9 5

9_4

9_6

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 5 at Knotilus!

[edit 9 5 Quick Notes]

[edit 9 5 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_14,6,15,5 X_18,8,1,7 X_16,10,17,9 X_10,16,11,15 X_8,18,9,17 X_2,14,3,13 X_12,4,13,3 X_4,12,5,11
Gauss code	1, -7, 8, -9, 2, -1, 3, -6, 4, -5, 9, -8, 7, -2, 5, -4, 6, -3
Dowker-Thistlethwaite code	6 12 14 18 16 4 2 10 8
Conway Notation	[513]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 12, width is 5,

Braid index is 5

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

[edit Notes on presentations of 9 5]

Knot 9_5.

A graph, knot 9_5.

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 5"];

In[4]:= PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= X₆₂₇₁ X_14,6,15,5 X_18,8,1,7 X_16,10,17,9 X_10,16,11,15 X_8,18,9,17 X_2,14,3,13 X_12,4,13,3 X_4,12,5,11

In[5]:= GaussCode[K]

Out[5]= 1, -7, 8, -9, 2, -1, 3, -6, 4, -5, 9, -8, 7, -2, 5, -4, 6, -3

In[6]:= DTCode[K]

Out[6]= 6 12 14 18 16 4 2 10 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [513]

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}(5,\{1,1,2,-1,2,2,3,-2,3,4,-3,4\}) }[/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]= { 5, 12, 5 }

In[11]:= Show[BraidPlot[br]]

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."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{3, 5}, {6, 4}, {5, 7}, {8, 6}, {7, 9}, {2, 8}, {10, 3}, {9, 11}, {1, 10}, {11, 2}, {4, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	1
Bridge index	2
Super bridge index	[math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index	1
Maximal Thurston-Bennequin number	[1][-12]
Hyperbolic Volume	5.69844
A-Polynomial	See Data:9 5/A-polynomial

[edit Notes for 9 5'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{ 1 }[/math]
Rasmussen s-Invariant	-2

[edit Notes for 9 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 6 t-11+6 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 23, 2 }
Jones polynomial	[math]\displaystyle{ -q^{10}+q^9-2 q^8+3 q^7-3 q^6+4 q^5-3 q^4+3 q^3-2 q^2+q }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^{-2} +2 z^2 a^{-4} +2 z^2 a^{-6} +z^2 a^{-8} + a^{-4} + a^{-6} - a^{-10} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +2 z^7 a^{-7} +3 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-6} -2 z^6 a^{-8} -5 z^6 a^{-10} +3 z^5 a^{-5} -5 z^5 a^{-7} -14 z^5 a^{-9} -6 z^5 a^{-11} +3 z^4 a^{-4} -7 z^4 a^{-6} -3 z^4 a^{-8} +7 z^4 a^{-10} +2 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +18 z^3 a^{-9} +11 z^3 a^{-11} +z^2 a^{-2} -3 z^2 a^{-4} +3 z^2 a^{-6} +4 z^2 a^{-8} -3 z^2 a^{-10} -6 z a^{-9} -6 z a^{-11} + a^{-4} - a^{-6} + a^{-10} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-2} - q^{-4} + q^{-8} + q^{-12} + q^{-14} + q^{-16} + q^{-18} + q^{-22} - q^{-26} - q^{-30} - q^{-32} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-22} +3 q^{-24} -2 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} - q^{-36} +3 q^{-38} -2 q^{-40} + q^{-42} +2 q^{-48} - q^{-50} + q^{-52} - q^{-54} + q^{-56} - q^{-60} + q^{-62} + q^{-66} + q^{-72} + q^{-74} +2 q^{-78} -2 q^{-80} +5 q^{-82} - q^{-84} - q^{-86} +5 q^{-88} -4 q^{-90} +6 q^{-92} -2 q^{-94} - q^{-96} +3 q^{-98} -2 q^{-100} +4 q^{-102} -3 q^{-104} - q^{-110} + q^{-112} -2 q^{-114} - q^{-116} + q^{-118} -3 q^{-120} -2 q^{-124} -2 q^{-126} +3 q^{-128} -6 q^{-130} +3 q^{-132} -2 q^{-134} -2 q^{-136} +4 q^{-138} -5 q^{-140} +3 q^{-142} - q^{-144} + q^{-148} -2 q^{-150} +2 q^{-152} + q^{-156} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_5.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{-1} - q^{-3} + q^{-5} + q^{-9} + q^{-11} + q^{-15} - q^{-17} - q^{-21} }[/math]
2	[math]\displaystyle{ q^{-2} - q^{-4} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} + q^{-20} +2 q^{-26} - q^{-30} + q^{-34} - q^{-36} - q^{-38} + q^{-40} - q^{-42} -2 q^{-44} + q^{-46} -2 q^{-50} + q^{-52} + q^{-54} - q^{-56} + q^{-60} }[/math]
3	[math]\displaystyle{ q^{-3} - q^{-5} + q^{-9} + q^{-11} - q^{-15} + q^{-17} + q^{-19} + q^{-21} - q^{-25} + q^{-27} +2 q^{-29} + q^{-31} -3 q^{-33} +2 q^{-37} +2 q^{-39} - q^{-43} - q^{-45} + q^{-47} +3 q^{-49} + q^{-51} -3 q^{-53} -2 q^{-55} +2 q^{-57} - q^{-59} -3 q^{-61} -2 q^{-63} + q^{-65} - q^{-71} + q^{-73} + q^{-75} -2 q^{-79} - q^{-81} + q^{-83} +2 q^{-85} - q^{-87} -2 q^{-89} +3 q^{-93} +2 q^{-95} -2 q^{-97} -2 q^{-99} + q^{-101} +3 q^{-103} -2 q^{-107} - q^{-109} + q^{-111} + q^{-113} - q^{-117} }[/math]
4	[math]\displaystyle{ q^{-4} - q^{-6} + q^{-10} +2 q^{-14} -2 q^{-16} + q^{-20} + q^{-22} +4 q^{-24} -4 q^{-26} -2 q^{-28} + q^{-30} +4 q^{-32} +5 q^{-34} -5 q^{-36} -5 q^{-38} +7 q^{-42} +8 q^{-44} -4 q^{-46} -8 q^{-48} -3 q^{-50} +6 q^{-52} +10 q^{-54} -7 q^{-58} -8 q^{-60} - q^{-62} +7 q^{-64} +4 q^{-66} +2 q^{-68} -3 q^{-70} -6 q^{-72} - q^{-74} +4 q^{-76} +6 q^{-78} +2 q^{-80} -6 q^{-82} -6 q^{-84} - q^{-86} +4 q^{-88} +3 q^{-90} -4 q^{-92} -6 q^{-94} +3 q^{-98} + q^{-100} -4 q^{-102} -5 q^{-104} +2 q^{-106} +5 q^{-108} +3 q^{-110} -3 q^{-112} -5 q^{-114} +2 q^{-116} +5 q^{-118} +5 q^{-120} + q^{-122} -5 q^{-124} -2 q^{-126} - q^{-128} +3 q^{-130} +4 q^{-132} -2 q^{-134} -2 q^{-136} -4 q^{-138} - q^{-140} +5 q^{-142} +3 q^{-144} +3 q^{-146} -3 q^{-148} -5 q^{-150} - q^{-152} + q^{-154} +6 q^{-156} +2 q^{-158} -3 q^{-160} -4 q^{-162} -4 q^{-164} +3 q^{-166} +4 q^{-168} +2 q^{-170} - q^{-172} -5 q^{-174} - q^{-176} + q^{-178} +2 q^{-180} +2 q^{-182} - q^{-184} - q^{-186} - q^{-188} + q^{-192} }[/math]
5	[math]\displaystyle{ q^{-5} - q^{-7} + q^{-11} + q^{-15} - q^{-19} +2 q^{-23} +2 q^{-25} -2 q^{-29} -3 q^{-31} + q^{-33} +5 q^{-35} +6 q^{-37} -2 q^{-39} -7 q^{-41} -5 q^{-43} +3 q^{-45} +11 q^{-47} +8 q^{-49} -4 q^{-51} -13 q^{-53} -7 q^{-55} +7 q^{-57} +14 q^{-59} +8 q^{-61} -7 q^{-63} -17 q^{-65} -11 q^{-67} +8 q^{-69} +21 q^{-71} +13 q^{-73} -4 q^{-75} -17 q^{-77} -18 q^{-79} - q^{-81} +17 q^{-83} +17 q^{-85} +5 q^{-87} -8 q^{-89} -17 q^{-91} -12 q^{-93} - q^{-95} +10 q^{-97} +14 q^{-99} +8 q^{-101} -2 q^{-103} -13 q^{-105} -15 q^{-107} -6 q^{-109} +8 q^{-111} +15 q^{-113} +10 q^{-115} -3 q^{-117} -13 q^{-119} -11 q^{-121} + q^{-123} +8 q^{-125} +7 q^{-127} -2 q^{-129} -8 q^{-131} -5 q^{-133} +2 q^{-135} +5 q^{-137} +4 q^{-139} -6 q^{-141} -10 q^{-143} -3 q^{-145} +7 q^{-147} +12 q^{-149} +8 q^{-151} -6 q^{-153} -14 q^{-155} -8 q^{-157} +5 q^{-159} +15 q^{-161} +14 q^{-163} + q^{-165} -13 q^{-167} -14 q^{-169} -3 q^{-171} +10 q^{-173} +17 q^{-175} +8 q^{-177} -6 q^{-179} -14 q^{-181} -12 q^{-183} + q^{-185} +10 q^{-187} +11 q^{-189} +4 q^{-191} -5 q^{-193} -11 q^{-195} -10 q^{-197} - q^{-199} +6 q^{-201} +8 q^{-203} +7 q^{-205} + q^{-207} -6 q^{-209} -8 q^{-211} -4 q^{-213} +6 q^{-217} +8 q^{-219} +5 q^{-221} -2 q^{-223} -7 q^{-225} -8 q^{-227} -5 q^{-229} +2 q^{-231} +8 q^{-233} +8 q^{-235} +3 q^{-237} -4 q^{-239} -8 q^{-241} -7 q^{-243} - q^{-245} +5 q^{-247} +8 q^{-249} +5 q^{-251} - q^{-253} -5 q^{-255} -6 q^{-257} -3 q^{-259} +2 q^{-261} +5 q^{-263} +3 q^{-265} + q^{-267} - q^{-269} -3 q^{-271} -2 q^{-273} + q^{-277} + q^{-279} + q^{-281} - q^{-285} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{-2} - q^{-4} + q^{-8} + q^{-12} + q^{-14} + q^{-16} + q^{-18} + q^{-22} - q^{-26} - q^{-30} - q^{-32} }[/math]
1,1	[math]\displaystyle{ q^{-4} -2 q^{-6} +2 q^{-8} -2 q^{-10} +5 q^{-12} -4 q^{-14} +4 q^{-16} -2 q^{-18} +5 q^{-20} -4 q^{-22} +4 q^{-24} +6 q^{-28} +4 q^{-32} +2 q^{-34} + q^{-36} +2 q^{-38} -4 q^{-40} +2 q^{-42} -11 q^{-44} +10 q^{-46} -16 q^{-48} +12 q^{-50} -16 q^{-52} +14 q^{-54} -10 q^{-56} +8 q^{-58} -3 q^{-60} +2 q^{-62} +4 q^{-64} -8 q^{-66} +7 q^{-68} -12 q^{-70} +10 q^{-72} -10 q^{-74} +6 q^{-76} -4 q^{-78} +4 q^{-80} + q^{-84} }[/math]
2,0	[math]\displaystyle{ q^{-4} - q^{-6} - q^{-8} +2 q^{-10} + q^{-12} - q^{-14} +2 q^{-18} + q^{-20} -2 q^{-22} +3 q^{-26} +2 q^{-28} + q^{-30} +2 q^{-32} + q^{-34} +2 q^{-36} + q^{-38} - q^{-42} + q^{-46} -2 q^{-50} - q^{-52} - q^{-56} -3 q^{-58} -2 q^{-60} -2 q^{-66} - q^{-68} + q^{-70} + q^{-72} - q^{-76} + q^{-78} + q^{-80} + q^{-82} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{-4} - q^{-6} - q^{-8} +2 q^{-10} - q^{-14} +2 q^{-16} + q^{-18} +2 q^{-22} +2 q^{-24} + q^{-28} + q^{-30} + q^{-32} +2 q^{-36} +3 q^{-38} + q^{-40} + q^{-44} -3 q^{-46} -3 q^{-48} -3 q^{-50} -4 q^{-52} -2 q^{-54} + q^{-60} +2 q^{-62} + q^{-66} }[/math]
1,0,0	[math]\displaystyle{ q^{-3} - q^{-5} + q^{-11} + q^{-15} + q^{-17} + q^{-19} + q^{-21} + q^{-23} + q^{-25} + q^{-29} - q^{-35} - q^{-39} - q^{-41} - q^{-43} }[/math]

B2 Invariants.

Weight

Invariant

0,1

[math]\displaystyle{ q^{-4} - q^{-6} + q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14} +2 q^{-16} - q^{-18} +2 q^{-20} +2 q^{-26} - q^{-28} +3 q^{-30} -3 q^{-32} +4 q^{-34} -4 q^{-36} +3 q^{-38} -3 q^{-40} +2 q^{-42} - q^{-44} + q^{-46} + q^{-48} - q^{-50} +2 q^{-52} -2 q^{-54} +2 q^{-56} -2 q^{-58} + q^{-60} -2 q^{-62} - q^{-66} }[/math]

1,0

[math]\displaystyle{ q^{-6} - q^{-10} - q^{-12} +2 q^{-16} + q^{-18} - q^{-20} - q^{-22} +2 q^{-26} + q^{-28} - q^{-32} + q^{-34} +2 q^{-36} + q^{-38} - q^{-40} + q^{-44} +2 q^{-46} - q^{-50} +2 q^{-54} + q^{-56} + q^{-60} +2 q^{-62} + q^{-64} - q^{-66} - q^{-68} + q^{-70} + q^{-72} - q^{-74} -4 q^{-76} -2 q^{-78} -2 q^{-84} -3 q^{-86} - q^{-88} + q^{-90} + q^{-92} - q^{-94} - q^{-96} + q^{-98} +2 q^{-100} + q^{-108} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-22} +3 q^{-24} -2 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} - q^{-36} +3 q^{-38} -2 q^{-40} + q^{-42} +2 q^{-48} - q^{-50} + q^{-52} - q^{-54} + q^{-56} - q^{-60} + q^{-62} + q^{-66} + q^{-72} + q^{-74} +2 q^{-78} -2 q^{-80} +5 q^{-82} - q^{-84} - q^{-86} +5 q^{-88} -4 q^{-90} +6 q^{-92} -2 q^{-94} - q^{-96} +3 q^{-98} -2 q^{-100} +4 q^{-102} -3 q^{-104} - q^{-110} + q^{-112} -2 q^{-114} - q^{-116} + q^{-118} -3 q^{-120} -2 q^{-124} -2 q^{-126} +3 q^{-128} -6 q^{-130} +3 q^{-132} -2 q^{-134} -2 q^{-136} +4 q^{-138} -5 q^{-140} +3 q^{-142} - q^{-144} + q^{-148} -2 q^{-150} +2 q^{-152} + q^{-156} }[/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.

In[3]:= K = Knot["9 5"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= [math]\displaystyle{ 6 t-11+6 t^{-1} }[/math]

In[5]:= Conway[K][z]

Out[5]= [math]\displaystyle{ 6 z^2+1 }[/math]

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= [math]\displaystyle{ \{1\} }[/math]

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 23, 2 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= [math]\displaystyle{ -q^{10}+q^9-2 q^8+3 q^7-3 q^6+4 q^5-3 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{ z^2 a^{-2} +2 z^2 a^{-4} +2 z^2 a^{-6} +z^2 a^{-8} + a^{-4} + a^{-6} - 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} +2 z^7 a^{-7} +3 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-6} -2 z^6 a^{-8} -5 z^6 a^{-10} +3 z^5 a^{-5} -5 z^5 a^{-7} -14 z^5 a^{-9} -6 z^5 a^{-11} +3 z^4 a^{-4} -7 z^4 a^{-6} -3 z^4 a^{-8} +7 z^4 a^{-10} +2 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +18 z^3 a^{-9} +11 z^3 a^{-11} +z^2 a^{-2} -3 z^2 a^{-4} +3 z^2 a^{-6} +4 z^2 a^{-8} -3 z^2 a^{-10} -6 z a^{-9} -6 z a^{-11} + a^{-4} - a^{-6} + 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.

In[3]:= K = Knot["9 5"];

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{ 6 t-11+6 t^{-1} }[/math], [math]\displaystyle{ -q^{10}+q^9-2 q^8+3 q^7-3 q^6+4 q^5-3 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]= {}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(6, 15)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

[math]\displaystyle{ 24 }[/math]

[math]\displaystyle{ 120 }[/math]

[math]\displaystyle{ 288 }[/math]

[math]\displaystyle{ 780 }[/math]

[math]\displaystyle{ 140 }[/math]

[math]\displaystyle{ 2880 }[/math]

[math]\displaystyle{ 5520 }[/math]

[math]\displaystyle{ 960 }[/math]

[math]\displaystyle{ 920 }[/math]

[math]\displaystyle{ 2304 }[/math]

[math]\displaystyle{ 7200 }[/math]

[math]\displaystyle{ 18720 }[/math]

[math]\displaystyle{ 3360 }[/math]

[math]\displaystyle{ \frac{196391}{5} }[/math]

[math]\displaystyle{ -\frac{14492}{15} }[/math]

[math]\displaystyle{ \frac{274084}{15} }[/math]

[math]\displaystyle{ \frac{1193}{3} }[/math]

[math]\displaystyle{ \frac{13031}{5} }[/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 9 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

0

1

2

3

4

5

6

7

8

9

χ

21

1

-1

19

0

17

2

1

-1

15

1

13

2

0

11

2

1

9

1

2

1

7

2

0

5

1

3

1

2

-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}^{2}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{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^{29}-q^{28}-q^{27}+3 q^{26}-q^{25}-4 q^{24}+5 q^{23}-7 q^{21}+6 q^{20}+2 q^{19}-9 q^{18}+6 q^{17}+4 q^{16}-10 q^{15}+5 q^{14}+5 q^{13}-8 q^{12}+3 q^{11}+5 q^{10}-7 q^9+3 q^8+3 q^7-5 q^6+3 q^5+q^4-2 q^3+q^2 }[/math]
3	[math]\displaystyle{ -q^{57}+q^{56}+q^{55}-3 q^{53}+3 q^{51}+3 q^{50}-5 q^{49}-3 q^{48}+3 q^{47}+7 q^{46}-4 q^{45}-6 q^{44}+q^{43}+8 q^{42}-q^{41}-7 q^{40}-q^{39}+7 q^{38}+q^{37}-6 q^{36}-q^{35}+5 q^{34}+2 q^{33}-6 q^{32}+2 q^{30}+q^{29}-4 q^{28}+3 q^{27}-2 q^{26}+5 q^{23}-4 q^{22}-2 q^{21}+6 q^{19}-2 q^{18}-2 q^{17}-2 q^{16}+3 q^{15}+2 q^{14}-q^{13}-3 q^{12}+q^{11}+3 q^{10}-3 q^8+q^7+q^6+q^5-2 q^4+q^3 }[/math]
4	[math]\displaystyle{ q^{94}-q^{93}-q^{92}+4 q^{89}-q^{88}-2 q^{87}-2 q^{86}-4 q^{85}+8 q^{84}+2 q^{83}-3 q^{81}-11 q^{80}+8 q^{79}+3 q^{78}+5 q^{77}+q^{76}-16 q^{75}+6 q^{74}-q^{73}+7 q^{72}+7 q^{71}-16 q^{70}+8 q^{69}-7 q^{68}+4 q^{67}+9 q^{66}-16 q^{65}+14 q^{64}-8 q^{63}+8 q^{61}-19 q^{60}+20 q^{59}-4 q^{58}+5 q^{56}-26 q^{55}+22 q^{54}+2 q^{53}+2 q^{52}+2 q^{51}-33 q^{50}+23 q^{49}+7 q^{48}+4 q^{47}-q^{46}-39 q^{45}+25 q^{44}+14 q^{43}+5 q^{42}-6 q^{41}-44 q^{40}+25 q^{39}+22 q^{38}+9 q^{37}-8 q^{36}-49 q^{35}+20 q^{34}+25 q^{33}+14 q^{32}-6 q^{31}-46 q^{30}+12 q^{29}+18 q^{28}+15 q^{27}+q^{26}-36 q^{25}+8 q^{24}+9 q^{23}+10 q^{22}+5 q^{21}-24 q^{20}+7 q^{19}+2 q^{18}+5 q^{17}+5 q^{16}-14 q^{15}+6 q^{14}-q^{13}+2 q^{12}+3 q^{11}-6 q^{10}+3 q^9-q^8+q^7+q^6-2 q^5+q^4 }[/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).

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Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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