9 10

9_9

9_11

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 10's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 10 at Knotilus!

Knot presentations

Planar diagram presentation	X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13
Gauss code	1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3
Dowker-Thistlethwaite code	8 12 14 16 18 2 6 4 10
Conway Notation	[333]

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

[edit Notes on presentations of 9 10]

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 10"];

In[4]:=

PD[K]

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

Out[4]=

X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

8 12 14 16 18 2 6 4 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[333]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{2,3\} }[/math] 3-genus 2 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [3][-14] Hyperbolic Volume 8.77346 A-Polynomial See Data:9 10/A-polynomial

[edit Notes for 9 10'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{ 2 }[/math] Rasmussen s-Invariant -4

[edit Notes for 9 10's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t^2-8 t+9-8 t^{-1} +4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^4+8 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 33, 4 }
Jones polynomial	[math]\displaystyle{ -q^{11}+q^{10}-3 q^9+5 q^8-5 q^7+6 q^6-5 q^5+4 q^4-2 q^3+q^2 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{-4} +2 z^4 a^{-6} +z^4 a^{-8} +2 z^2 a^{-4} +5 z^2 a^{-6} +2 z^2 a^{-8} -z^2 a^{-10} +2 a^{-6} + a^{-8} -2 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} -z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} +2 z^5 a^{-5} -3 z^5 a^{-7} -7 z^5 a^{-9} -z^5 a^{-11} +z^5 a^{-13} +z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +9 z^4 a^{-10} -2 z^4 a^{-12} -3 z^3 a^{-5} +3 z^3 a^{-7} +9 z^3 a^{-9} -z^3 a^{-11} -4 z^3 a^{-13} -2 z^2 a^{-4} +7 z^2 a^{-6} -2 z^2 a^{-8} -11 z^2 a^{-10} -4 z a^{-9} +4 z a^{-13} -2 a^{-6} + a^{-8} +2 a^{-10} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-6} - q^{-8} + q^{-10} +2 q^{-16} +2 q^{-20} + q^{-22} + q^{-24} + q^{-26} -2 q^{-28} - q^{-30} - q^{-32} - q^{-34} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +7 q^{-44} -9 q^{-46} +11 q^{-48} -8 q^{-50} +3 q^{-52} +5 q^{-54} -13 q^{-56} +21 q^{-58} -19 q^{-60} +12 q^{-62} -2 q^{-64} -10 q^{-66} +18 q^{-68} -17 q^{-70} +14 q^{-72} -2 q^{-74} -7 q^{-76} +13 q^{-78} -9 q^{-80} - q^{-82} +12 q^{-84} -19 q^{-86} +18 q^{-88} -9 q^{-90} -4 q^{-92} +21 q^{-94} -28 q^{-96} +31 q^{-98} -20 q^{-100} +6 q^{-102} +11 q^{-104} -21 q^{-106} +26 q^{-108} -18 q^{-110} +12 q^{-112} +2 q^{-114} -10 q^{-116} +15 q^{-118} -10 q^{-120} -2 q^{-122} +9 q^{-124} -16 q^{-126} +10 q^{-128} -3 q^{-130} -12 q^{-132} +18 q^{-134} -20 q^{-136} +15 q^{-138} -10 q^{-140} -9 q^{-142} +13 q^{-144} -16 q^{-146} +13 q^{-148} -9 q^{-150} +2 q^{-152} +3 q^{-154} -4 q^{-156} +6 q^{-158} -6 q^{-160} +4 q^{-162} - q^{-164} + q^{-168} -2 q^{-170} +2 q^{-172} + q^{-176} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_10.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{-3} - q^{-5} +2 q^{-7} - q^{-9} + q^{-11} + q^{-13} +2 q^{-17} -2 q^{-19} - q^{-23} }[/math]
2	[math]\displaystyle{ q^{-6} - q^{-8} +4 q^{-12} -2 q^{-14} -3 q^{-16} +7 q^{-18} - q^{-20} -6 q^{-22} +7 q^{-24} + q^{-26} -5 q^{-28} +2 q^{-30} +2 q^{-32} -3 q^{-36} +4 q^{-38} +3 q^{-40} -7 q^{-42} + q^{-44} +5 q^{-46} -7 q^{-48} -2 q^{-50} +4 q^{-52} -3 q^{-54} - q^{-56} +2 q^{-58} + q^{-64} }[/math]
3	[math]\displaystyle{ q^{-9} - q^{-11} +2 q^{-15} +2 q^{-17} -2 q^{-19} -3 q^{-21} +4 q^{-23} +7 q^{-25} -4 q^{-27} -10 q^{-29} +2 q^{-31} +17 q^{-33} +2 q^{-35} -20 q^{-37} -6 q^{-39} +21 q^{-41} +12 q^{-43} -20 q^{-45} -14 q^{-47} +17 q^{-49} +15 q^{-51} -9 q^{-53} -14 q^{-55} +3 q^{-57} +9 q^{-59} +3 q^{-61} -9 q^{-63} -9 q^{-65} +6 q^{-67} +15 q^{-69} -2 q^{-71} -20 q^{-73} + q^{-75} +20 q^{-77} +3 q^{-79} -23 q^{-81} -9 q^{-83} +16 q^{-85} +13 q^{-87} -16 q^{-89} -13 q^{-91} +8 q^{-93} +14 q^{-95} -2 q^{-97} -10 q^{-99} + q^{-101} +7 q^{-103} +2 q^{-105} -3 q^{-107} + q^{-111} + q^{-113} - q^{-115} - q^{-123} }[/math]
4	[math]\displaystyle{ q^{-12} - q^{-14} +2 q^{-18} +2 q^{-22} -3 q^{-24} - q^{-26} +5 q^{-28} +5 q^{-32} -8 q^{-34} -6 q^{-36} +9 q^{-38} +6 q^{-40} +14 q^{-42} -17 q^{-44} -24 q^{-46} + q^{-48} +19 q^{-50} +47 q^{-52} -10 q^{-54} -51 q^{-56} -33 q^{-58} +14 q^{-60} +83 q^{-62} +23 q^{-64} -54 q^{-66} -70 q^{-68} -17 q^{-70} +90 q^{-72} +55 q^{-74} -28 q^{-76} -73 q^{-78} -42 q^{-80} +56 q^{-82} +55 q^{-84} +6 q^{-86} -43 q^{-88} -44 q^{-90} +11 q^{-92} +35 q^{-94} +26 q^{-96} -10 q^{-98} -34 q^{-100} -28 q^{-102} +15 q^{-104} +44 q^{-106} +17 q^{-108} -28 q^{-110} -58 q^{-112} +60 q^{-116} +42 q^{-118} -19 q^{-120} -84 q^{-122} -19 q^{-124} +59 q^{-126} +62 q^{-128} +5 q^{-130} -86 q^{-132} -45 q^{-134} +27 q^{-136} +66 q^{-138} +43 q^{-140} -53 q^{-142} -52 q^{-144} -10 q^{-146} +38 q^{-148} +55 q^{-150} -8 q^{-152} -29 q^{-154} -28 q^{-156} +2 q^{-158} +32 q^{-160} +9 q^{-162} -4 q^{-164} -16 q^{-166} -9 q^{-168} +8 q^{-170} +3 q^{-172} +4 q^{-174} -3 q^{-176} -4 q^{-178} + q^{-180} -2 q^{-182} +2 q^{-184} - q^{-188} + q^{-190} - q^{-192} + q^{-200} }[/math]
5	[math]\displaystyle{ q^{-15} - q^{-17} +2 q^{-21} + q^{-27} - q^{-29} - q^{-31} +3 q^{-33} +3 q^{-35} -2 q^{-37} - q^{-41} +4 q^{-45} +6 q^{-47} - q^{-49} -9 q^{-51} -11 q^{-53} -4 q^{-55} +14 q^{-57} +26 q^{-59} +22 q^{-61} -9 q^{-63} -49 q^{-65} -52 q^{-67} -8 q^{-69} +62 q^{-71} +99 q^{-73} +56 q^{-75} -60 q^{-77} -153 q^{-79} -118 q^{-81} +31 q^{-83} +186 q^{-85} +199 q^{-87} +30 q^{-89} -197 q^{-91} -273 q^{-93} -104 q^{-95} +171 q^{-97} +314 q^{-99} +183 q^{-101} -115 q^{-103} -323 q^{-105} -243 q^{-107} +53 q^{-109} +287 q^{-111} +266 q^{-113} +15 q^{-115} -227 q^{-117} -259 q^{-119} -71 q^{-121} +158 q^{-123} +220 q^{-125} +96 q^{-127} -80 q^{-129} -173 q^{-131} -115 q^{-133} +26 q^{-135} +121 q^{-137} +115 q^{-139} +26 q^{-141} -78 q^{-143} -122 q^{-145} -65 q^{-147} +45 q^{-149} +124 q^{-151} +100 q^{-153} -22 q^{-155} -140 q^{-157} -139 q^{-159} +5 q^{-161} +164 q^{-163} +177 q^{-165} +17 q^{-167} -179 q^{-169} -224 q^{-171} -48 q^{-173} +193 q^{-175} +264 q^{-177} +89 q^{-179} -176 q^{-181} -302 q^{-183} -140 q^{-185} +144 q^{-187} +300 q^{-189} +204 q^{-191} -76 q^{-193} -287 q^{-195} -239 q^{-197} +2 q^{-199} +221 q^{-201} +257 q^{-203} +86 q^{-205} -151 q^{-207} -235 q^{-209} -128 q^{-211} +58 q^{-213} +184 q^{-215} +157 q^{-217} +16 q^{-219} -118 q^{-221} -140 q^{-223} -62 q^{-225} +48 q^{-227} +102 q^{-229} +76 q^{-231} -3 q^{-233} -64 q^{-235} -64 q^{-237} -23 q^{-239} +22 q^{-241} +44 q^{-243} +25 q^{-245} -3 q^{-247} -20 q^{-249} -20 q^{-251} -7 q^{-253} +9 q^{-255} +11 q^{-257} +6 q^{-259} +2 q^{-261} -5 q^{-263} -6 q^{-265} + q^{-269} + q^{-271} +4 q^{-273} -2 q^{-277} - q^{-283} + q^{-285} + q^{-287} - q^{-295} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{-6} - q^{-8} + q^{-10} +2 q^{-16} +2 q^{-20} + q^{-22} + q^{-24} + q^{-26} -2 q^{-28} - q^{-30} - q^{-32} - q^{-34} }[/math]
1,1	[math]\displaystyle{ q^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +17 q^{-20} -22 q^{-22} +32 q^{-24} -42 q^{-26} +58 q^{-28} -56 q^{-30} +60 q^{-32} -54 q^{-34} +43 q^{-36} -16 q^{-38} -12 q^{-40} +44 q^{-42} -69 q^{-44} +100 q^{-46} -114 q^{-48} +118 q^{-50} -119 q^{-52} +94 q^{-54} -80 q^{-56} +46 q^{-58} -24 q^{-60} -8 q^{-62} +36 q^{-64} -46 q^{-66} +56 q^{-68} -56 q^{-70} +56 q^{-72} -44 q^{-74} +30 q^{-76} -28 q^{-78} +16 q^{-80} -12 q^{-82} +8 q^{-84} -4 q^{-86} +4 q^{-88} + q^{-92} }[/math]
2,0	[math]\displaystyle{ q^{-12} - q^{-14} - q^{-16} +3 q^{-18} +2 q^{-20} -3 q^{-22} - q^{-24} +4 q^{-26} +3 q^{-28} -2 q^{-30} + q^{-32} +4 q^{-34} - q^{-36} -2 q^{-38} +2 q^{-40} + q^{-42} +4 q^{-46} +4 q^{-48} +2 q^{-54} -6 q^{-58} - q^{-60} -3 q^{-64} -6 q^{-66} -3 q^{-68} - q^{-72} - q^{-74} +2 q^{-78} + q^{-80} +2 q^{-82} + q^{-84} + q^{-86} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{-12} - q^{-14} +2 q^{-18} -3 q^{-20} + q^{-22} +6 q^{-24} -4 q^{-26} + q^{-28} +7 q^{-30} -2 q^{-32} + q^{-34} +6 q^{-36} + q^{-38} +2 q^{-44} -2 q^{-46} -4 q^{-48} +3 q^{-50} -8 q^{-54} +2 q^{-56} - q^{-58} -7 q^{-60} + q^{-62} - q^{-66} +2 q^{-68} +2 q^{-70} + q^{-74} }[/math]
1,0,0	[math]\displaystyle{ q^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17} +2 q^{-21} + q^{-23} + q^{-25} +2 q^{-27} + q^{-29} +2 q^{-31} + q^{-35} -2 q^{-37} - q^{-39} -2 q^{-41} - q^{-43} - q^{-45} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{-18} - q^{-20} + q^{-24} - q^{-26} - q^{-28} +3 q^{-30} +2 q^{-32} -2 q^{-34} + q^{-36} +5 q^{-38} +2 q^{-40} -2 q^{-42} +4 q^{-44} +8 q^{-46} + q^{-50} +6 q^{-52} +3 q^{-54} - q^{-56} +4 q^{-58} +5 q^{-60} - q^{-64} +2 q^{-66} -4 q^{-68} -11 q^{-70} -7 q^{-72} -5 q^{-74} -10 q^{-76} -8 q^{-78} +2 q^{-82} + q^{-84} +2 q^{-86} +5 q^{-88} +3 q^{-90} + q^{-92} + q^{-94} + q^{-96} }[/math]
1,0,0,0	[math]\displaystyle{ q^{-12} - q^{-14} + q^{-16} - q^{-18} + q^{-22} +2 q^{-26} + q^{-28} +2 q^{-30} + q^{-32} +2 q^{-34} + q^{-36} +2 q^{-38} + q^{-40} + q^{-44} -2 q^{-46} - q^{-48} -2 q^{-50} -2 q^{-52} - q^{-54} - q^{-56} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{-12} - q^{-14} +2 q^{-16} -4 q^{-18} +5 q^{-20} -5 q^{-22} +6 q^{-24} -4 q^{-26} +5 q^{-28} - q^{-30} +5 q^{-34} -6 q^{-36} +9 q^{-38} -10 q^{-40} +10 q^{-42} -10 q^{-44} +8 q^{-46} -6 q^{-48} +3 q^{-50} -2 q^{-54} +4 q^{-56} -5 q^{-58} +5 q^{-60} -5 q^{-62} +4 q^{-64} -3 q^{-66} +2 q^{-68} -2 q^{-70} - q^{-74} }[/math]
1,0	[math]\displaystyle{ q^{-18} - q^{-22} - q^{-24} + q^{-26} +3 q^{-28} -4 q^{-32} -2 q^{-34} +4 q^{-36} +6 q^{-38} - q^{-40} -5 q^{-42} -2 q^{-44} +6 q^{-46} +5 q^{-48} -2 q^{-50} -4 q^{-52} +3 q^{-54} +5 q^{-56} + q^{-58} -3 q^{-60} +4 q^{-64} +2 q^{-66} -3 q^{-68} -3 q^{-70} +2 q^{-72} +3 q^{-74} -2 q^{-76} -4 q^{-78} + q^{-80} +5 q^{-82} -6 q^{-86} -5 q^{-88} +3 q^{-90} +4 q^{-92} -3 q^{-94} -7 q^{-96} -3 q^{-98} +3 q^{-100} +2 q^{-102} - q^{-104} -2 q^{-106} +2 q^{-110} +2 q^{-112} + q^{-120} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{-18} - q^{-20} + q^{-22} -2 q^{-24} +3 q^{-26} -4 q^{-28} +4 q^{-30} -3 q^{-32} +6 q^{-34} -3 q^{-36} +4 q^{-38} - q^{-40} +5 q^{-42} +2 q^{-44} - q^{-46} +5 q^{-48} -2 q^{-50} +9 q^{-52} -5 q^{-54} +8 q^{-56} -7 q^{-58} +9 q^{-60} -7 q^{-62} +5 q^{-64} -8 q^{-66} +2 q^{-68} -3 q^{-70} - q^{-72} -2 q^{-74} -4 q^{-76} +2 q^{-78} -5 q^{-80} +2 q^{-82} -6 q^{-84} +3 q^{-86} -4 q^{-88} +2 q^{-90} -2 q^{-92} +3 q^{-94} +2 q^{-98} + q^{-102} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +7 q^{-44} -9 q^{-46} +11 q^{-48} -8 q^{-50} +3 q^{-52} +5 q^{-54} -13 q^{-56} +21 q^{-58} -19 q^{-60} +12 q^{-62} -2 q^{-64} -10 q^{-66} +18 q^{-68} -17 q^{-70} +14 q^{-72} -2 q^{-74} -7 q^{-76} +13 q^{-78} -9 q^{-80} - q^{-82} +12 q^{-84} -19 q^{-86} +18 q^{-88} -9 q^{-90} -4 q^{-92} +21 q^{-94} -28 q^{-96} +31 q^{-98} -20 q^{-100} +6 q^{-102} +11 q^{-104} -21 q^{-106} +26 q^{-108} -18 q^{-110} +12 q^{-112} +2 q^{-114} -10 q^{-116} +15 q^{-118} -10 q^{-120} -2 q^{-122} +9 q^{-124} -16 q^{-126} +10 q^{-128} -3 q^{-130} -12 q^{-132} +18 q^{-134} -20 q^{-136} +15 q^{-138} -10 q^{-140} -9 q^{-142} +13 q^{-144} -16 q^{-146} +13 q^{-148} -9 q^{-150} +2 q^{-152} +3 q^{-154} -4 q^{-156} +6 q^{-158} -6 q^{-160} +4 q^{-162} - q^{-164} + q^{-168} -2 q^{-170} +2 q^{-172} + q^{-176} }[/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 10"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 4 t^2-8 t+9-8 t^{-1} +4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^4 a^{-4} +2 z^4 a^{-6} +z^4 a^{-8} +2 z^2 a^{-4} +5 z^2 a^{-6} +2 z^2 a^{-8} -z^2 a^{-10} +2 a^{-6} + a^{-8} -2 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} -z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} +2 z^5 a^{-5} -3 z^5 a^{-7} -7 z^5 a^{-9} -z^5 a^{-11} +z^5 a^{-13} +z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +9 z^4 a^{-10} -2 z^4 a^{-12} -3 z^3 a^{-5} +3 z^3 a^{-7} +9 z^3 a^{-9} -z^3 a^{-11} -4 z^3 a^{-13} -2 z^2 a^{-4} +7 z^2 a^{-6} -2 z^2 a^{-8} -11 z^2 a^{-10} -4 z a^{-9} +4 z a^{-13} -2 a^{-6} + a^{-8} +2 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 10"];

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

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

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{}

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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:

(8, 22)

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{ 32 }[/math] [math]\displaystyle{ 176 }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{3856}{3} }[/math] [math]\displaystyle{ \frac{656}{3} }[/math] [math]\displaystyle{ 5632 }[/math] [math]\displaystyle{ \frac{30752}{3} }[/math] [math]\displaystyle{ \frac{5504}{3} }[/math] [math]\displaystyle{ 1520 }[/math] [math]\displaystyle{ \frac{16384}{3} }[/math] [math]\displaystyle{ 15488 }[/math] [math]\displaystyle{ \frac{123392}{3} }[/math] [math]\displaystyle{ \frac{20992}{3} }[/math] [math]\displaystyle{ \frac{1241164}{15} }[/math] [math]\displaystyle{ \frac{6064}{15} }[/math] [math]\displaystyle{ \frac{1589776}{45} }[/math] [math]\displaystyle{ \frac{5108}{9} }[/math] [math]\displaystyle{ \frac{72844}{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 10. 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 23                   1 -1 21                     0 19               3 1   -2 17             2       2 15           3 3       0 13         3 2         1 11       2 3           1 9     2 3             -1 7     2               2 5 1 2                 -1 3 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=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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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^{31}-q^{30}+3 q^{28}-4 q^{27}-2 q^{26}+10 q^{25}-10 q^{24}-7 q^{23}+22 q^{22}-14 q^{21}-15 q^{20}+32 q^{19}-13 q^{18}-22 q^{17}+35 q^{16}-11 q^{15}-22 q^{14}+28 q^{13}-5 q^{12}-16 q^{11}+15 q^{10}-8 q^8+5 q^7+q^6-2 q^5+q^4 }[/math]
3	[math]\displaystyle{ -q^{60}+q^{59}-2 q^{56}+3 q^{55}-q^{53}-5 q^{52}+8 q^{51}+5 q^{50}-7 q^{49}-16 q^{48}+16 q^{47}+21 q^{46}-13 q^{45}-37 q^{44}+13 q^{43}+50 q^{42}-10 q^{41}-62 q^{40}-q^{39}+76 q^{38}+7 q^{37}-81 q^{36}-22 q^{35}+94 q^{34}+24 q^{33}-90 q^{32}-37 q^{31}+94 q^{30}+36 q^{29}-84 q^{28}-43 q^{27}+77 q^{26}+41 q^{25}-60 q^{24}-41 q^{23}+46 q^{22}+35 q^{21}-28 q^{20}-32 q^{19}+19 q^{18}+21 q^{17}-6 q^{16}-17 q^{15}+4 q^{14}+9 q^{13}-6 q^{11}+q^{10}+2 q^9+q^8-2 q^7+q^6 }[/math]
4	[math]\displaystyle{ q^{98}-q^{97}-q^{94}+3 q^{93}-3 q^{92}+q^{91}+2 q^{90}-5 q^{89}+6 q^{88}-8 q^{87}+2 q^{86}+9 q^{85}-6 q^{84}+11 q^{83}-25 q^{82}-5 q^{81}+21 q^{80}+7 q^{79}+34 q^{78}-55 q^{77}-35 q^{76}+20 q^{75}+28 q^{74}+97 q^{73}-72 q^{72}-83 q^{71}-22 q^{70}+27 q^{69}+193 q^{68}-49 q^{67}-122 q^{66}-94 q^{65}-14 q^{64}+284 q^{63}+8 q^{62}-125 q^{61}-172 q^{60}-79 q^{59}+349 q^{58}+69 q^{57}-107 q^{56}-232 q^{55}-137 q^{54}+379 q^{53}+114 q^{52}-80 q^{51}-261 q^{50}-180 q^{49}+373 q^{48}+138 q^{47}-44 q^{46}-252 q^{45}-204 q^{44}+318 q^{43}+139 q^{42}+5 q^{41}-203 q^{40}-203 q^{39}+220 q^{38}+108 q^{37}+50 q^{36}-120 q^{35}-168 q^{34}+113 q^{33}+55 q^{32}+66 q^{31}-43 q^{30}-108 q^{29}+44 q^{28}+8 q^{27}+48 q^{26}-2 q^{25}-51 q^{24}+16 q^{23}-10 q^{22}+23 q^{21}+5 q^{20}-20 q^{19}+8 q^{18}-7 q^{17}+8 q^{16}+3 q^{15}-7 q^{14}+3 q^{13}-2 q^{12}+2 q^{11}+q^{10}-2 q^9+q^8 }[/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_9 9_11