9 11

9_10

9_12

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 11's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 11 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8
Gauss code	-1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code	4 10 14 16 12 2 18 6 8
Conway Notation	[4122]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{11, 7}, {8, 6}, {7, 10}, {1, 8}, {9, 11}, {10, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 1}, {2, 9}]

[edit Notes on presentations of 9 11]

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 11"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5

In[6]:=

DTCode[K]

Out[6]=

4 10 14 16 12 2 18 6 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[4122]

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,\{1,1,1,1,-2,1,3,-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]]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [1][-12] Hyperbolic Volume 8.28859 A-Polynomial See Data:9 11/A-polynomial

[edit Notes for 9 11'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{ 3 }[/math] Rasmussen s-Invariant 4

[edit Notes for 9 11's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-7 t+7-7 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 33, 4 }
Jones polynomial	[math]\displaystyle{ -q^9+2 q^8-4 q^7+5 q^6-5 q^5+6 q^4-4 q^3+3 q^2-2 q+1 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -4 z^4 a^{-4} +2 z^4 a^{-6} +3 z^2 a^{-2} -4 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} + a^{-2} - a^{-4} +3 a^{-6} -2 a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +4 z^7 a^{-5} +2 z^7 a^{-7} +z^6 a^{-2} -z^6 a^{-4} +z^6 a^{-6} +3 z^6 a^{-8} -8 z^5 a^{-3} -12 z^5 a^{-5} -z^5 a^{-7} +3 z^5 a^{-9} -4 z^4 a^{-2} -5 z^4 a^{-4} -7 z^4 a^{-6} -4 z^4 a^{-8} +2 z^4 a^{-10} +8 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +4 z^2 a^{-2} +5 z^2 a^{-4} +6 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} -z a^{-3} -2 z a^{-5} +2 z a^{-7} +2 z a^{-9} -z a^{-11} - a^{-2} - a^{-4} -3 a^{-6} -2 a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ 1- q^{-8} +2 q^{-10} +2 q^{-14} + q^{-16} + q^{-20} - q^{-22} - q^{-26} - q^{-28} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-2} - q^{-4} +3 q^{-6} -4 q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} +10 q^{-16} -12 q^{-18} +13 q^{-20} -7 q^{-22} -2 q^{-24} +10 q^{-26} -17 q^{-28} +17 q^{-30} -12 q^{-32} +2 q^{-34} +8 q^{-36} -13 q^{-38} +12 q^{-40} -7 q^{-42} -2 q^{-44} +7 q^{-46} -8 q^{-48} +4 q^{-50} - q^{-52} -5 q^{-54} +16 q^{-56} -12 q^{-58} +10 q^{-60} - q^{-62} -8 q^{-64} +19 q^{-66} -20 q^{-68} +18 q^{-70} -9 q^{-72} +16 q^{-76} -19 q^{-78} +17 q^{-80} -8 q^{-82} -2 q^{-84} +8 q^{-86} -10 q^{-88} +4 q^{-90} -4 q^{-94} +7 q^{-96} -6 q^{-98} +3 q^{-102} -10 q^{-104} +10 q^{-106} -9 q^{-108} +4 q^{-110} - q^{-112} -4 q^{-114} +8 q^{-116} -11 q^{-118} +11 q^{-120} -7 q^{-122} +2 q^{-124} + q^{-126} -6 q^{-128} +6 q^{-130} -6 q^{-132} +5 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_11.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q- q^{-1} + q^{-3} - q^{-5} +2 q^{-7} + q^{-9} + q^{-13} -2 q^{-15} + q^{-17} - q^{-19} }[/math]
2	[math]\displaystyle{ q^6-q^4-2 q^2+3+ q^{-2} -4 q^{-4} +3 q^{-6} +4 q^{-8} -4 q^{-10} + q^{-12} +4 q^{-14} -4 q^{-16} - q^{-18} +4 q^{-20} -2 q^{-24} +2 q^{-26} +3 q^{-28} -2 q^{-30} -4 q^{-32} +4 q^{-34} - q^{-36} -5 q^{-38} +4 q^{-40} - q^{-42} -3 q^{-44} +3 q^{-46} - q^{-50} + q^{-52} }[/math]
3	[math]\displaystyle{ q^{15}-q^{13}-2 q^{11}+4 q^7+3 q^5-5 q^3-6 q+4 q^{-1} +9 q^{-3} -11 q^{-7} -4 q^{-9} +11 q^{-11} +8 q^{-13} -7 q^{-15} -10 q^{-17} +4 q^{-19} +13 q^{-21} + q^{-23} -9 q^{-25} -3 q^{-27} +10 q^{-29} +4 q^{-31} -8 q^{-33} -8 q^{-35} +8 q^{-37} +8 q^{-39} -7 q^{-41} -9 q^{-43} +4 q^{-45} +10 q^{-47} -8 q^{-51} -6 q^{-53} +6 q^{-55} +8 q^{-57} - q^{-59} -14 q^{-61} -2 q^{-63} +10 q^{-65} +5 q^{-67} -9 q^{-69} -4 q^{-71} +6 q^{-73} +2 q^{-75} -3 q^{-77} +3 q^{-81} - q^{-83} -2 q^{-85} +2 q^{-87} +2 q^{-89} - q^{-91} - q^{-93} + q^{-97} - q^{-99} }[/math]
4	[math]\displaystyle{ q^{28}-q^{26}-2 q^{24}+q^{20}+6 q^{18}+q^{16}-5 q^{14}-7 q^{12}-7 q^{10}+12 q^8+13 q^6+3 q^4-10 q^2-24+16 q^{-4} +23 q^{-6} +9 q^{-8} -29 q^{-10} -21 q^{-12} -4 q^{-14} +25 q^{-16} +33 q^{-18} -5 q^{-20} -21 q^{-22} -31 q^{-24} +35 q^{-28} +21 q^{-30} +3 q^{-32} -36 q^{-34} -28 q^{-36} +14 q^{-38} +30 q^{-40} +30 q^{-42} -20 q^{-44} -37 q^{-46} -6 q^{-48} +23 q^{-50} +36 q^{-52} -9 q^{-54} -36 q^{-56} -13 q^{-58} +20 q^{-60} +30 q^{-62} -7 q^{-64} -32 q^{-66} -16 q^{-68} +19 q^{-70} +29 q^{-72} +3 q^{-74} -25 q^{-76} -27 q^{-78} +6 q^{-80} +22 q^{-82} +24 q^{-84} + q^{-86} -29 q^{-88} -25 q^{-90} -6 q^{-92} +38 q^{-94} +35 q^{-96} -11 q^{-98} -37 q^{-100} -38 q^{-102} +22 q^{-104} +47 q^{-106} +14 q^{-108} -18 q^{-110} -45 q^{-112} - q^{-114} +28 q^{-116} +17 q^{-118} +5 q^{-120} -26 q^{-122} -7 q^{-124} +7 q^{-126} +5 q^{-128} +11 q^{-130} -8 q^{-132} -2 q^{-134} -2 q^{-136} -3 q^{-138} +7 q^{-140} -2 q^{-142} + q^{-144} - q^{-146} -3 q^{-148} +2 q^{-150} - q^{-152} + q^{-154} - q^{-158} + q^{-160} }[/math]
5	[math]\displaystyle{ q^{45}-q^{43}-2 q^{41}+q^{37}+3 q^{35}+4 q^{33}+q^{31}-7 q^{29}-9 q^{27}-5 q^{25}+3 q^{23}+15 q^{21}+18 q^{19}+5 q^{17}-18 q^{15}-28 q^{13}-20 q^{11}+5 q^9+35 q^7+41 q^5+16 q^3-27 q-53 q^{-1} -43 q^{-3} - q^{-5} +51 q^{-7} +67 q^{-9} +35 q^{-11} -27 q^{-13} -69 q^{-15} -66 q^{-17} -16 q^{-19} +51 q^{-21} +84 q^{-23} +59 q^{-25} -9 q^{-27} -73 q^{-29} -89 q^{-31} -43 q^{-33} +41 q^{-35} +100 q^{-37} +92 q^{-39} +10 q^{-41} -83 q^{-43} -118 q^{-45} -67 q^{-47} +48 q^{-49} +132 q^{-51} +111 q^{-53} -7 q^{-55} -120 q^{-57} -139 q^{-59} -44 q^{-61} +98 q^{-63} +156 q^{-65} +73 q^{-67} -67 q^{-69} -145 q^{-71} -97 q^{-73} +40 q^{-75} +135 q^{-77} +98 q^{-79} -23 q^{-81} -112 q^{-83} -94 q^{-85} +12 q^{-87} +99 q^{-89} +81 q^{-91} -16 q^{-93} -90 q^{-95} -73 q^{-97} +16 q^{-99} +86 q^{-101} +69 q^{-103} -18 q^{-105} -88 q^{-107} -75 q^{-109} +4 q^{-111} +75 q^{-113} +87 q^{-115} +31 q^{-117} -52 q^{-119} -90 q^{-121} -74 q^{-123} +5 q^{-125} +91 q^{-127} +114 q^{-129} +55 q^{-131} -56 q^{-133} -148 q^{-135} -122 q^{-137} +18 q^{-139} +149 q^{-141} +166 q^{-143} +51 q^{-145} -127 q^{-147} -199 q^{-149} -96 q^{-151} +86 q^{-153} +190 q^{-155} +132 q^{-157} -37 q^{-159} -167 q^{-161} -141 q^{-163} - q^{-165} +127 q^{-167} +128 q^{-169} +23 q^{-171} -82 q^{-173} -104 q^{-175} -34 q^{-177} +51 q^{-179} +74 q^{-181} +33 q^{-183} -25 q^{-185} -47 q^{-187} -29 q^{-189} +8 q^{-191} +28 q^{-193} +22 q^{-195} - q^{-197} -15 q^{-199} -14 q^{-201} -4 q^{-203} +6 q^{-205} +9 q^{-207} +4 q^{-209} -4 q^{-211} -3 q^{-213} - q^{-215} - q^{-217} +2 q^{-219} +2 q^{-221} - q^{-223} + q^{-227} - q^{-229} + q^{-233} - q^{-235} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ 1- q^{-8} +2 q^{-10} +2 q^{-14} + q^{-16} + q^{-20} - q^{-22} - q^{-26} - q^{-28} }[/math]
1,1	[math]\displaystyle{ q^4-2 q^2+6-12 q^{-2} +19 q^{-4} -30 q^{-6} +40 q^{-8} -44 q^{-10} +49 q^{-12} -44 q^{-14} +38 q^{-16} -18 q^{-18} +3 q^{-20} +18 q^{-22} -34 q^{-24} +54 q^{-26} -68 q^{-28} +74 q^{-30} -76 q^{-32} +74 q^{-34} -63 q^{-36} +50 q^{-38} -32 q^{-40} +14 q^{-42} -2 q^{-44} -10 q^{-46} +14 q^{-48} -22 q^{-50} +25 q^{-52} -26 q^{-54} +24 q^{-56} -28 q^{-58} +26 q^{-60} -22 q^{-62} +18 q^{-64} -14 q^{-66} +11 q^{-68} -6 q^{-70} +4 q^{-72} -2 q^{-74} + q^{-76} }[/math]
2,0	[math]\displaystyle{ q^4-1- q^{-2} + q^{-4} + q^{-6} -2 q^{-8} +4 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} + q^{-28} -2 q^{-30} +3 q^{-32} +3 q^{-34} + q^{-38} +4 q^{-40} -4 q^{-44} - q^{-46} -2 q^{-50} -4 q^{-52} - q^{-54} - q^{-58} + q^{-66} + q^{-68} + q^{-70} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ 1- q^{-2} + q^{-4} + q^{-6} -2 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} -4 q^{-20} +3 q^{-22} +3 q^{-24} -2 q^{-26} +5 q^{-28} +4 q^{-30} +2 q^{-32} - q^{-34} -6 q^{-40} -3 q^{-42} +2 q^{-44} -4 q^{-46} -2 q^{-48} +5 q^{-50} -2 q^{-52} -2 q^{-54} +3 q^{-56} - q^{-60} + q^{-62} }[/math]
1,0,0	[math]\displaystyle{ q^{-1} + q^{-5} - q^{-7} + q^{-9} - q^{-11} + q^{-13} + q^{-17} + q^{-19} + q^{-21} +2 q^{-23} +2 q^{-27} - q^{-29} -2 q^{-33} - q^{-35} - q^{-37} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ 1- q^{-2} +3 q^{-4} -3 q^{-6} +4 q^{-8} -5 q^{-10} +5 q^{-12} -4 q^{-14} +3 q^{-16} -2 q^{-18} +3 q^{-22} -5 q^{-24} +8 q^{-26} -7 q^{-28} +10 q^{-30} -8 q^{-32} +9 q^{-34} -6 q^{-36} +4 q^{-38} -2 q^{-40} - q^{-42} +2 q^{-44} -4 q^{-46} +4 q^{-48} -5 q^{-50} +4 q^{-52} -4 q^{-54} +3 q^{-56} -2 q^{-58} + q^{-60} - q^{-62} }[/math]
1,0	[math]\displaystyle{ q^2- q^{-2} - q^{-4} +2 q^{-6} +2 q^{-8} -2 q^{-10} -3 q^{-12} + q^{-14} +5 q^{-16} +2 q^{-18} -4 q^{-20} -3 q^{-22} +3 q^{-24} +5 q^{-26} -5 q^{-30} -3 q^{-32} +3 q^{-34} +4 q^{-36} -3 q^{-40} +2 q^{-42} +4 q^{-44} +2 q^{-46} -2 q^{-48} + q^{-50} +4 q^{-52} +2 q^{-54} -4 q^{-56} -3 q^{-58} +2 q^{-60} +2 q^{-62} -3 q^{-64} -5 q^{-66} - q^{-68} +3 q^{-70} + q^{-72} -4 q^{-74} -4 q^{-76} + q^{-78} +5 q^{-80} + q^{-82} -3 q^{-84} -3 q^{-86} + q^{-88} +3 q^{-90} + q^{-92} - q^{-94} - q^{-96} + q^{-100} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{-2} - q^{-4} +2 q^{-6} -2 q^{-8} +4 q^{-10} -3 q^{-12} +4 q^{-14} -3 q^{-16} +5 q^{-18} -3 q^{-20} +2 q^{-22} -2 q^{-24} + q^{-26} -2 q^{-30} +3 q^{-32} -3 q^{-34} +7 q^{-36} -4 q^{-38} +10 q^{-40} -3 q^{-42} +10 q^{-44} -5 q^{-46} +7 q^{-48} -5 q^{-50} +2 q^{-52} -5 q^{-54} -3 q^{-56} -3 q^{-58} -3 q^{-60} + q^{-62} -4 q^{-64} +2 q^{-66} -3 q^{-68} +5 q^{-70} -3 q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-2} - q^{-4} +3 q^{-6} -4 q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} +10 q^{-16} -12 q^{-18} +13 q^{-20} -7 q^{-22} -2 q^{-24} +10 q^{-26} -17 q^{-28} +17 q^{-30} -12 q^{-32} +2 q^{-34} +8 q^{-36} -13 q^{-38} +12 q^{-40} -7 q^{-42} -2 q^{-44} +7 q^{-46} -8 q^{-48} +4 q^{-50} - q^{-52} -5 q^{-54} +16 q^{-56} -12 q^{-58} +10 q^{-60} - q^{-62} -8 q^{-64} +19 q^{-66} -20 q^{-68} +18 q^{-70} -9 q^{-72} +16 q^{-76} -19 q^{-78} +17 q^{-80} -8 q^{-82} -2 q^{-84} +8 q^{-86} -10 q^{-88} +4 q^{-90} -4 q^{-94} +7 q^{-96} -6 q^{-98} +3 q^{-102} -10 q^{-104} +10 q^{-106} -9 q^{-108} +4 q^{-110} - q^{-112} -4 q^{-114} +8 q^{-116} -11 q^{-118} +11 q^{-120} -7 q^{-122} +2 q^{-124} + q^{-126} -6 q^{-128} +6 q^{-130} -6 q^{-132} +5 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + 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["9 11"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-7 t+7-7 t^{-1} +5 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-z^4+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]=

{ 33, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -4 z^4 a^{-4} +2 z^4 a^{-6} +3 z^2 a^{-2} -4 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} + a^{-2} - a^{-4} +3 a^{-6} -2 a^{-8} }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^8 a^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +4 z^7 a^{-5} +2 z^7 a^{-7} +z^6 a^{-2} -z^6 a^{-4} +z^6 a^{-6} +3 z^6 a^{-8} -8 z^5 a^{-3} -12 z^5 a^{-5} -z^5 a^{-7} +3 z^5 a^{-9} -4 z^4 a^{-2} -5 z^4 a^{-4} -7 z^4 a^{-6} -4 z^4 a^{-8} +2 z^4 a^{-10} +8 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +4 z^2 a^{-2} +5 z^2 a^{-4} +6 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} -z a^{-3} -2 z a^{-5} +2 z a^{-7} +2 z a^{-9} -z a^{-11} - a^{-2} - a^{-4} -3 a^{-6} -2 a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n95,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 11"];

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

{K11n95,}

In[6]:=

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

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

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{}

Vassiliev invariants

V2 and V3:

(4, 9)

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{ 72 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1160}{3} }[/math] [math]\displaystyle{ \frac{208}{3} }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 2288 }[/math] [math]\displaystyle{ 384 }[/math] [math]\displaystyle{ 392 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 2592 }[/math] [math]\displaystyle{ \frac{18560}{3} }[/math] [math]\displaystyle{ \frac{3328}{3} }[/math] [math]\displaystyle{ \frac{204662}{15} }[/math] [math]\displaystyle{ -\frac{5408}{15} }[/math] [math]\displaystyle{ \frac{284288}{45} }[/math] [math]\displaystyle{ \frac{1450}{9} }[/math] [math]\displaystyle{ \frac{13382}{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]4 is the signature of 9 11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 19                   1 -1 17                 1   1 15               3 1   -2 13             2 1     1 11           3 3       0 9         3 2         1 7       1 3           2 5     2 3             -1 3   1 2               1 1   1                 -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=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}-2 q^{24}+q^{23}+4 q^{22}-8 q^{21}+3 q^{20}+9 q^{19}-17 q^{18}+7 q^{17}+14 q^{16}-25 q^{15}+9 q^{14}+19 q^{13}-26 q^{12}+5 q^{11}+21 q^{10}-22 q^9+18 q^7-14 q^6-3 q^5+13 q^4-6 q^3-4 q^2+6 q-1-2 q^{-1} + q^{-2} }[/math]
3	[math]\displaystyle{ -q^{48}+2 q^{47}-q^{46}-q^{45}-q^{44}+5 q^{43}-q^{42}-5 q^{41}+9 q^{39}-4 q^{38}-8 q^{37}+5 q^{36}+13 q^{35}-14 q^{34}-13 q^{33}+19 q^{32}+18 q^{31}-26 q^{30}-25 q^{29}+32 q^{28}+27 q^{27}-28 q^{26}-37 q^{25}+30 q^{24}+35 q^{23}-18 q^{22}-43 q^{21}+17 q^{20}+37 q^{19}-3 q^{18}-43 q^{17}+q^{16}+37 q^{15}+9 q^{14}-37 q^{13}-12 q^{12}+31 q^{11}+19 q^{10}-25 q^9-21 q^8+17 q^7+22 q^6-10 q^5-18 q^4+2 q^3+15 q^2+q-9-3 q^{-1} +5 q^{-2} +2 q^{-3} - q^{-4} -2 q^{-5} + q^{-6} }[/math]
4	[math]\displaystyle{ q^{78}-2 q^{77}+q^{76}+q^{75}-2 q^{74}+4 q^{73}-7 q^{72}+3 q^{71}+3 q^{70}-5 q^{69}+13 q^{68}-17 q^{67}+4 q^{66}+3 q^{65}-11 q^{64}+32 q^{63}-23 q^{62}+6 q^{61}-11 q^{60}-30 q^{59}+63 q^{58}-11 q^{57}+17 q^{56}-40 q^{55}-74 q^{54}+90 q^{53}+21 q^{52}+50 q^{51}-65 q^{50}-134 q^{49}+91 q^{48}+47 q^{47}+96 q^{46}-62 q^{45}-178 q^{44}+72 q^{43}+43 q^{42}+126 q^{41}-39 q^{40}-180 q^{39}+56 q^{38}+10 q^{37}+128 q^{36}-11 q^{35}-154 q^{34}+46 q^{33}-25 q^{32}+112 q^{31}+14 q^{30}-117 q^{29}+36 q^{28}-58 q^{27}+89 q^{26}+41 q^{25}-72 q^{24}+23 q^{23}-87 q^{22}+58 q^{21}+58 q^{20}-22 q^{19}+23 q^{18}-103 q^{17}+16 q^{16}+50 q^{15}+17 q^{14}+41 q^{13}-89 q^{12}-19 q^{11}+19 q^{10}+27 q^9+57 q^8-51 q^7-27 q^6-10 q^5+10 q^4+49 q^3-13 q^2-13 q-17-6 q^{-1} +25 q^{-2} + q^{-3} -7 q^{-5} -7 q^{-6} +6 q^{-7} + q^{-8} +2 q^{-9} - q^{-10} -2 q^{-11} + q^{-12} }[/math]
5	[math]\displaystyle{ -q^{115}+2 q^{114}-q^{113}-q^{112}+2 q^{111}-q^{110}-2 q^{109}+5 q^{108}-q^{107}-4 q^{106}+2 q^{105}-3 q^{104}-3 q^{103}+13 q^{102}+4 q^{101}-7 q^{100}-8 q^{99}-13 q^{98}-4 q^{97}+27 q^{96}+27 q^{95}-q^{94}-28 q^{93}-50 q^{92}-22 q^{91}+49 q^{90}+85 q^{89}+40 q^{88}-51 q^{87}-135 q^{86}-92 q^{85}+71 q^{84}+190 q^{83}+145 q^{82}-52 q^{81}-263 q^{80}-232 q^{79}+45 q^{78}+320 q^{77}+314 q^{76}+6 q^{75}-367 q^{74}-414 q^{73}-58 q^{72}+392 q^{71}+492 q^{70}+121 q^{69}-384 q^{68}-545 q^{67}-198 q^{66}+366 q^{65}+584 q^{64}+232 q^{63}-325 q^{62}-568 q^{61}-284 q^{60}+287 q^{59}+568 q^{58}+270 q^{57}-242 q^{56}-512 q^{55}-296 q^{54}+216 q^{53}+489 q^{52}+257 q^{51}-172 q^{50}-425 q^{49}-279 q^{48}+146 q^{47}+400 q^{46}+240 q^{45}-98 q^{44}-328 q^{43}-261 q^{42}+59 q^{41}+294 q^{40}+222 q^{39}-9 q^{38}-207 q^{37}-224 q^{36}-36 q^{35}+157 q^{34}+174 q^{33}+69 q^{32}-67 q^{31}-141 q^{30}-94 q^{29}+15 q^{28}+79 q^{27}+88 q^{26}+46 q^{25}-23 q^{24}-73 q^{23}-69 q^{22}-36 q^{21}+37 q^{20}+81 q^{19}+70 q^{18}+9 q^{17}-61 q^{16}-95 q^{15}-47 q^{14}+35 q^{13}+86 q^{12}+73 q^{11}+7 q^{10}-70 q^9-80 q^8-32 q^7+36 q^6+70 q^5+49 q^4-8 q^3-48 q^2-48 q-16+28 q^{-1} +39 q^{-2} +18 q^{-3} -5 q^{-4} -23 q^{-5} -22 q^{-6} -2 q^{-7} +14 q^{-8} +10 q^{-9} +5 q^{-10} -9 q^{-12} -5 q^{-13} +2 q^{-14} +2 q^{-15} + q^{-16} +2 q^{-17} - q^{-18} -2 q^{-19} + q^{-20} }[/math]
6	[math]\displaystyle{ q^{159}-2 q^{158}+q^{157}+q^{156}-2 q^{155}+q^{154}-q^{153}+4 q^{152}-7 q^{151}+2 q^{150}+7 q^{149}-6 q^{148}+4 q^{147}-3 q^{146}+4 q^{145}-20 q^{144}+5 q^{143}+24 q^{142}-7 q^{141}+12 q^{140}-8 q^{139}-5 q^{138}-54 q^{137}+7 q^{136}+58 q^{135}+7 q^{134}+40 q^{133}-14 q^{132}-37 q^{131}-130 q^{130}-6 q^{129}+116 q^{128}+63 q^{127}+116 q^{126}-12 q^{125}-120 q^{124}-291 q^{123}-64 q^{122}+209 q^{121}+215 q^{120}+297 q^{119}+20 q^{118}-292 q^{117}-608 q^{116}-232 q^{115}+323 q^{114}+511 q^{113}+664 q^{112}+152 q^{111}-524 q^{110}-1111 q^{109}-601 q^{108}+354 q^{107}+881 q^{106}+1227 q^{105}+488 q^{104}-661 q^{103}-1664 q^{102}-1161 q^{101}+161 q^{100}+1111 q^{99}+1803 q^{98}+1003 q^{97}-535 q^{96}-2011 q^{95}-1700 q^{94}-220 q^{93}+1046 q^{92}+2132 q^{91}+1460 q^{90}-213 q^{89}-2027 q^{88}-1968 q^{87}-553 q^{86}+788 q^{85}+2129 q^{84}+1654 q^{83}+73 q^{82}-1842 q^{81}-1932 q^{80}-684 q^{79}+546 q^{78}+1940 q^{77}+1608 q^{76}+225 q^{75}-1630 q^{74}-1755 q^{73}-687 q^{72}+371 q^{71}+1717 q^{70}+1490 q^{69}+337 q^{68}-1415 q^{67}-1568 q^{66}-707 q^{65}+179 q^{64}+1469 q^{63}+1388 q^{62}+513 q^{61}-1124 q^{60}-1363 q^{59}-775 q^{58}-91 q^{57}+1136 q^{56}+1257 q^{55}+729 q^{54}-726 q^{53}-1067 q^{52}-805 q^{51}-391 q^{50}+697 q^{49}+1007 q^{48}+865 q^{47}-291 q^{46}-644 q^{45}-685 q^{44}-594 q^{43}+226 q^{42}+607 q^{41}+804 q^{40}+36 q^{39}-180 q^{38}-384 q^{37}-572 q^{36}-118 q^{35}+159 q^{34}+522 q^{33}+117 q^{32}+149 q^{31}-21 q^{30}-318 q^{29}-196 q^{28}-138 q^{27}+171 q^{26}-44 q^{25}+198 q^{24}+193 q^{23}-8 q^{22}-42 q^{21}-155 q^{20}-24 q^{19}-240 q^{18}+32 q^{17}+150 q^{16}+133 q^{15}+124 q^{14}+5 q^{13}+12 q^{12}-266 q^{11}-113 q^{10}-15 q^9+67 q^8+129 q^7+110 q^6+121 q^5-133 q^4-103 q^3-94 q^2-36 q+25+78 q^{-1} +135 q^{-2} -13 q^{-3} -19 q^{-4} -58 q^{-5} -49 q^{-6} -38 q^{-7} +8 q^{-8} +71 q^{-9} +13 q^{-10} +19 q^{-11} -8 q^{-12} -15 q^{-13} -29 q^{-14} -14 q^{-15} +19 q^{-16} +2 q^{-17} +11 q^{-18} +4 q^{-19} +3 q^{-20} -9 q^{-21} -7 q^{-22} +4 q^{-23} -2 q^{-24} +2 q^{-25} + q^{-26} +2 q^{-27} - q^{-28} -2 q^{-29} + q^{-30} }[/math]
7	[math]\displaystyle{ -q^{210}+2 q^{209}-q^{208}-q^{207}+2 q^{206}-q^{205}+q^{204}-q^{203}-2 q^{202}+6 q^{201}-5 q^{200}-3 q^{199}+5 q^{198}-4 q^{197}+5 q^{196}+q^{195}-2 q^{194}+13 q^{193}-17 q^{192}-12 q^{191}+7 q^{190}-7 q^{189}+18 q^{188}+9 q^{187}+6 q^{186}+24 q^{185}-40 q^{184}-38 q^{183}-q^{182}-16 q^{181}+48 q^{180}+38 q^{179}+26 q^{178}+43 q^{177}-80 q^{176}-89 q^{175}-39 q^{174}-29 q^{173}+115 q^{172}+113 q^{171}+64 q^{170}+49 q^{169}-172 q^{168}-205 q^{167}-109 q^{166}-16 q^{165}+291 q^{164}+307 q^{163}+161 q^{162}-16 q^{161}-454 q^{160}-521 q^{159}-270 q^{158}+94 q^{157}+726 q^{156}+830 q^{155}+451 q^{154}-162 q^{153}-1113 q^{152}-1304 q^{151}-738 q^{150}+222 q^{149}+1559 q^{148}+1928 q^{147}+1239 q^{146}-185 q^{145}-2093 q^{144}-2729 q^{143}-1871 q^{142}+23 q^{141}+2534 q^{140}+3585 q^{139}+2730 q^{138}+379 q^{137}-2886 q^{136}-4463 q^{135}-3683 q^{134}-935 q^{133}+3000 q^{132}+5166 q^{131}+4652 q^{130}+1697 q^{129}-2859 q^{128}-5677 q^{127}-5533 q^{126}-2501 q^{125}+2526 q^{124}+5901 q^{123}+6162 q^{122}+3261 q^{121}-2021 q^{120}-5841 q^{119}-6578 q^{118}-3889 q^{117}+1516 q^{116}+5635 q^{115}+6687 q^{114}+4265 q^{113}-1042 q^{112}-5271 q^{111}-6611 q^{110}-4495 q^{109}+705 q^{108}+4953 q^{107}+6390 q^{106}+4489 q^{105}-469 q^{104}-4589 q^{103}-6136 q^{102}-4461 q^{101}+335 q^{100}+4346 q^{99}+5844 q^{98}+4336 q^{97}-184 q^{96}-4049 q^{95}-5621 q^{94}-4306 q^{93}+62 q^{92}+3797 q^{91}+5348 q^{90}+4260 q^{89}+198 q^{88}-3430 q^{87}-5137 q^{86}-4318 q^{85}-473 q^{84}+3046 q^{83}+4820 q^{82}+4332 q^{81}+887 q^{80}-2499 q^{79}-4495 q^{78}-4406 q^{77}-1303 q^{76}+1945 q^{75}+4033 q^{74}+4347 q^{73}+1766 q^{72}-1225 q^{71}-3499 q^{70}-4289 q^{69}-2176 q^{68}+559 q^{67}+2853 q^{66}+4003 q^{65}+2500 q^{64}+193 q^{63}-2103 q^{62}-3659 q^{61}-2703 q^{60}-801 q^{59}+1334 q^{58}+3083 q^{57}+2678 q^{56}+1345 q^{55}-519 q^{54}-2433 q^{53}-2499 q^{52}-1659 q^{51}-152 q^{50}+1671 q^{49}+2072 q^{48}+1755 q^{47}+727 q^{46}-940 q^{45}-1537 q^{44}-1615 q^{43}-1031 q^{42}+302 q^{41}+913 q^{40}+1275 q^{39}+1114 q^{38}+143 q^{37}-345 q^{36}-807 q^{35}-963 q^{34}-378 q^{33}-83 q^{32}+347 q^{31}+661 q^{30}+353 q^{29}+327 q^{28}+48 q^{27}-307 q^{26}-204 q^{25}-356 q^{24}-254 q^{23}+4 q^{22}-43 q^{21}+226 q^{20}+308 q^{19}+179 q^{18}+251 q^{17}-33 q^{16}-215 q^{15}-208 q^{14}-365 q^{13}-148 q^{12}+45 q^{11}+132 q^{10}+370 q^9+259 q^8+93 q^7-10 q^6-268 q^5-257 q^4-185 q^3-124 q^2+145 q+207+194 q^{-1} +171 q^{-2} -32 q^{-3} -96 q^{-4} -140 q^{-5} -188 q^{-6} -54 q^{-7} +30 q^{-8} +86 q^{-9} +140 q^{-10} +56 q^{-11} +27 q^{-12} -9 q^{-13} -91 q^{-14} -63 q^{-15} -43 q^{-16} -5 q^{-17} +48 q^{-18} +26 q^{-19} +28 q^{-20} +29 q^{-21} -10 q^{-22} -18 q^{-23} -25 q^{-24} -18 q^{-25} +10 q^{-26} +4 q^{-28} +13 q^{-29} +4 q^{-30} +2 q^{-31} -6 q^{-32} -7 q^{-33} +2 q^{-34} -2 q^{-36} +2 q^{-37} + q^{-38} +2 q^{-39} - q^{-40} -2 q^{-41} + q^{-42} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

9_10 9_12