9 15

9_14

9_16

Visit 9 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 15's page at Knotilus! Visit 9 15's page at the original Knot Atlas! 9 15 Quick Notes

9 15 Further Notes and Views

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]

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]

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]

Computer Talk

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

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 15]]
Out[2]=	9
In[3]:=	PD[Knot[9, 15]]
Out[3]=	PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[13, 17, 14, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[11, 1, 12, 18], X[17, 13, 18, 12]]
In[4]:=	GaussCode[Knot[9, 15]]
Out[4]=	GaussCode[-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8]
In[5]:=	BR[Knot[9, 15]]
Out[5]=	BR[5, {1, 1, 1, 2, -1, -3, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[9, 15]][t]
Out[6]=	2 10 2 -15 - -- + -- + 10 t - 2 t 2 t t
In[7]:=	Conway[Knot[9, 15]][z]
Out[7]=	2 4 1 + 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 15], Knot[10, 165], Knot[11, NonAlternating, 63], Knot[11, NonAlternating, 101]}
In[9]:=	{KnotDet[Knot[9, 15]], KnotSignature[Knot[9, 15]]}
Out[9]=	{39, 2}
In[10]:=	J=Jones[Knot[9, 15]][q]
Out[10]=	1 2 3 4 5 6 7 8 -2 + - + 4 q - 6 q + 7 q - 6 q + 6 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 15]}
In[12]:=	A2Invariant[Knot[9, 15]][q]
Out[12]=	-4 2 4 12 16 20 22 24 26 q + 2 q - 2 q + 2 q + 2 q - q + q - q - q
In[13]:=	Kauffman[Knot[9, 15]][a, z]
Out[13]=	2 -8 -6 -4 -2 2 z z z z z 2 3 z 1 - a - a + a + a + --- + -- - -- + -- + - - 2 z + ---- + 9 7 5 3 a 8 a a a a a 2 2 2 3 3 3 3 4 4 2 z 2 z 3 z 3 z z 5 z 3 z 4 5 z 4 z ---- - ---- - ---- - ---- - -- + ---- - ---- + z - ---- - ---- + 6 4 2 9 7 5 a 8 6 a a a a a a a a 5 5 5 5 5 6 6 6 6 7 7 z 3 z 7 z z 2 z 2 z z z 2 z 2 z 4 z -- - ---- - ---- - -- + ---- + ---- + -- + -- + ---- + ---- + ---- + 9 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a a 7 8 8 2 z z z ---- + -- + -- 3 6 4 a a a
In[14]:=	{Vassiliev[2][Knot[9, 15]], Vassiliev[3][Knot[9, 15]]}
Out[14]=	{0, 5}
In[15]:=	Kh[Knot[9, 15]][q, t]
Out[15]=	3 1 1 q 3 5 5 2 7 2 3 q + 2 q + ----- + --- + - + 4 q t + 2 q t + 3 q t + 4 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 3 q t + 3 q t + 3 q t + 3 q t + q t + 3 q t + q t + 15 6 17 7 q t + q t