7 4

7_3

7_5

Visit 7 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 4's page at Knotilus! Visit 7 4's page at the original Knot Atlas! Simplest version of Endless knot symbol.

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]

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]

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]

V_2,1 through V_6,9 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   \ 0 1 2 3 4 5 6 7 χ 17               1 -1 15                 0 13           2 1   -1 11         1       1 9       1 2       1 7     2 1         1 5     1           1 3 1 2             -1 1 1               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]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[7, 4]]
Out[2]=	7
In[3]:=	PD[Knot[7, 4]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 6, 13, 5], X[14, 8, 1, 7], X[8, 14, 9, 13], X[2, 12, 3, 11], X[10, 4, 11, 3], X[4, 10, 5, 9]]
In[4]:=	GaussCode[Knot[7, 4]]
Out[4]=	GaussCode[1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3]
In[5]:=	BR[Knot[7, 4]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[7, 4]][t]
Out[6]=	4 -7 + - + 4 t t
In[7]:=	Conway[Knot[7, 4]][z]
Out[7]=	2 1 + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 4], Knot[9, 2]}
In[9]:=	{KnotDet[Knot[7, 4]], KnotSignature[Knot[7, 4]]}
Out[9]=	{15, 2}
In[10]:=	J=Jones[Knot[7, 4]][q]
Out[10]=	2 3 4 5 6 7 8 q - 2 q + 3 q - 2 q + 3 q - 2 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[7, 4]}
In[12]:=	A2Invariant[Knot[7, 4]][q]
Out[12]=	2 4 8 10 12 14 16 20 24 26 q - q + q + q + 2 q + q + q - q - q - q
In[13]:=	Kauffman[Knot[7, 4]][a, z]
Out[13]=	2 2 2 2 3 3 3 -8 2 4 z 4 z 2 z 3 z 4 z z 4 z 8 z 2 z -a + -- + --- + --- + ---- - ---- - ---- + -- - ---- - ---- - ---- + 4 9 7 8 6 4 2 9 7 5 a a a a a a a a a a 3 4 4 5 5 5 6 6 2 z 3 z 3 z z 3 z 2 z z z ---- - ---- + ---- + -- + ---- + ---- + -- + -- 3 8 4 9 7 5 8 6 a a a a a a a a
In[14]:=	{Vassiliev[2][Knot[7, 4]], Vassiliev[3][Knot[7, 4]]}
Out[14]=	{0, 8}
In[15]:=	Kh[Knot[7, 4]][q, t]
Out[15]=	3 3 5 2 7 2 7 3 9 3 9 4 11 4 q + q + 2 q t + q t + 2 q t + q t + q t + 2 q t + q t + 13 5 13 6 17 7 2 q t + q t + q t