7 4

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7 3.gif

7_3

7 5.gif

7_5

7 4.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 7 4 at Knotilus!

[edit 7 4 Quick Notes]

Simplest version of Endless knot symbol.

[edit 7 4 Further Notes and Views]
Celtic or pseudo-Celtic knot
Mongolian ornament
Susan Williams' medallion [1], the "Endless knot" of Buddhism [2]
Ornamental "Endless knot"
a knot seen at the Castle of Kornik [3]
A 7-4 knot reduced from TakaraMusubi with 9 crossings [4]
TakaraMusubi knot seen in Japanese symbols, or Kolam in South India [5]
Buddhist Endless Knot
Ornamental Endless Knot
Albrecht Dürer knot, 16th-century
A laser cut by Tom Longtin [6]
Unicursal hexagram of occultism
Logo of the raelian sect
Lissajous curve : x=cos 3t , y=sin 2t, z=sin 7t
French europa stamp 2023


Knot presentations

Planar diagram presentation X6271 X12,6,13,5 X14,8,1,7 X8,14,9,13 X2,12,3,11 X10,4,11,3 X4,10,5,9
Gauss code 1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3
Dowker-Thistlethwaite code 6 10 12 14 4 2 8
Conway Notation [313]


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.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 9, width is 4,

Braid index is 4

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

[edit Notes on presentations of 7 4]

Knot 7_4.
A graph, knot 7_4.
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 4"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X12,6,13,5 X14,8,1,7 X8,14,9,13 X2,12,3,11 X10,4,11,3 X4,10,5,9
In[5]:= GaussCode[K]
Out[5]= 1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3
In[6]:= DTCode[K]
Out[6]= 6 10 12 14 4 2 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [313]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= [math]\displaystyle{ \textrm{BR}(4,\{1,1,2,-1,2,2,3,-2,3\}) }[/math]
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 4, 9, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
7 4 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 5}, {6, 4}, {5, 7}, {2, 6}, {8, 3}, {7, 9}, {1, 8}, {9, 2}, {4, 1}]
In[14]:= Draw[ap]
7 4 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 1
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 5.13794
A-Polynomial See Data:7 4/A-polynomial

[edit Notes for 7 4'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{ \textrm{ConcordanceGenus}(\textrm{Knot}(7,4)) }[/math]
Rasmussen s-Invariant -2

[edit Notes for 7 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 4 t+4 t^{-1} -7 }[/math]
Conway polynomial [math]\displaystyle{ 4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 15, 2 }
Jones polynomial [math]\displaystyle{ -q^8+q^7-2 q^6+3 q^5-2 q^4+3 q^3-2 q^2+q }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ - a^{-8} +z^2 a^{-6} +2 z^2 a^{-4} +2 a^{-4} +z^2 a^{-2} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +2 z^5 a^{-5} +3 z^5 a^{-7} +z^5 a^{-9} +3 z^4 a^{-4} -3 z^4 a^{-8} +2 z^3 a^{-3} -2 z^3 a^{-5} -8 z^3 a^{-7} -4 z^3 a^{-9} +z^2 a^{-2} -4 z^2 a^{-4} -3 z^2 a^{-6} +2 z^2 a^{-8} +4 z a^{-7} +4 z a^{-9} +2 a^{-4} - a^{-8} }[/math]
The A2 invariant [math]\displaystyle{ q^{-2} - q^{-4} + q^{-8} + q^{-10} +2 q^{-12} + q^{-14} + q^{-16} - q^{-20} - q^{-24} - q^{-26} }[/math]
The G2 invariant [math]\displaystyle{ q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-22} +4 q^{-24} -2 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} -2 q^{-36} +3 q^{-38} -2 q^{-40} + q^{-44} +2 q^{-48} + q^{-50} - q^{-52} +2 q^{-54} - q^{-56} +3 q^{-58} + q^{-60} -2 q^{-62} +6 q^{-64} -3 q^{-66} +4 q^{-68} +2 q^{-70} -3 q^{-72} +4 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} - q^{-82} + q^{-84} -2 q^{-86} +2 q^{-88} -3 q^{-92} - q^{-94} -2 q^{-96} -4 q^{-102} +2 q^{-104} -3 q^{-106} + q^{-108} + q^{-110} -4 q^{-112} +3 q^{-114} -2 q^{-116} + q^{-118} -2 q^{-122} +2 q^{-124} + q^{-128} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 7_4.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{-1} - q^{-3} + q^{-5} + q^{-7} + q^{-9} + q^{-11} - q^{-13} - q^{-17} }[/math]
2 [math]\displaystyle{ q^{-2} - q^{-4} +3 q^{-8} - q^{-10} +2 q^{-14} - q^{-16} + q^{-20} +2 q^{-22} + q^{-28} - q^{-30} -3 q^{-32} -2 q^{-38} + q^{-40} + q^{-42} - q^{-44} + q^{-48} }[/math]
3 [math]\displaystyle{ q^{-3} - q^{-5} +2 q^{-9} + q^{-11} - q^{-13} - q^{-15} +3 q^{-17} + q^{-19} -3 q^{-21} + q^{-23} +4 q^{-25} +2 q^{-27} -3 q^{-29} - q^{-31} + q^{-33} +2 q^{-35} - q^{-39} - q^{-41} + q^{-43} +3 q^{-45} -2 q^{-47} -3 q^{-49} +2 q^{-53} -2 q^{-55} -3 q^{-57} -2 q^{-59} +2 q^{-61} -2 q^{-65} - q^{-67} +3 q^{-69} +3 q^{-71} - q^{-73} -2 q^{-75} + q^{-77} +3 q^{-79} -2 q^{-83} - q^{-85} + q^{-87} + q^{-89} - q^{-93} }[/math]
4 [math]\displaystyle{ q^{-4} - q^{-6} +2 q^{-10} + q^{-14} -2 q^{-16} + q^{-18} +3 q^{-20} -2 q^{-22} + q^{-24} -2 q^{-26} +4 q^{-28} +6 q^{-30} -3 q^{-32} -4 q^{-34} -5 q^{-36} +5 q^{-38} +9 q^{-40} -4 q^{-44} -6 q^{-46} +6 q^{-50} +3 q^{-52} -3 q^{-56} -3 q^{-58} - q^{-60} +2 q^{-62} +4 q^{-64} - q^{-66} -4 q^{-68} -5 q^{-70} + q^{-72} +4 q^{-74} +2 q^{-76} -4 q^{-78} -6 q^{-80} + q^{-82} +4 q^{-84} +2 q^{-86} -5 q^{-88} -5 q^{-90} +2 q^{-92} +3 q^{-94} +4 q^{-96} -2 q^{-98} -5 q^{-100} +2 q^{-102} +3 q^{-104} +7 q^{-106} + q^{-108} -5 q^{-110} -3 q^{-112} - q^{-114} +6 q^{-116} +4 q^{-118} -2 q^{-120} -4 q^{-122} -5 q^{-124} +2 q^{-126} +4 q^{-128} +2 q^{-130} - q^{-132} -5 q^{-134} - q^{-136} + q^{-138} +2 q^{-140} +2 q^{-142} - q^{-144} - q^{-146} - q^{-148} + q^{-152} }[/math]
5 [math]\displaystyle{ q^{-5} - q^{-7} +2 q^{-11} + q^{-21} + q^{-23} -2 q^{-27} + q^{-29} +5 q^{-31} +5 q^{-33} -2 q^{-35} -6 q^{-37} -7 q^{-39} +11 q^{-43} +12 q^{-45} +3 q^{-47} -11 q^{-49} -15 q^{-51} -5 q^{-53} +9 q^{-55} +18 q^{-57} +12 q^{-59} -7 q^{-61} -16 q^{-63} -12 q^{-65} - q^{-67} +11 q^{-69} +13 q^{-71} +3 q^{-73} -7 q^{-75} -8 q^{-77} -7 q^{-79} - q^{-81} +6 q^{-83} +7 q^{-85} +4 q^{-87} - q^{-89} -7 q^{-91} -8 q^{-93} -4 q^{-95} +5 q^{-97} +9 q^{-99} +2 q^{-101} -6 q^{-103} -10 q^{-105} -5 q^{-107} +5 q^{-109} +9 q^{-111} + q^{-113} -4 q^{-115} -9 q^{-117} -3 q^{-119} +8 q^{-121} +10 q^{-123} +2 q^{-125} -6 q^{-127} -10 q^{-129} -3 q^{-131} +9 q^{-133} +13 q^{-135} +3 q^{-137} -6 q^{-139} -11 q^{-141} -7 q^{-143} +7 q^{-145} +13 q^{-147} +9 q^{-149} -9 q^{-153} -12 q^{-155} -4 q^{-157} +6 q^{-159} +11 q^{-161} +6 q^{-163} -2 q^{-165} -10 q^{-167} -11 q^{-169} -4 q^{-171} +6 q^{-173} +10 q^{-175} +7 q^{-177} - q^{-179} -8 q^{-181} -9 q^{-183} -3 q^{-185} +4 q^{-187} +8 q^{-189} +6 q^{-191} -5 q^{-195} -6 q^{-197} -3 q^{-199} +2 q^{-201} +5 q^{-203} +3 q^{-205} + q^{-207} - q^{-209} -3 q^{-211} -2 q^{-213} + q^{-217} + q^{-219} + q^{-221} - q^{-225} }[/math]
6 [math]\displaystyle{ q^{-6} - q^{-8} +2 q^{-12} - q^{-18} +2 q^{-20} - q^{-24} +3 q^{-26} -2 q^{-28} - q^{-30} +2 q^{-32} +7 q^{-34} +2 q^{-36} -3 q^{-38} -3 q^{-40} -10 q^{-42} -4 q^{-44} +8 q^{-46} +21 q^{-48} +10 q^{-50} -5 q^{-52} -13 q^{-54} -25 q^{-56} -11 q^{-58} +13 q^{-60} +34 q^{-62} +25 q^{-64} + q^{-66} -19 q^{-68} -40 q^{-70} -28 q^{-72} +4 q^{-74} +35 q^{-76} +39 q^{-78} +19 q^{-80} -5 q^{-82} -36 q^{-84} -38 q^{-86} -17 q^{-88} +15 q^{-90} +32 q^{-92} +28 q^{-94} +13 q^{-96} -14 q^{-98} -26 q^{-100} -26 q^{-102} -8 q^{-104} +7 q^{-106} +15 q^{-108} +18 q^{-110} +8 q^{-112} -2 q^{-114} -12 q^{-116} -14 q^{-118} -12 q^{-120} -2 q^{-122} +10 q^{-124} +17 q^{-126} +12 q^{-128} -12 q^{-132} -19 q^{-134} -9 q^{-136} +3 q^{-138} +15 q^{-140} +13 q^{-142} -12 q^{-146} -18 q^{-148} -3 q^{-150} +8 q^{-152} +18 q^{-154} +12 q^{-156} -6 q^{-158} -16 q^{-160} -14 q^{-162} +2 q^{-164} +14 q^{-166} +24 q^{-168} +12 q^{-170} -9 q^{-172} -23 q^{-174} -19 q^{-176} -2 q^{-178} +14 q^{-180} +29 q^{-182} +20 q^{-184} -4 q^{-186} -26 q^{-188} -27 q^{-190} -12 q^{-192} +6 q^{-194} +29 q^{-196} +28 q^{-198} +10 q^{-200} -17 q^{-202} -27 q^{-204} -23 q^{-206} -11 q^{-208} +15 q^{-210} +27 q^{-212} +23 q^{-214} + q^{-216} -13 q^{-218} -23 q^{-220} -25 q^{-222} -5 q^{-224} +12 q^{-226} +24 q^{-228} +18 q^{-230} +9 q^{-232} -8 q^{-234} -23 q^{-236} -18 q^{-238} -9 q^{-240} +8 q^{-242} +15 q^{-244} +21 q^{-246} +12 q^{-248} -5 q^{-250} -13 q^{-252} -17 q^{-254} -10 q^{-256} -2 q^{-258} +12 q^{-260} +15 q^{-262} +9 q^{-264} +2 q^{-266} -6 q^{-268} -10 q^{-270} -12 q^{-272} -2 q^{-274} +4 q^{-276} +7 q^{-278} +7 q^{-280} +4 q^{-282} -6 q^{-286} -4 q^{-288} -3 q^{-290} - q^{-292} + q^{-294} +3 q^{-296} +3 q^{-298} - q^{-304} - q^{-306} - q^{-308} + q^{-312} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-2} - q^{-4} + q^{-8} + q^{-10} +2 q^{-12} + q^{-14} + q^{-16} - q^{-20} - q^{-24} - q^{-26} }[/math]
1,1 [math]\displaystyle{ q^{-4} -2 q^{-6} +2 q^{-8} -2 q^{-10} +7 q^{-12} -4 q^{-14} +4 q^{-16} -4 q^{-18} +7 q^{-20} +4 q^{-24} +4 q^{-26} + q^{-28} +6 q^{-30} -8 q^{-32} +8 q^{-34} -15 q^{-36} +8 q^{-38} -12 q^{-40} +6 q^{-42} -7 q^{-44} +4 q^{-46} +2 q^{-48} -2 q^{-50} +3 q^{-52} -6 q^{-54} +6 q^{-56} -8 q^{-58} +4 q^{-60} -4 q^{-62} +4 q^{-64} + q^{-68} }[/math]
2,0 [math]\displaystyle{ q^{-4} - q^{-6} - q^{-8} +2 q^{-10} +2 q^{-12} - q^{-16} + q^{-18} +2 q^{-20} + q^{-24} +2 q^{-26} +2 q^{-28} +2 q^{-30} +3 q^{-32} + q^{-34} + q^{-36} - q^{-40} -4 q^{-42} -4 q^{-44} -2 q^{-46} - q^{-48} -2 q^{-50} - q^{-52} + q^{-54} + q^{-56} - q^{-60} + q^{-62} + q^{-64} + q^{-66} }[/math]
3,0 [math]\displaystyle{ q^{-6} - q^{-8} - q^{-10} + q^{-12} +3 q^{-14} +2 q^{-16} -4 q^{-18} -2 q^{-20} +3 q^{-22} +5 q^{-24} +3 q^{-26} -4 q^{-28} - q^{-30} +2 q^{-32} +6 q^{-34} +4 q^{-36} - q^{-38} - q^{-40} +2 q^{-42} +3 q^{-44} +2 q^{-46} + q^{-48} +2 q^{-50} +2 q^{-52} + q^{-54} + q^{-56} +2 q^{-58} + q^{-60} -4 q^{-62} -5 q^{-64} -4 q^{-66} - q^{-68} -4 q^{-70} -7 q^{-72} -7 q^{-74} -4 q^{-76} + q^{-78} -2 q^{-82} -2 q^{-84} + q^{-86} +6 q^{-88} +5 q^{-90} +2 q^{-92} +4 q^{-98} +3 q^{-100} + q^{-102} -2 q^{-104} -3 q^{-106} + q^{-110} +2 q^{-112} - q^{-116} - q^{-118} - q^{-120} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{-4} - q^{-6} - q^{-8} +2 q^{-10} +3 q^{-16} + q^{-18} + q^{-20} +2 q^{-22} +2 q^{-24} + q^{-26} +2 q^{-28} +2 q^{-30} +2 q^{-32} -2 q^{-34} - q^{-36} -2 q^{-38} -5 q^{-40} -3 q^{-42} - q^{-44} - q^{-46} + q^{-48} +2 q^{-50} + q^{-54} }[/math]
1,0,0 [math]\displaystyle{ q^{-3} - q^{-5} + q^{-11} + q^{-13} +2 q^{-15} +2 q^{-17} + q^{-19} + q^{-21} - q^{-27} - q^{-31} - q^{-33} - q^{-35} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{-6} - q^{-8} - q^{-10} + q^{-12} + q^{-14} +2 q^{-20} +2 q^{-22} + q^{-26} +3 q^{-28} +3 q^{-30} +2 q^{-32} +5 q^{-34} +4 q^{-36} +4 q^{-38} +3 q^{-40} +3 q^{-42} -3 q^{-46} -3 q^{-48} -5 q^{-50} -8 q^{-52} -7 q^{-54} -3 q^{-56} -2 q^{-58} - q^{-60} + q^{-62} +3 q^{-64} +2 q^{-66} + q^{-68} + q^{-70} + q^{-72} }[/math]
1,0,0,0 [math]\displaystyle{ q^{-4} - q^{-6} + q^{-14} + q^{-16} +2 q^{-18} +2 q^{-20} +2 q^{-22} + q^{-24} + q^{-26} - q^{-34} - q^{-38} - q^{-40} - q^{-42} - q^{-44} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{-4} - q^{-6} + q^{-8} -2 q^{-10} +2 q^{-12} + q^{-16} + q^{-18} + q^{-20} +2 q^{-22} -2 q^{-24} +3 q^{-26} -2 q^{-28} +2 q^{-30} -2 q^{-32} +2 q^{-34} - q^{-36} + q^{-40} - q^{-42} + q^{-44} - q^{-46} + q^{-48} -2 q^{-50} - q^{-54} }[/math]
1,0 [math]\displaystyle{ q^{-6} - q^{-10} - q^{-12} +2 q^{-16} + q^{-18} - q^{-20} + q^{-24} +2 q^{-26} +2 q^{-34} + q^{-36} +2 q^{-42} +2 q^{-44} + q^{-46} + q^{-50} + q^{-52} -2 q^{-56} - q^{-58} - q^{-62} -3 q^{-64} -3 q^{-66} - q^{-68} - q^{-74} - q^{-76} + q^{-78} +2 q^{-80} + q^{-88} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{-6} - q^{-8} - q^{-12} +2 q^{-14} - q^{-16} + q^{-18} +2 q^{-22} +2 q^{-24} +2 q^{-26} +2 q^{-28} + q^{-30} +3 q^{-32} +3 q^{-36} - q^{-38} +3 q^{-40} +3 q^{-44} - q^{-46} + q^{-48} -2 q^{-50} -2 q^{-52} -3 q^{-54} -4 q^{-56} -2 q^{-58} -3 q^{-60} - q^{-62} - q^{-64} +2 q^{-66} +2 q^{-70} + q^{-74} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-22} +4 q^{-24} -2 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} -2 q^{-36} +3 q^{-38} -2 q^{-40} + q^{-44} +2 q^{-48} + q^{-50} - q^{-52} +2 q^{-54} - q^{-56} +3 q^{-58} + q^{-60} -2 q^{-62} +6 q^{-64} -3 q^{-66} +4 q^{-68} +2 q^{-70} -3 q^{-72} +4 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} - q^{-82} + q^{-84} -2 q^{-86} +2 q^{-88} -3 q^{-92} - q^{-94} -2 q^{-96} -4 q^{-102} +2 q^{-104} -3 q^{-106} + q^{-108} + q^{-110} -4 q^{-112} +3 q^{-114} -2 q^{-116} + q^{-118} -2 q^{-122} +2 q^{-124} + q^{-128} }[/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 4"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 4 t+4 t^{-1} -7 }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 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]= { 15, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^8+q^7-2 q^6+3 q^5-2 q^4+3 q^3-2 q^2+q }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ - a^{-8} +z^2 a^{-6} +2 z^2 a^{-4} +2 a^{-4} +z^2 a^{-2} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +2 z^5 a^{-5} +3 z^5 a^{-7} +z^5 a^{-9} +3 z^4 a^{-4} -3 z^4 a^{-8} +2 z^3 a^{-3} -2 z^3 a^{-5} -8 z^3 a^{-7} -4 z^3 a^{-9} +z^2 a^{-2} -4 z^2 a^{-4} -3 z^2 a^{-6} +2 z^2 a^{-8} +4 z a^{-7} +4 z a^{-9} +2 a^{-4} - a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_2,}

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 4"];
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{ 4 t+4 t^{-1} -7 }[/math], [math]\displaystyle{ -q^8+q^7-2 q^6+3 q^5-2 q^4+3 q^3-2 q^2+q }[/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]= {9_2,}
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: (4, 8)
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{ 16 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1016}{3} }[/math] [math]\displaystyle{ \frac{184}{3} }[/math] [math]\displaystyle{ 1024 }[/math] [math]\displaystyle{ \frac{5824}{3} }[/math] [math]\displaystyle{ \frac{1024}{3} }[/math] [math]\displaystyle{ 320 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 2048 }[/math] [math]\displaystyle{ \frac{16256}{3} }[/math] [math]\displaystyle{ \frac{2944}{3} }[/math] [math]\displaystyle{ \frac{168062}{15} }[/math] [math]\displaystyle{ -\frac{1176}{5} }[/math] [math]\displaystyle{ \frac{233288}{45} }[/math] [math]\displaystyle{ \frac{898}{9} }[/math] [math]\displaystyle{ \frac{11102}{15} }[/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]2 is the signature of 7 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 01234567χ
17       1-1
15        0
13     21 -1
11    1   1
9   12   1
7  21    1
5  1     1
312      -1
11       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=3 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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^{23}-q^{22}-q^{21}+3 q^{20}-q^{19}-4 q^{18}+5 q^{17}-q^{16}-7 q^{15}+7 q^{14}+q^{13}-8 q^{12}+7 q^{11}+3 q^{10}-9 q^9+6 q^8+2 q^7-6 q^6+4 q^5+q^4-2 q^3+q^2 }[/math]
3 [math]\displaystyle{ -q^{45}+q^{44}+q^{43}-3 q^{41}+3 q^{39}+3 q^{38}-5 q^{37}-3 q^{36}+4 q^{35}+7 q^{34}-5 q^{33}-7 q^{32}+3 q^{31}+9 q^{30}-3 q^{29}-11 q^{28}+2 q^{27}+10 q^{26}+q^{25}-13 q^{24}-q^{23}+11 q^{22}+6 q^{21}-15 q^{20}-3 q^{19}+11 q^{18}+7 q^{17}-13 q^{16}-4 q^{15}+9 q^{14}+5 q^{13}-8 q^{12}-2 q^{11}+6 q^{10}+q^9-4 q^8+2 q^6+q^5-2 q^4+q^3 }[/math]
4 [math]\displaystyle{ q^{74}-q^{73}-q^{72}+4 q^{69}-q^{68}-2 q^{67}-2 q^{66}-4 q^{65}+8 q^{64}+2 q^{63}-4 q^{61}-11 q^{60}+9 q^{59}+4 q^{58}+6 q^{57}-2 q^{56}-18 q^{55}+7 q^{54}+2 q^{53}+12 q^{52}+4 q^{51}-22 q^{50}+6 q^{49}-5 q^{48}+15 q^{47}+10 q^{46}-23 q^{45}+5 q^{44}-12 q^{43}+15 q^{42}+17 q^{41}-21 q^{40}+2 q^{39}-19 q^{38}+17 q^{37}+23 q^{36}-19 q^{35}-q^{34}-25 q^{33}+18 q^{32}+26 q^{31}-14 q^{30}-3 q^{29}-28 q^{28}+16 q^{27}+26 q^{26}-11 q^{25}-25 q^{23}+10 q^{22}+20 q^{21}-9 q^{20}+4 q^{19}-16 q^{18}+6 q^{17}+10 q^{16}-8 q^{15}+5 q^{14}-7 q^{13}+4 q^{12}+4 q^{11}-5 q^{10}+2 q^9-2 q^8+2 q^7+q^6-2 q^5+q^4 }[/math]
5 [math]\displaystyle{ -q^{110}+q^{109}+q^{108}-q^{105}-3 q^{104}+3 q^{102}+2 q^{101}+2 q^{100}+q^{99}-6 q^{98}-5 q^{97}+3 q^{95}+7 q^{94}+7 q^{93}-4 q^{92}-9 q^{91}-7 q^{90}-3 q^{89}+8 q^{88}+14 q^{87}+4 q^{86}-6 q^{85}-11 q^{84}-13 q^{83}+q^{82}+15 q^{81}+12 q^{80}+2 q^{79}-6 q^{78}-18 q^{77}-9 q^{76}+7 q^{75}+15 q^{74}+11 q^{73}+3 q^{72}-14 q^{71}-15 q^{70}-7 q^{69}+11 q^{68}+16 q^{67}+12 q^{66}-4 q^{65}-19 q^{64}-19 q^{63}+4 q^{62}+20 q^{61}+20 q^{60}+4 q^{59}-21 q^{58}-30 q^{57}-2 q^{56}+25 q^{55}+25 q^{54}+12 q^{53}-25 q^{52}-40 q^{51}-7 q^{50}+29 q^{49}+33 q^{48}+19 q^{47}-29 q^{46}-49 q^{45}-11 q^{44}+30 q^{43}+39 q^{42}+24 q^{41}-26 q^{40}-50 q^{39}-18 q^{38}+24 q^{37}+38 q^{36}+25 q^{35}-16 q^{34}-40 q^{33}-20 q^{32}+12 q^{31}+27 q^{30}+21 q^{29}-7 q^{28}-21 q^{27}-14 q^{26}+3 q^{25}+13 q^{24}+11 q^{23}-3 q^{22}-7 q^{21}-5 q^{20}+2 q^{19}+2 q^{18}+4 q^{17}-2 q^{16}-3 q^{15}+2 q^{14}+2 q^{13}-2 q^{12}+q^{11}-2 q^9+2 q^8+q^7-2 q^6+q^5 }[/math]
6 [math]\displaystyle{ q^{153}-q^{152}-q^{151}+q^{148}+4 q^{146}-q^{145}-3 q^{144}-2 q^{143}-2 q^{142}-2 q^{140}+10 q^{139}+3 q^{138}-2 q^{136}-5 q^{135}-6 q^{134}-12 q^{133}+12 q^{132}+7 q^{131}+8 q^{130}+5 q^{129}+q^{128}-9 q^{127}-26 q^{126}+4 q^{125}+12 q^{123}+13 q^{122}+18 q^{121}-32 q^{119}-3 q^{118}-17 q^{117}+3 q^{116}+8 q^{115}+33 q^{114}+17 q^{113}-23 q^{112}+3 q^{111}-29 q^{110}-14 q^{109}-12 q^{108}+35 q^{107}+27 q^{106}-9 q^{105}+25 q^{104}-25 q^{103}-26 q^{102}-38 q^{101}+23 q^{100}+23 q^{99}+q^{98}+52 q^{97}-7 q^{96}-25 q^{95}-61 q^{94}+5 q^{93}+8 q^{92}+2 q^{91}+74 q^{90}+17 q^{89}-16 q^{88}-76 q^{87}-11 q^{86}-9 q^{85}-2 q^{84}+88 q^{83}+38 q^{82}-4 q^{81}-86 q^{80}-23 q^{79}-25 q^{78}-4 q^{77}+98 q^{76}+56 q^{75}+2 q^{74}-96 q^{73}-34 q^{72}-40 q^{71}+2 q^{70}+110 q^{69}+69 q^{68}+4 q^{67}-108 q^{66}-46 q^{65}-50 q^{64}+9 q^{63}+122 q^{62}+81 q^{61}+9 q^{60}-115 q^{59}-58 q^{58}-60 q^{57}+7 q^{56}+124 q^{55}+91 q^{54}+19 q^{53}-105 q^{52}-61 q^{51}-68 q^{50}-8 q^{49}+106 q^{48}+91 q^{47}+31 q^{46}-78 q^{45}-46 q^{44}-64 q^{43}-25 q^{42}+74 q^{41}+70 q^{40}+33 q^{39}-47 q^{38}-22 q^{37}-44 q^{36}-29 q^{35}+43 q^{34}+38 q^{33}+21 q^{32}-26 q^{31}-2 q^{30}-20 q^{29}-20 q^{28}+22 q^{27}+14 q^{26}+7 q^{25}-14 q^{24}+6 q^{23}-5 q^{22}-9 q^{21}+9 q^{20}+2 q^{19}+q^{18}-7 q^{17}+6 q^{16}-4 q^{14}+4 q^{13}-q^{12}-2 q^{10}+2 q^9+q^8-2 q^7+q^6 }[/math]
7 [math]\displaystyle{ -q^{203}+q^{202}+q^{201}-q^{198}-q^{196}-3 q^{195}+q^{194}+3 q^{193}+2 q^{192}+3 q^{191}-q^{190}-9 q^{187}-4 q^{186}+2 q^{184}+8 q^{183}+3 q^{182}+7 q^{181}+8 q^{180}-10 q^{179}-10 q^{178}-10 q^{177}-10 q^{176}+5 q^{175}+2 q^{174}+13 q^{173}+24 q^{172}+3 q^{171}-q^{170}-12 q^{169}-25 q^{168}-10 q^{167}-15 q^{166}+3 q^{165}+31 q^{164}+17 q^{163}+22 q^{162}+9 q^{161}-22 q^{160}-15 q^{159}-36 q^{158}-24 q^{157}+14 q^{156}+9 q^{155}+34 q^{154}+37 q^{153}+5 q^{152}+7 q^{151}-36 q^{150}-45 q^{149}-12 q^{148}-23 q^{147}+14 q^{146}+44 q^{145}+29 q^{144}+47 q^{143}-6 q^{142}-37 q^{141}-22 q^{140}-61 q^{139}-26 q^{138}+18 q^{137}+25 q^{136}+78 q^{135}+40 q^{134}-3 q^{133}-5 q^{132}-80 q^{131}-65 q^{130}-25 q^{129}-10 q^{128}+86 q^{127}+79 q^{126}+41 q^{125}+34 q^{124}-77 q^{123}-88 q^{122}-69 q^{121}-58 q^{120}+70 q^{119}+99 q^{118}+81 q^{117}+79 q^{116}-55 q^{115}-97 q^{114}-101 q^{113}-104 q^{112}+41 q^{111}+105 q^{110}+112 q^{109}+120 q^{108}-29 q^{107}-96 q^{106}-124 q^{105}-144 q^{104}+15 q^{103}+108 q^{102}+136 q^{101}+150 q^{100}-10 q^{99}-99 q^{98}-143 q^{97}-174 q^{96}+q^{95}+118 q^{94}+157 q^{93}+169 q^{92}-2 q^{91}-110 q^{90}-165 q^{89}-194 q^{88}-3 q^{87}+137 q^{86}+179 q^{85}+185 q^{84}+3 q^{83}-129 q^{82}-188 q^{81}-212 q^{80}-7 q^{79}+151 q^{78}+199 q^{77}+204 q^{76}+14 q^{75}-133 q^{74}-202 q^{73}-229 q^{72}-30 q^{71}+138 q^{70}+204 q^{69}+220 q^{68}+40 q^{67}-107 q^{66}-187 q^{65}-233 q^{64}-60 q^{63}+93 q^{62}+173 q^{61}+213 q^{60}+67 q^{59}-59 q^{58}-142 q^{57}-200 q^{56}-72 q^{55}+39 q^{54}+113 q^{53}+167 q^{52}+69 q^{51}-21 q^{50}-83 q^{49}-138 q^{48}-52 q^{47}+10 q^{46}+53 q^{45}+103 q^{44}+43 q^{43}-5 q^{42}-37 q^{41}-75 q^{40}-21 q^{39}+4 q^{38}+18 q^{37}+48 q^{36}+13 q^{35}-q^{34}-10 q^{33}-34 q^{32}-3 q^{31}+5 q^{30}+4 q^{29}+15 q^{28}-q^{27}+q^{26}+2 q^{25}-14 q^{24}+2 q^{23}+4 q^{22}+2 q^{20}-4 q^{19}+2 q^{18}+4 q^{17}-6 q^{16}+2 q^{15}+2 q^{14}-q^{13}-2 q^{11}+2 q^{10}+q^9-2 q^8+q^7 }[/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 3.gif

7_3

7 5.gif

7_5

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