9 15

9_14

9_16

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 15 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X7,10,8,11 X3948 X9,3,10,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X11,1,12,18 X17,13,18,12
Gauss code	-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 14 10 2 18 16 6 12
Conway Notation	[2322]

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

[edit Notes on presentations of 9 15]

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 15"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X7,10,8,11 X3948 X9,3,10,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X11,1,12,18 X17,13,18,12

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 14 10 2 18 16 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2322]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,5\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 9.8855 A-Polynomial See Data:9 15/A-polynomial

[edit Notes for 9 15'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{ 2 }[/math] Rasmussen s-Invariant 2

[edit Notes for 9 15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+10 t-15+10 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 39, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+2 q^7-4 q^6+6 q^5-6 q^4+7 q^3-6 q^2+4 q-2+ q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4 a^{-2} -z^4 a^{-4} -z^2 a^{-2} +2 z^2 a^{-6} +z^2- a^{-2} + a^{-4} + a^{-6} - a^{-8} +1 }[/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} +2 z^6 a^{-2} +z^6 a^{-4} +z^6 a^{-6} +2 z^6 a^{-8} +2 z^5 a^{-1} -z^5 a^{-3} -7 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} -4 z^4 a^{-6} -5 z^4 a^{-8} +z^4-3 z^3 a^{-1} +5 z^3 a^{-5} -z^3 a^{-7} -3 z^3 a^{-9} -3 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2 a^{-6} +3 z^2 a^{-8} -2 z^2+z a^{-1} +z a^{-3} -z a^{-5} +z a^{-7} +2 z a^{-9} + a^{-2} + a^{-4} - a^{-6} - a^{-8} +1 }[/math]
The A2 invariant	[math]\displaystyle{ q^4+2 q^{-2} -2 q^{-4} +2 q^{-12} +2 q^{-16} - q^{-20} + q^{-22} - q^{-24} - q^{-26} }[/math]
The G2 invariant	[math]\displaystyle{ q^{18}-q^{16}+3 q^{14}-3 q^{12}+2 q^{10}-3 q^6+8 q^4-9 q^2+11-9 q^{-2} +5 q^{-4} +4 q^{-6} -13 q^{-8} +23 q^{-10} -26 q^{-12} +21 q^{-14} -13 q^{-16} -5 q^{-18} +20 q^{-20} -31 q^{-22} +33 q^{-24} -20 q^{-26} +2 q^{-28} +14 q^{-30} -23 q^{-32} +15 q^{-34} -2 q^{-36} -13 q^{-38} +24 q^{-40} -23 q^{-42} +9 q^{-44} +17 q^{-46} -34 q^{-48} +46 q^{-50} -42 q^{-52} +21 q^{-54} +6 q^{-56} -28 q^{-58} +43 q^{-60} -44 q^{-62} +36 q^{-64} -11 q^{-66} -11 q^{-68} +26 q^{-70} -30 q^{-72} +20 q^{-74} - q^{-76} -15 q^{-78} +21 q^{-80} -15 q^{-82} +3 q^{-84} +18 q^{-86} -31 q^{-88} +33 q^{-90} -23 q^{-92} +18 q^{-96} -32 q^{-98} +33 q^{-100} -24 q^{-102} +10 q^{-104} +3 q^{-106} -15 q^{-108} +17 q^{-110} -15 q^{-112} +9 q^{-114} -3 q^{-116} -2 q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_15.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^3-q+2 q^{-1} -2 q^{-3} + q^{-5} + q^{-7} +2 q^{-11} -2 q^{-13} + q^{-15} - q^{-17} }[/math]
2	[math]\displaystyle{ q^{10}-q^8-q^6+3 q^4-3 q^2-2+9 q^{-2} -4 q^{-4} -6 q^{-6} +11 q^{-8} - q^{-10} -7 q^{-12} +5 q^{-14} +2 q^{-16} -3 q^{-18} -3 q^{-20} +5 q^{-22} +2 q^{-24} -8 q^{-26} +5 q^{-28} +6 q^{-30} -10 q^{-32} + q^{-34} +7 q^{-36} -6 q^{-38} -2 q^{-40} +4 q^{-42} - q^{-44} - q^{-46} + q^{-48} }[/math]
3	[math]\displaystyle{ q^{21}-q^{19}-q^{17}+2 q^{13}-q^{11}-2 q^9+4 q^7+4 q^5-7 q^3-8 q+11 q^{-1} +16 q^{-3} -14 q^{-5} -25 q^{-7} +13 q^{-9} +33 q^{-11} -5 q^{-13} -38 q^{-15} +38 q^{-19} +8 q^{-21} -28 q^{-23} -14 q^{-25} +21 q^{-27} +16 q^{-29} -9 q^{-31} -19 q^{-33} -2 q^{-35} +17 q^{-37} +11 q^{-39} -19 q^{-41} -21 q^{-43} +19 q^{-45} +27 q^{-47} -14 q^{-49} -33 q^{-51} +10 q^{-53} +36 q^{-55} -37 q^{-59} -7 q^{-61} +28 q^{-63} +15 q^{-65} -21 q^{-67} -18 q^{-69} +12 q^{-71} +16 q^{-73} -3 q^{-75} -12 q^{-77} - q^{-79} +7 q^{-81} +2 q^{-83} -3 q^{-85} - q^{-87} + q^{-89} + q^{-91} - q^{-93} }[/math]
4	[math]\displaystyle{ q^{36}-q^{34}-q^{32}-q^{28}+4 q^{26}-q^{24}+2 q^{20}-5 q^{18}+3 q^{16}-7 q^{14}+2 q^{12}+15 q^{10}-q^8-2 q^6-32 q^4-9 q^2+41+34 q^{-2} +13 q^{-4} -78 q^{-6} -62 q^{-8} +46 q^{-10} +92 q^{-12} +74 q^{-14} -95 q^{-16} -135 q^{-18} -7 q^{-20} +113 q^{-22} +151 q^{-24} -49 q^{-26} -159 q^{-28} -75 q^{-30} +68 q^{-32} +162 q^{-34} +20 q^{-36} -103 q^{-38} -97 q^{-40} -2 q^{-42} +107 q^{-44} +62 q^{-46} -25 q^{-48} -78 q^{-50} -50 q^{-52} +33 q^{-54} +76 q^{-56} +40 q^{-58} -56 q^{-60} -85 q^{-62} -25 q^{-64} +91 q^{-66} +95 q^{-68} -35 q^{-70} -115 q^{-72} -81 q^{-74} +96 q^{-76} +143 q^{-78} +5 q^{-80} -118 q^{-82} -136 q^{-84} +55 q^{-86} +152 q^{-88} +67 q^{-90} -67 q^{-92} -161 q^{-94} -20 q^{-96} +101 q^{-98} +99 q^{-100} +15 q^{-102} -113 q^{-104} -64 q^{-106} +17 q^{-108} +70 q^{-110} +61 q^{-112} -37 q^{-114} -47 q^{-116} -28 q^{-118} +16 q^{-120} +44 q^{-122} +4 q^{-124} -9 q^{-126} -21 q^{-128} -7 q^{-130} +14 q^{-132} +4 q^{-134} +3 q^{-136} -5 q^{-138} -4 q^{-140} +3 q^{-142} + q^{-146} - q^{-148} - q^{-150} + q^{-152} }[/math]
5	[math]\displaystyle{ q^{55}-q^{53}-q^{51}-q^{47}+q^{45}+4 q^{43}+q^{41}-2 q^{39}-5 q^{35}-5 q^{33}+3 q^{31}+6 q^{29}+7 q^{27}+6 q^{25}-5 q^{23}-20 q^{21}-19 q^{19}+2 q^{17}+30 q^{15}+45 q^{13}+21 q^{11}-37 q^9-88 q^7-68 q^5+34 q^3+137 q+146 q^{-1} +8 q^{-3} -182 q^{-5} -260 q^{-7} -97 q^{-9} +206 q^{-11} +382 q^{-13} +235 q^{-15} -169 q^{-17} -487 q^{-19} -410 q^{-21} +67 q^{-23} +553 q^{-25} +581 q^{-27} +84 q^{-29} -529 q^{-31} -704 q^{-33} -266 q^{-35} +436 q^{-37} +765 q^{-39} +412 q^{-41} -287 q^{-43} -721 q^{-45} -510 q^{-47} +116 q^{-49} +604 q^{-51} +547 q^{-53} +28 q^{-55} -451 q^{-57} -501 q^{-59} -150 q^{-61} +277 q^{-63} +434 q^{-65} +222 q^{-67} -136 q^{-69} -336 q^{-71} -267 q^{-73} +5 q^{-75} +263 q^{-77} +298 q^{-79} +91 q^{-81} -198 q^{-83} -334 q^{-85} -181 q^{-87} +156 q^{-89} +385 q^{-91} +269 q^{-93} -125 q^{-95} -443 q^{-97} -366 q^{-99} +79 q^{-101} +496 q^{-103} +476 q^{-105} -16 q^{-107} -530 q^{-109} -576 q^{-111} -88 q^{-113} +522 q^{-115} +667 q^{-117} +208 q^{-119} -441 q^{-121} -711 q^{-123} -352 q^{-125} +317 q^{-127} +695 q^{-129} +462 q^{-131} -137 q^{-133} -595 q^{-135} -548 q^{-137} -43 q^{-139} +449 q^{-141} +534 q^{-143} +195 q^{-145} -252 q^{-147} -461 q^{-149} -292 q^{-151} +79 q^{-153} +328 q^{-155} +302 q^{-157} +61 q^{-159} -178 q^{-161} -252 q^{-163} -137 q^{-165} +57 q^{-167} +168 q^{-169} +141 q^{-171} +26 q^{-173} -80 q^{-175} -110 q^{-177} -59 q^{-179} +21 q^{-181} +63 q^{-183} +54 q^{-185} +9 q^{-187} -25 q^{-189} -34 q^{-191} -19 q^{-193} +7 q^{-195} +18 q^{-197} +11 q^{-199} -4 q^{-203} -7 q^{-205} -3 q^{-207} +4 q^{-209} +2 q^{-211} - q^{-213} - q^{-219} + q^{-221} + q^{-223} - q^{-225} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^4+2 q^{-2} -2 q^{-4} +2 q^{-12} +2 q^{-16} - q^{-20} + q^{-22} - q^{-24} - q^{-26} }[/math]
1,1	[math]\displaystyle{ q^{12}-2 q^{10}+6 q^8-10 q^6+17 q^4-26 q^2+36-52 q^{-2} +68 q^{-4} -82 q^{-6} +98 q^{-8} -102 q^{-10} +106 q^{-12} -82 q^{-14} +52 q^{-16} -8 q^{-18} -47 q^{-20} +102 q^{-22} -156 q^{-24} +194 q^{-26} -217 q^{-28} +220 q^{-30} -204 q^{-32} +174 q^{-34} -126 q^{-36} +72 q^{-38} -12 q^{-40} -38 q^{-42} +78 q^{-44} -108 q^{-46} +118 q^{-48} -116 q^{-50} +98 q^{-52} -80 q^{-54} +58 q^{-56} -38 q^{-58} +23 q^{-60} -12 q^{-62} +6 q^{-64} -2 q^{-66} + q^{-68} }[/math]
2,0	[math]\displaystyle{ q^{12}-q^8+2 q^4-4+7 q^{-4} - q^{-6} -6 q^{-8} +3 q^{-10} +7 q^{-12} + q^{-14} -4 q^{-16} +3 q^{-18} +3 q^{-20} -3 q^{-22} - q^{-24} -3 q^{-28} - q^{-30} +5 q^{-32} - q^{-34} -2 q^{-36} +3 q^{-38} +6 q^{-40} -2 q^{-42} -6 q^{-44} +2 q^{-46} +3 q^{-48} -3 q^{-50} -5 q^{-52} +3 q^{-56} -2 q^{-60} + q^{-64} + q^{-66} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^8-q^6+q^4+3 q^2-3+6 q^{-4} -6 q^{-6} -2 q^{-8} +9 q^{-10} -5 q^{-12} -2 q^{-14} +6 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-22} +4 q^{-24} -2 q^{-28} +6 q^{-30} +3 q^{-32} -8 q^{-34} +4 q^{-36} +2 q^{-38} -9 q^{-40} +2 q^{-42} + q^{-44} -4 q^{-46} +2 q^{-48} + q^{-50} - q^{-52} + q^{-54} }[/math]
1,0,0	[math]\displaystyle{ q^5+q+2 q^{-3} -2 q^{-5} - q^{-9} + q^{-15} +2 q^{-17} +2 q^{-21} + q^{-25} - q^{-27} + q^{-29} - q^{-31} - q^{-33} - q^{-35} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^8-q^6+3 q^4-3 q^2+5-6 q^{-2} +8 q^{-4} -8 q^{-6} +6 q^{-8} -5 q^{-10} + q^{-12} +2 q^{-14} -6 q^{-16} +10 q^{-18} -12 q^{-20} +15 q^{-22} -14 q^{-24} +14 q^{-26} -10 q^{-28} +8 q^{-30} -3 q^{-32} +4 q^{-36} -6 q^{-38} +7 q^{-40} -8 q^{-42} +7 q^{-44} -6 q^{-46} +4 q^{-48} -3 q^{-50} + q^{-52} - q^{-54} }[/math]
1,0	[math]\displaystyle{ q^{14}-q^{10}-q^8+2 q^6+3 q^4-4-3 q^{-2} +3 q^{-4} +7 q^{-6} -8 q^{-10} -4 q^{-12} +6 q^{-14} +8 q^{-16} -2 q^{-18} -7 q^{-20} +7 q^{-24} +2 q^{-26} -6 q^{-28} -3 q^{-30} +4 q^{-32} +4 q^{-34} -3 q^{-36} -4 q^{-38} +2 q^{-40} +6 q^{-42} -4 q^{-46} +7 q^{-50} +3 q^{-52} -6 q^{-54} -7 q^{-56} +4 q^{-58} +8 q^{-60} - q^{-62} -9 q^{-64} -5 q^{-66} +5 q^{-68} +5 q^{-70} -2 q^{-72} -5 q^{-74} - q^{-76} +3 q^{-78} +2 q^{-80} - q^{-82} - q^{-84} + q^{-88} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{18}-q^{16}+3 q^{14}-3 q^{12}+2 q^{10}-3 q^6+8 q^4-9 q^2+11-9 q^{-2} +5 q^{-4} +4 q^{-6} -13 q^{-8} +23 q^{-10} -26 q^{-12} +21 q^{-14} -13 q^{-16} -5 q^{-18} +20 q^{-20} -31 q^{-22} +33 q^{-24} -20 q^{-26} +2 q^{-28} +14 q^{-30} -23 q^{-32} +15 q^{-34} -2 q^{-36} -13 q^{-38} +24 q^{-40} -23 q^{-42} +9 q^{-44} +17 q^{-46} -34 q^{-48} +46 q^{-50} -42 q^{-52} +21 q^{-54} +6 q^{-56} -28 q^{-58} +43 q^{-60} -44 q^{-62} +36 q^{-64} -11 q^{-66} -11 q^{-68} +26 q^{-70} -30 q^{-72} +20 q^{-74} - q^{-76} -15 q^{-78} +21 q^{-80} -15 q^{-82} +3 q^{-84} +18 q^{-86} -31 q^{-88} +33 q^{-90} -23 q^{-92} +18 q^{-96} -32 q^{-98} +33 q^{-100} -24 q^{-102} +10 q^{-104} +3 q^{-106} -15 q^{-108} +17 q^{-110} -15 q^{-112} +9 q^{-114} -3 q^{-116} -2 q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + 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["9 15"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^2+10 t-15+10 t^{-1} -2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^4+2 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]=

{ 39, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_165, K11n63, K11n101,}

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 15"];

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{ -2 t^2+10 t-15+10 t^{-1} -2 t^{-2} }[/math], [math]\displaystyle{ -q^8+2 q^7-4 q^6+6 q^5-6 q^4+7 q^3-6 q^2+4 q-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]=

{10_165, K11n63, K11n101,}

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:

(2, 5)

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{ 8 }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{508}{3} }[/math] [math]\displaystyle{ \frac{92}{3} }[/math] [math]\displaystyle{ 320 }[/math] [math]\displaystyle{ \frac{2416}{3} }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ 168 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ \frac{4064}{3} }[/math] [math]\displaystyle{ \frac{736}{3} }[/math] [math]\displaystyle{ \frac{59071}{15} }[/math] [math]\displaystyle{ -\frac{1628}{5} }[/math] [math]\displaystyle{ \frac{92164}{45} }[/math] [math]\displaystyle{ \frac{881}{9} }[/math] [math]\displaystyle{ \frac{4351}{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 9 15. 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 17                   1 -1 15                 1   1 13               3 1   -2 11             3 1     2 9           3 3       0 7         4 3         1 5       2 3           1 3     2 4             -2 1   1 3               2 -1   1                 -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=1 }[/math] [math]\displaystyle{ i=3 }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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^{23}-2 q^{22}+6 q^{20}-8 q^{19}-4 q^{18}+19 q^{17}-14 q^{16}-15 q^{15}+35 q^{14}-15 q^{13}-28 q^{12}+45 q^{11}-12 q^{10}-36 q^9+45 q^8-7 q^7-33 q^6+33 q^5-q^4-21 q^3+16 q^2+q-8+5 q^{-1} -2 q^{-3} + q^{-4} }[/math]
3	[math]\displaystyle{ -q^{45}+2 q^{44}-2 q^{42}-3 q^{41}+7 q^{40}+5 q^{39}-10 q^{38}-14 q^{37}+16 q^{36}+24 q^{35}-14 q^{34}-44 q^{33}+13 q^{32}+60 q^{31}-q^{30}-79 q^{29}-17 q^{28}+97 q^{27}+35 q^{26}-105 q^{25}-60 q^{24}+116 q^{23}+76 q^{22}-113 q^{21}-100 q^{20}+118 q^{19}+106 q^{18}-107 q^{17}-119 q^{16}+101 q^{15}+116 q^{14}-82 q^{13}-114 q^{12}+66 q^{11}+102 q^{10}-46 q^9-84 q^8+28 q^7+64 q^6-13 q^5-46 q^4+8 q^3+26 q^2-2 q-16+3 q^{-1} +7 q^{-2} - q^{-3} -5 q^{-4} +3 q^{-5} + q^{-6} -2 q^{-8} + q^{-9} }[/math]
4	[math]\displaystyle{ q^{74}-2 q^{73}+2 q^{71}-q^{70}+4 q^{69}-9 q^{68}-q^{67}+10 q^{66}+14 q^{64}-30 q^{63}-15 q^{62}+22 q^{61}+13 q^{60}+54 q^{59}-58 q^{58}-59 q^{57}+3 q^{56}+23 q^{55}+152 q^{54}-49 q^{53}-112 q^{52}-78 q^{51}-26 q^{50}+280 q^{49}+35 q^{48}-110 q^{47}-199 q^{46}-167 q^{45}+374 q^{44}+169 q^{43}-25 q^{42}-296 q^{41}-358 q^{40}+392 q^{39}+292 q^{38}+113 q^{37}-343 q^{36}-535 q^{35}+358 q^{34}+372 q^{33}+243 q^{32}-347 q^{31}-651 q^{30}+298 q^{29}+401 q^{28}+339 q^{27}-311 q^{26}-694 q^{25}+215 q^{24}+373 q^{23}+392 q^{22}-224 q^{21}-649 q^{20}+106 q^{19}+278 q^{18}+386 q^{17}-101 q^{16}-507 q^{15}+12 q^{14}+135 q^{13}+302 q^{12}+9 q^{11}-307 q^{10}-26 q^9+15 q^8+174 q^7+49 q^6-138 q^5-8 q^4-31 q^3+66 q^2+33 q-47+13 q^{-1} -24 q^{-2} +16 q^{-3} +10 q^{-4} -17 q^{-5} +14 q^{-6} -8 q^{-7} +3 q^{-8} + q^{-9} -7 q^{-10} +6 q^{-11} - q^{-12} + q^{-13} -2 q^{-15} + q^{-16} }[/math]
5	[math]\displaystyle{ -q^{110}+2 q^{109}-2 q^{107}+q^{106}-2 q^{104}+5 q^{103}+2 q^{102}-9 q^{101}-3 q^{100}+3 q^{99}+2 q^{98}+16 q^{97}+9 q^{96}-20 q^{95}-29 q^{94}-12 q^{93}+11 q^{92}+50 q^{91}+54 q^{90}-11 q^{89}-71 q^{88}-92 q^{87}-40 q^{86}+80 q^{85}+160 q^{84}+104 q^{83}-44 q^{82}-203 q^{81}-234 q^{80}-35 q^{79}+234 q^{78}+343 q^{77}+197 q^{76}-177 q^{75}-483 q^{74}-406 q^{73}+65 q^{72}+552 q^{71}+644 q^{70}+162 q^{69}-568 q^{68}-898 q^{67}-440 q^{66}+505 q^{65}+1102 q^{64}+761 q^{63}-335 q^{62}-1276 q^{61}-1109 q^{60}+146 q^{59}+1372 q^{58}+1410 q^{57}+124 q^{56}-1421 q^{55}-1719 q^{54}-342 q^{53}+1418 q^{52}+1924 q^{51}+616 q^{50}-1401 q^{49}-2136 q^{48}-787 q^{47}+1341 q^{46}+2242 q^{45}+1010 q^{44}-1285 q^{43}-2365 q^{42}-1124 q^{41}+1188 q^{40}+2378 q^{39}+1299 q^{38}-1078 q^{37}-2400 q^{36}-1382 q^{35}+916 q^{34}+2309 q^{33}+1499 q^{32}-720 q^{31}-2188 q^{30}-1539 q^{29}+489 q^{28}+1958 q^{27}+1549 q^{26}-241 q^{25}-1669 q^{24}-1482 q^{23}+q^{22}+1332 q^{21}+1338 q^{20}+193 q^{19}-970 q^{18}-1129 q^{17}-328 q^{16}+630 q^{15}+900 q^{14}+368 q^{13}-357 q^{12}-632 q^{11}-356 q^{10}+144 q^9+423 q^8+291 q^7-39 q^6-228 q^5-209 q^4-32 q^3+120 q^2+128 q+39-38 q^{-1} -71 q^{-2} -41 q^{-3} +17 q^{-4} +26 q^{-5} +19 q^{-6} +13 q^{-7} -13 q^{-8} -17 q^{-9} +2 q^{-10} -2 q^{-11} -2 q^{-12} +12 q^{-13} +2 q^{-14} -6 q^{-15} +3 q^{-16} -3 q^{-17} -5 q^{-18} +4 q^{-19} +2 q^{-20} - q^{-21} + q^{-22} -2 q^{-24} + q^{-25} }[/math]
6	[math]\displaystyle{ q^{153}-2 q^{152}+2 q^{150}-q^{149}-2 q^{147}+6 q^{146}-6 q^{145}-3 q^{144}+11 q^{143}-q^{142}-2 q^{141}-13 q^{140}+12 q^{139}-14 q^{138}-8 q^{137}+36 q^{136}+14 q^{135}+4 q^{134}-42 q^{133}+9 q^{132}-56 q^{131}-40 q^{130}+78 q^{129}+73 q^{128}+74 q^{127}-47 q^{126}+18 q^{125}-169 q^{124}-189 q^{123}+32 q^{122}+127 q^{121}+245 q^{120}+106 q^{119}+225 q^{118}-239 q^{117}-465 q^{116}-296 q^{115}-103 q^{114}+287 q^{113}+379 q^{112}+870 q^{111}+139 q^{110}-498 q^{109}-797 q^{108}-869 q^{107}-338 q^{106}+251 q^{105}+1722 q^{104}+1220 q^{103}+356 q^{102}-797 q^{101}-1808 q^{100}-1854 q^{99}-973 q^{98}+1961 q^{97}+2517 q^{96}+2241 q^{95}+419 q^{94}-2012 q^{93}-3651 q^{92}-3328 q^{91}+884 q^{90}+3095 q^{89}+4451 q^{88}+2782 q^{87}-882 q^{86}-4793 q^{85}-6017 q^{84}-1336 q^{83}+2457 q^{82}+6087 q^{81}+5471 q^{80}+1273 q^{79}-4876 q^{78}-8170 q^{77}-3863 q^{76}+959 q^{75}+6803 q^{74}+7686 q^{73}+3621 q^{72}-4237 q^{71}-9459 q^{70}-5964 q^{69}-679 q^{68}+6860 q^{67}+9134 q^{66}+5537 q^{65}-3414 q^{64}-10060 q^{63}-7406 q^{62}-2014 q^{61}+6618 q^{60}+9940 q^{59}+6889 q^{58}-2632 q^{57}-10192 q^{56}-8327 q^{55}-3070 q^{54}+6131 q^{53}+10254 q^{52}+7867 q^{51}-1723 q^{50}-9805 q^{49}-8855 q^{48}-4092 q^{47}+5144 q^{46}+9955 q^{45}+8556 q^{44}-408 q^{43}-8580 q^{42}-8795 q^{41}-5116 q^{40}+3405 q^{39}+8681 q^{38}+8653 q^{37}+1237 q^{36}-6307 q^{35}-7713 q^{34}-5715 q^{33}+1143 q^{32}+6283 q^{31}+7664 q^{30}+2591 q^{29}-3415 q^{28}-5504 q^{27}-5290 q^{26}-813 q^{25}+3352 q^{24}+5548 q^{23}+2923 q^{22}-933 q^{21}-2863 q^{20}-3801 q^{19}-1647 q^{18}+976 q^{17}+3078 q^{16}+2182 q^{15}+326 q^{14}-831 q^{13}-2006 q^{12}-1377 q^{11}-174 q^{10}+1227 q^9+1096 q^8+480 q^7+99 q^6-724 q^5-722 q^4-356 q^3+333 q^2+346 q+231+252 q^{-1} -159 q^{-2} -254 q^{-3} -206 q^{-4} +67 q^{-5} +50 q^{-6} +41 q^{-7} +159 q^{-8} -11 q^{-9} -62 q^{-10} -79 q^{-11} +18 q^{-12} -10 q^{-13} -17 q^{-14} +69 q^{-15} +7 q^{-16} -8 q^{-17} -25 q^{-18} +10 q^{-19} -10 q^{-20} -18 q^{-21} +24 q^{-22} +3 q^{-23} +3 q^{-24} -7 q^{-25} +5 q^{-26} -3 q^{-27} -9 q^{-28} +6 q^{-29} +2 q^{-31} - q^{-32} + q^{-33} -2 q^{-35} + q^{-36} }[/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_14 9_16