3 q^{-106} +2 q^{-108} +3 q^{-110} +2 q^{-112} +2 q^{-114} - q^{-118} -2 q^{-120} -2 q^{-122} - q^{-124} - q^{-126} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{-12} + q^{-16} + q^{-18} +2 q^{-22} +3 q^{-24} + q^{-26} +3 q^{-28} +3 q^{-30} + q^{-32} -2 q^{-38} -2 q^{-40} -3 q^{-42} - q^{-44} -2 q^{-46} -2 q^{-48} + q^{-50} - q^{-52} - q^{-54} + q^{-56} + q^{-58} + q^{-62} }[/math]
1,0,0 [math]\displaystyle{ q^{-9} + q^{-13} + q^{-17} + q^{-19} +2 q^{-21} +2 q^{-23} + q^{-25} + q^{-27} - q^{-29} - q^{-31} -2 q^{-33} - q^{-35} - q^{-37} }[/math]
1,0,1 [math]\displaystyle{ q^{-18} +2 q^{-22} + q^{-26} +5 q^{-28} +2 q^{-30} +9 q^{-32} +5 q^{-34} +6 q^{-36} +8 q^{-38} +9 q^{-42} - q^{-44} + q^{-48} -8 q^{-50} -4 q^{-52} -7 q^{-54} -8 q^{-56} -7 q^{-58} -3 q^{-60} -6 q^{-62} - q^{-66} +7 q^{-70} -2 q^{-72} +7 q^{-74} +2 q^{-76} -3 q^{-78} +3 q^{-80} -4 q^{-82} - q^{-84} + q^{-86} -2 q^{-88} + q^{-90} - q^{-92} +2 q^{-96} + q^{-100} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{-12} + q^{-16} - q^{-18} +2 q^{-20} + q^{-24} + q^{-26} + q^{-28} + q^{-30} - q^{-32} +2 q^{-34} -2 q^{-36} +2 q^{-38} -2 q^{-40} + q^{-42} - q^{-44} - q^{-50} + q^{-52} - q^{-54} + q^{-56} - q^{-58} - q^{-62} }[/math]
1,0 [math]\displaystyle{ q^{-18} + q^{-26} + q^{-28} - q^{-32} + q^{-34} +2 q^{-36} +2 q^{-38} + q^{-42} + q^{-44} +2 q^{-46} + q^{-48} + q^{-54} - q^{-58} - q^{-60} - q^{-66} -2 q^{-68} - q^{-70} - q^{-74} -2 q^{-76} - q^{-78} + q^{-80} - q^{-84} - q^{-86} + q^{-90} + q^{-92} + q^{-100} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{-18} + q^{-22} +2 q^{-26} +2 q^{-30} + q^{-32} +2 q^{-34} +2 q^{-36} +2 q^{-38} +3 q^{-40} +3 q^{-42} +3 q^{-44} +2 q^{-48} -2 q^{-50} -4 q^{-54} -2 q^{-56} -4 q^{-58} - q^{-60} -2 q^{-62} - q^{-64} - q^{-66} - q^{-68} + q^{-70} - q^{-72} - q^{-76} + q^{-78} + q^{-82} + q^{-86} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-30} + q^{-34} - q^{-36} + q^{-38} + q^{-40} - q^{-42} +3 q^{-44} - q^{-46} +2 q^{-48} - q^{-52} +2 q^{-54} -2 q^{-56} +2 q^{-58} - q^{-62} +2 q^{-64} + q^{-68} +2 q^{-70} - q^{-72} +2 q^{-74} +3 q^{-80} -2 q^{-82} +3 q^{-84} +2 q^{-88} + q^{-90} -3 q^{-92} +3 q^{-94} -3 q^{-96} +2 q^{-98} - q^{-100} -3 q^{-102} + q^{-104} - q^{-106} - q^{-108} -3 q^{-112} - q^{-114} - q^{-116} -2 q^{-118} +2 q^{-120} -3 q^{-122} + q^{-124} - q^{-128} + q^{-130} - q^{-132} + q^{-134} - q^{-136} + q^{-138} - q^{-142} + q^{-144} + q^{-148} }[/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["7 3"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^2+2 t^{-2} -3 t-3 t^{-1} +3 }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^4+5 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]= { 13, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^9+q^8-2 q^7+3 q^6-2 q^5+2 q^4-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^2 a^{-8} -2 a^{-8} +z^4 a^{-6} +3 z^2 a^{-6} +2 a^{-6} +z^4 a^{-4} +3 z^2 a^{-4} + a^{-4} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^3 a^{-11} -2 z a^{-11} +z^4 a^{-10} -z^2 a^{-10} +z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +z^6 a^{-8} -3 z^4 a^{-8} +6 z^2 a^{-8} -2 a^{-8} +2 z^5 a^{-7} -4 z^3 a^{-7} +3 z a^{-7} +z^6 a^{-6} -3 z^4 a^{-6} +4 z^2 a^{-6} -2 a^{-6} +z^5 a^{-5} -2 z^3 a^{-5} +z^4 a^{-4} -3 z^2 a^{-4} + 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["7 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^2+2 t^{-2} -3 t-3 t^{-1} +3 }[/math], [math]\displaystyle{ -q^9+q^8-2 q^7+3 q^6-2 q^5+2 q^4-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: (5, 11)
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{ 20 }[/math] [math]\displaystyle{ 88 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1510}{3} }[/math] [math]\displaystyle{ \frac{242}{3} }[/math] [math]\displaystyle{ 1760 }[/math] [math]\displaystyle{ \frac{9520}{3} }[/math] [math]\displaystyle{ \frac{1696}{3} }[/math] [math]\displaystyle{ 440 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 3872 }[/math] [math]\displaystyle{ \frac{30200}{3} }[/math] [math]\displaystyle{ \frac{4840}{3} }[/math] [math]\displaystyle{ \frac{121855}{6} }[/math] [math]\displaystyle{ 382 }[/math] [math]\displaystyle{ \frac{73862}{9} }[/math] [math]\displaystyle{ \frac{2437}{18} }[/math] [math]\displaystyle{ \frac{6559}{6} }[/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 7 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 01234567χ
19       1-1
17        0
15     21 -1
13    1   1
11   12   1
9  11    0
7  1     1
511      0
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} }[/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}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=7 }[/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^{25}-q^{24}+2 q^{22}-3 q^{21}-q^{20}+5 q^{19}-5 q^{18}-2 q^{17}+8 q^{16}-6 q^{15}-2 q^{14}+7 q^{13}-4 q^{12}-2 q^{11}+5 q^{10}-2 q^9-2 q^8+3 q^7-q^5+q^4 }[/math]
3 [math]\displaystyle{ -q^{48}+q^{47}-q^{44}+2 q^{43}-q^{41}-2 q^{40}+4 q^{39}+2 q^{38}-4 q^{37}-5 q^{36}+6 q^{35}+6 q^{34}-7 q^{33}-7 q^{32}+6 q^{31}+10 q^{30}-9 q^{29}-8 q^{28}+6 q^{27}+9 q^{26}-7 q^{25}-7 q^{24}+4 q^{23}+8 q^{22}-4 q^{21}-6 q^{20}+q^{19}+7 q^{18}-q^{17}-4 q^{16}-2 q^{15}+5 q^{14}+q^{13}-2 q^{12}-2 q^{11}+2 q^{10}+q^9-q^7+q^6 }[/math]
4 [math]\displaystyle{ q^{78}-q^{77}-q^{74}+2 q^{73}-2 q^{72}+q^{71}+q^{70}-3 q^{69}+3 q^{68}-4 q^{67}+2 q^{66}+4 q^{65}-3 q^{64}+4 q^{63}-10 q^{62}+q^{61}+7 q^{60}+q^{59}+8 q^{58}-18 q^{57}-3 q^{56}+10 q^{55}+4 q^{54}+14 q^{53}-24 q^{52}-6 q^{51}+10 q^{50}+5 q^{49}+19 q^{48}-27 q^{47}-8 q^{46}+10 q^{45}+5 q^{44}+18 q^{43}-25 q^{42}-7 q^{41}+8 q^{40}+4 q^{39}+17 q^{38}-20 q^{37}-6 q^{36}+4 q^{35}+2 q^{34}+16 q^{33}-13 q^{32}-5 q^{31}-q^{30}-q^{29}+15 q^{28}-6 q^{27}-2 q^{26}-4 q^{25}-4 q^{24}+11 q^{23}-q^{22}+q^{21}-4 q^{20}-5 q^{19}+6 q^{18}+2 q^{16}-q^{15}-3 q^{14}+2 q^{13}+q^{11}-q^9+q^8 }[/math]
5 [math]\displaystyle{ -q^{115}+q^{114}+q^{111}-2 q^{109}+q^{108}-q^{106}+2 q^{105}+2 q^{104}-3 q^{103}-q^{101}-3 q^{100}+3 q^{99}+4 q^{98}+q^{96}-3 q^{95}-8 q^{94}+4 q^{92}+7 q^{91}+7 q^{90}-q^{89}-14 q^{88}-10 q^{87}-q^{86}+12 q^{85}+18 q^{84}+6 q^{83}-17 q^{82}-21 q^{81}-9 q^{80}+16 q^{79}+25 q^{78}+13 q^{77}-14 q^{76}-29 q^{75}-15 q^{74}+17 q^{73}+27 q^{72}+15 q^{71}-9 q^{70}-33 q^{69}-18 q^{68}+17 q^{67}+27 q^{66}+16 q^{65}-10 q^{64}-31 q^{63}-17 q^{62}+15 q^{61}+26 q^{60}+14 q^{59}-8 q^{58}-27 q^{57}-16 q^{56}+11 q^{55}+21 q^{54}+13 q^{53}-3 q^{52}-21 q^{51}-14 q^{50}+4 q^{49}+13 q^{48}+12 q^{47}+4 q^{46}-12 q^{45}-12 q^{44}-2 q^{43}+3 q^{42}+8 q^{41}+10 q^{40}-3 q^{39}-7 q^{38}-5 q^{37}-4 q^{36}+q^{35}+10 q^{34}+3 q^{33}-3 q^{31}-7 q^{30}-3 q^{29}+5 q^{28}+3 q^{27}+4 q^{26}-4 q^{24}-4 q^{23}+2 q^{22}+2 q^{20}+2 q^{19}-q^{18}-2 q^{17}+q^{16}+q^{13}-q^{11}+q^{10} }[/math]
6 [math]\displaystyle{ q^{159}-q^{158}-q^{155}+3 q^{152}-2 q^{151}+q^{149}-2 q^{148}-q^{147}-q^{146}+6 q^{145}-2 q^{144}+3 q^{142}-4 q^{141}-4 q^{140}-3 q^{139}+9 q^{138}-2 q^{137}+2 q^{136}+8 q^{135}-4 q^{134}-9 q^{133}-10 q^{132}+8 q^{131}-4 q^{130}+7 q^{129}+20 q^{128}+2 q^{127}-10 q^{126}-20 q^{125}-q^{124}-18 q^{123}+9 q^{122}+36 q^{121}+18 q^{120}-28 q^{118}-13 q^{117}-42 q^{116}+q^{115}+49 q^{114}+37 q^{113}+16 q^{112}-29 q^{111}-20 q^{110}-65 q^{109}-11 q^{108}+56 q^{107}+50 q^{106}+28 q^{105}-26 q^{104}-18 q^{103}-80 q^{102}-18 q^{101}+57 q^{100}+54 q^{99}+33 q^{98}-25 q^{97}-13 q^{96}-86 q^{95}-22 q^{94}+57 q^{93}+54 q^{92}+35 q^{91}-25 q^{90}-13 q^{89}-84 q^{88}-21 q^{87}+55 q^{86}+53 q^{85}+33 q^{84}-23 q^{83}-13 q^{82}-79 q^{81}-20 q^{80}+48 q^{79}+49 q^{78}+30 q^{77}-18 q^{76}-8 q^{75}-70 q^{74}-20 q^{73}+35 q^{72}+40 q^{71}+27 q^{70}-9 q^{69}+2 q^{68}-58 q^{67}-21 q^{66}+18 q^{65}+26 q^{64}+21 q^{63}+q^{62}+15 q^{61}-41 q^{60}-19 q^{59}+2 q^{58}+11 q^{57}+10 q^{56}+6 q^{55}+25 q^{54}-23 q^{53}-10 q^{52}-6 q^{51}-q^{50}-3 q^{49}+2 q^{48}+25 q^{47}-8 q^{46}+q^{45}-3 q^{44}-4 q^{43}-11 q^{42}-5 q^{41}+16 q^{40}-2 q^{39}+7 q^{38}+2 q^{37}+q^{36}-10 q^{35}-8 q^{34}+7 q^{33}-3 q^{32}+5 q^{31}+3 q^{30}+4 q^{29}-4 q^{28}-5 q^{27}+3 q^{26}-3 q^{25}+q^{24}+q^{23}+3 q^{22}-q^{21}-2 q^{20}+2 q^{19}-q^{18}+q^{15}-q^{13}+q^{12} }[/math]
7 [math]\displaystyle{ -q^{210}+q^{209}+q^{206}-q^{203}-2 q^{202}+2 q^{201}-q^{199}+2 q^{198}+q^{196}-q^{195}-5 q^{194}+3 q^{193}+q^{192}-2 q^{191}+3 q^{190}+4 q^{188}-2 q^{187}-9 q^{186}+3 q^{185}+q^{184}-2 q^{183}+5 q^{182}+q^{181}+9 q^{180}-q^{179}-13 q^{178}-q^{177}-6 q^{176}-6 q^{175}+8 q^{174}+6 q^{173}+19 q^{172}+9 q^{171}-11 q^{170}-5 q^{169}-22 q^{168}-23 q^{167}+q^{166}+5 q^{165}+35 q^{164}+34 q^{163}+8 q^{162}+3 q^{161}-36 q^{160}-53 q^{159}-27 q^{158}-12 q^{157}+43 q^{156}+66 q^{155}+40 q^{154}+32 q^{153}-38 q^{152}-82 q^{151}-63 q^{150}-46 q^{149}+39 q^{148}+88 q^{147}+74 q^{146}+65 q^{145}-30 q^{144}-96 q^{143}-88 q^{142}-79 q^{141}+25 q^{140}+101 q^{139}+93 q^{138}+88 q^{137}-22 q^{136}-98 q^{135}-95 q^{134}-98 q^{133}+14 q^{132}+104 q^{131}+100 q^{130}+97 q^{129}-19 q^{128}-97 q^{127}-92 q^{126}-106 q^{125}+9 q^{124}+104 q^{123}+100 q^{122}+100 q^{121}-19 q^{120}-95 q^{119}-93 q^{118}-104 q^{117}+10 q^{116}+102 q^{115}+99 q^{114}+98 q^{113}-17 q^{112}-95 q^{111}-91 q^{110}-99 q^{109}+8 q^{108}+96 q^{107}+94 q^{106}+95 q^{105}-13 q^{104}-88 q^{103}-83 q^{102}-93 q^{101}+2 q^{100}+82 q^{99}+83 q^{98}+88 q^{97}-2 q^{96}-69 q^{95}-69 q^{94}-87 q^{93}-11 q^{92}+60 q^{91}+63 q^{90}+79 q^{89}+14 q^{88}-41 q^{87}-46 q^{86}-78 q^{85}-25 q^{84}+30 q^{83}+37 q^{82}+64 q^{81}+26 q^{80}-12 q^{79}-16 q^{78}-58 q^{77}-34 q^{76}+4 q^{75}+8 q^{74}+41 q^{73}+25 q^{72}+7 q^{71}+12 q^{70}-31 q^{69}-26 q^{68}-8 q^{67}-13 q^{66}+14 q^{65}+12 q^{64}+8 q^{63}+24 q^{62}-7 q^{61}-8 q^{60}-q^{59}-18 q^{58}-3 q^{57}-4 q^{56}-2 q^{55}+18 q^{54}+q^{53}+5 q^{52}+11 q^{51}-8 q^{50}-6 q^{49}-8 q^{48}-10 q^{47}+7 q^{46}-3 q^{45}+5 q^{44}+13 q^{43}+q^{42}+q^{41}-5 q^{40}-8 q^{39}+2 q^{38}-5 q^{37}-q^{36}+6 q^{35}+3 q^{34}+3 q^{33}-q^{32}-4 q^{31}+3 q^{30}-3 q^{29}-2 q^{28}+q^{27}+q^{26}+2 q^{25}-2 q^{23}+2 q^{22}-q^{20}+q^{17}-q^{15}+q^{14} }[/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.

7 2.gif

7_2

7 4.gif

7_4

Retrieved from "https://katlas.org/index.php?title=7_3&oldid=61050"