9 47

9_46

9_48

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 47's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 47 at Knotilus!

Simple square depiction

Threefold symmetrical depiction

Ornate threefold symmetrical depiction

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X11,1,12,18 X17,13,18,12
Gauss code	1, -4, 3, -7, 6, -1, 2, -3, 5, -6, -8, 9, 7, -5, 4, -2, -9, 8
Dowker-Thistlethwaite code	6 8 10 16 14 -18 4 2 -12
Conway Notation	[8*-20]

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

[edit Notes on presentations of 9 47]

Knot 9_47.

A graph, knot 9_47.

A part of a link and a part of a graph.

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 47"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X11,1,12,18 X17,13,18,12

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 8 10 16 14 -18 4 2 -12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[8*-20]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 2 Maximal Thurston-Bennequin number [-2][-7] Hyperbolic Volume 10.05 A-Polynomial See Data:9 47/A-polynomial

[edit Notes for 9 47'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{ 3 }[/math] Rasmussen s-Invariant -2

[edit Notes for 9 47's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-4 t^2+6 t-5+6 t^{-1} -4 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{3,t+1\} }[/math]
Determinant and Signature	{ 27, 2 }
Jones polynomial	[math]\displaystyle{ 2 q^5-4 q^4+4 q^3-5 q^2+5 q-3+3 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+4 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^7 a^{-1} +2 z^7 a^{-3} +6 z^6 a^{-2} +3 z^6 a^{-4} +3 z^6+a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} -16 z^4 a^{-2} -7 z^4 a^{-4} -9 z^4-2 a z^3+z^3 a^{-1} +6 z^3 a^{-3} +3 z^3 a^{-5} +11 z^2 a^{-2} +9 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^6+q^4+q^2+2+2 q^{-2} - q^{-4} + q^{-6} -2 q^{-8} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
The G2 invariant	[math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+4 q^{24}-q^{22}-8 q^{20}+16 q^{18}-16 q^{16}+16 q^{14}-4 q^{12}-11 q^{10}+20 q^8-18 q^6+14 q^4-2 q^2-13+21 q^{-2} -8 q^{-4} +13 q^{-8} -19 q^{-10} +18 q^{-12} -4 q^{-14} -9 q^{-16} +12 q^{-18} -21 q^{-20} +25 q^{-22} -13 q^{-24} +9 q^{-28} -19 q^{-30} +23 q^{-32} -18 q^{-34} +4 q^{-36} +3 q^{-38} -14 q^{-40} +19 q^{-42} -11 q^{-44} -3 q^{-46} +15 q^{-48} -19 q^{-50} +12 q^{-52} + q^{-54} -17 q^{-56} +20 q^{-58} -17 q^{-60} +11 q^{-62} +2 q^{-64} -11 q^{-66} +14 q^{-68} -12 q^{-70} +8 q^{-72} -5 q^{-76} +2 q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_47.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^5+2 q^3+2 q^{-1} - q^{-5} -2 q^{-9} +2 q^{-11} }[/math]
2	[math]\displaystyle{ q^{16}-2 q^{14}-3 q^{12}+5 q^{10}+2 q^8-5 q^6+3 q^4+6 q^2-4- q^{-2} +4 q^{-4} -3 q^{-6} -4 q^{-8} +2 q^{-10} +2 q^{-12} -3 q^{-14} - q^{-16} +7 q^{-18} - q^{-20} -5 q^{-22} +6 q^{-24} -5 q^{-28} + q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{33}+2 q^{31}+3 q^{29}-2 q^{27}-8 q^{25}-5 q^{23}+11 q^{21}+11 q^{19}-5 q^{17}-17 q^{15}-2 q^{13}+20 q^{11}+15 q^9-14 q^7-20 q^5+6 q^3+24 q- q^{-1} -24 q^{-3} -7 q^{-5} +19 q^{-7} +10 q^{-9} -18 q^{-11} -8 q^{-13} +13 q^{-15} +12 q^{-17} -9 q^{-19} -10 q^{-21} +4 q^{-23} +16 q^{-25} +3 q^{-27} -16 q^{-29} -13 q^{-31} +13 q^{-33} +21 q^{-35} -10 q^{-37} -27 q^{-39} + q^{-41} +27 q^{-43} +5 q^{-45} -17 q^{-47} -12 q^{-49} +11 q^{-51} +10 q^{-53} -3 q^{-55} -4 q^{-57} -2 q^{-59} +2 q^{-61} }[/math]
4	[math]\displaystyle{ q^{56}-2 q^{54}-3 q^{52}+2 q^{50}+5 q^{48}+11 q^{46}-2 q^{44}-16 q^{42}-18 q^{40}-10 q^{38}+29 q^{36}+33 q^{34}+10 q^{32}-26 q^{30}-58 q^{28}-16 q^{26}+35 q^{24}+70 q^{22}+43 q^{20}-51 q^{18}-83 q^{16}-45 q^{14}+56 q^{12}+109 q^{10}+32 q^8-71 q^6-116 q^4-24 q^2+98+97 q^{-2} -5 q^{-4} -109 q^{-6} -72 q^{-8} +49 q^{-10} +97 q^{-12} +32 q^{-14} -69 q^{-16} -68 q^{-18} +17 q^{-20} +70 q^{-22} +30 q^{-24} -42 q^{-26} -53 q^{-28} +54 q^{-32} +32 q^{-34} -20 q^{-36} -52 q^{-38} -34 q^{-40} +32 q^{-42} +60 q^{-44} +38 q^{-46} -47 q^{-48} -95 q^{-50} -23 q^{-52} +75 q^{-54} +116 q^{-56} +9 q^{-58} -120 q^{-60} -102 q^{-62} +18 q^{-64} +135 q^{-66} +84 q^{-68} -58 q^{-70} -108 q^{-72} -55 q^{-74} +60 q^{-76} +84 q^{-78} +16 q^{-80} -38 q^{-82} -49 q^{-84} -6 q^{-86} +22 q^{-88} +17 q^{-90} +4 q^{-92} -8 q^{-94} -5 q^{-96} + q^{-98} + q^{-100} }[/math]
5	[math]\displaystyle{ -q^{85}+2 q^{83}+3 q^{81}-2 q^{79}-5 q^{77}-8 q^{75}-4 q^{73}+7 q^{71}+24 q^{69}+24 q^{67}+3 q^{65}-27 q^{63}-51 q^{61}-48 q^{59}-5 q^{57}+66 q^{55}+95 q^{53}+67 q^{51}-13 q^{49}-112 q^{47}-156 q^{45}-90 q^{43}+58 q^{41}+190 q^{39}+220 q^{37}+93 q^{35}-137 q^{33}-303 q^{31}-264 q^{29}-17 q^{27}+278 q^{25}+409 q^{23}+230 q^{21}-146 q^{19}-452 q^{17}-434 q^{15}-65 q^{13}+380 q^{11}+548 q^9+284 q^7-221 q^5-572 q^3-451 q+40 q^{-1} +495 q^{-3} +539 q^{-5} +130 q^{-7} -371 q^{-9} -539 q^{-11} -236 q^{-13} +254 q^{-15} +480 q^{-17} +282 q^{-19} -146 q^{-21} -402 q^{-23} -276 q^{-25} +86 q^{-27} +317 q^{-29} +229 q^{-31} -48 q^{-33} -254 q^{-35} -192 q^{-37} +48 q^{-39} +206 q^{-41} +151 q^{-43} -34 q^{-45} -183 q^{-47} -159 q^{-49} +13 q^{-51} +173 q^{-53} +191 q^{-55} +55 q^{-57} -150 q^{-59} -255 q^{-61} -160 q^{-63} +92 q^{-65} +319 q^{-67} +312 q^{-69} +19 q^{-71} -345 q^{-73} -469 q^{-75} -191 q^{-77} +296 q^{-79} +590 q^{-81} +398 q^{-83} -168 q^{-85} -624 q^{-87} -572 q^{-89} -30 q^{-91} +534 q^{-93} +663 q^{-95} +242 q^{-97} -364 q^{-99} -619 q^{-101} -370 q^{-103} +129 q^{-105} +472 q^{-107} +404 q^{-109} +42 q^{-111} -276 q^{-113} -319 q^{-115} -135 q^{-117} +99 q^{-119} +199 q^{-121} +136 q^{-123} + q^{-125} -89 q^{-127} -78 q^{-129} -31 q^{-131} +18 q^{-133} +35 q^{-135} +18 q^{-137} -3 q^{-139} -6 q^{-141} -4 q^{-143} -2 q^{-145} +2 q^{-147} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^6+q^4+q^2+2+2 q^{-2} - q^{-4} + q^{-6} -2 q^{-8} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
1,1	[math]\displaystyle{ q^{20}-4 q^{18}+10 q^{16}-22 q^{14}+32 q^{12}-46 q^{10}+56 q^8-52 q^6+48 q^4-22 q^2+4+36 q^{-2} -58 q^{-4} +74 q^{-6} -94 q^{-8} +90 q^{-10} -94 q^{-12} +70 q^{-14} -52 q^{-16} +28 q^{-18} +5 q^{-20} -24 q^{-22} +50 q^{-24} -56 q^{-26} +58 q^{-28} -46 q^{-30} +32 q^{-32} -22 q^{-34} +10 q^{-36} -2 q^{-38} -4 q^{-40} +2 q^{-46} }[/math]
2,0	[math]\displaystyle{ q^{18}-q^{16}-3 q^{14}-q^{12}+2 q^{10}+2 q^8+4 q^4+6 q^2+3-2 q^{-2} + q^{-4} -3 q^{-6} -3 q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} -2 q^{-16} +3 q^{-18} +3 q^{-24} +5 q^{-26} - q^{-28} - q^{-30} + q^{-32} -2 q^{-38} - q^{-48} + q^{-52} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{14}-2 q^{12}+q^8-4 q^6+5 q^4+3 q^2+1+7 q^{-2} +3 q^{-4} -4 q^{-6} -2 q^{-8} -3 q^{-10} -5 q^{-12} -2 q^{-14} +5 q^{-18} + q^{-20} + q^{-22} +5 q^{-24} -4 q^{-26} -3 q^{-28} +3 q^{-30} -3 q^{-32} - q^{-34} +3 q^{-36} }[/math]
1,0,0	[math]\displaystyle{ -q^7+q^5+3 q+ q^{-1} +2 q^{-3} - q^{-11} -2 q^{-15} -2 q^{-19} + q^{-21} + q^{-25} + q^{-27} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{16}-q^{14}-q^{12}-2 q^8+4 q^4+3 q^2+3+9 q^{-2} +7 q^{-4} + q^{-6} -2 q^{-8} + q^{-10} -6 q^{-12} -12 q^{-14} -5 q^{-16} -6 q^{-20} + q^{-22} +9 q^{-24} +2 q^{-26} +2 q^{-28} +6 q^{-30} +3 q^{-32} -4 q^{-34} - q^{-36} +2 q^{-38} -3 q^{-40} -5 q^{-42} + q^{-44} +2 q^{-46} -2 q^{-48} + q^{-50} +2 q^{-52} }[/math]
1,0,0,0	[math]\displaystyle{ -q^8+q^6+2 q^2+2+ q^{-2} +2 q^{-4} + q^{-8} - q^{-10} + q^{-12} - q^{-14} -2 q^{-18} - q^{-20} - q^{-22} -2 q^{-24} + q^{-26} + q^{-30} + q^{-32} + q^{-34} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{14}+2 q^{12}-4 q^{10}+5 q^8-4 q^6+5 q^4-3 q^2+3+ q^{-2} - q^{-4} +6 q^{-6} -6 q^{-8} +9 q^{-10} -9 q^{-12} +8 q^{-14} -8 q^{-16} +3 q^{-18} -3 q^{-20} - q^{-22} + q^{-24} -4 q^{-26} +5 q^{-28} -5 q^{-30} +5 q^{-32} -3 q^{-34} +3 q^{-36} }[/math]
1,0	[math]\displaystyle{ q^{24}-2 q^{20}-2 q^{18}+2 q^{16}+3 q^{14}-3 q^{12}-4 q^{10}+q^8+7 q^6+3 q^4-3 q^2-1+6 q^{-2} +5 q^{-4} + q^{-6} -4 q^{-8} - q^{-10} + q^{-12} - q^{-14} -6 q^{-16} -4 q^{-18} + q^{-20} + q^{-22} -2 q^{-24} -4 q^{-26} +3 q^{-28} +6 q^{-30} +2 q^{-32} -3 q^{-34} + q^{-36} +5 q^{-38} +3 q^{-40} -4 q^{-42} -5 q^{-44} + q^{-46} +4 q^{-48} - q^{-50} -4 q^{-52} -2 q^{-54} + q^{-56} +3 q^{-58} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{18}-2 q^{16}+2 q^{14}-4 q^{12}+4 q^{10}-4 q^8+4 q^6-2 q^4+6 q^2+2+3 q^{-2} +4 q^{-4} +3 q^{-8} -6 q^{-10} +4 q^{-12} -9 q^{-14} +4 q^{-16} -9 q^{-18} +5 q^{-20} -5 q^{-22} +6 q^{-24} -2 q^{-26} +2 q^{-28} + q^{-30} +2 q^{-34} -4 q^{-36} +2 q^{-38} -5 q^{-40} +5 q^{-42} -3 q^{-44} +2 q^{-46} -2 q^{-48} +3 q^{-50} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+4 q^{24}-q^{22}-8 q^{20}+16 q^{18}-16 q^{16}+16 q^{14}-4 q^{12}-11 q^{10}+20 q^8-18 q^6+14 q^4-2 q^2-13+21 q^{-2} -8 q^{-4} +13 q^{-8} -19 q^{-10} +18 q^{-12} -4 q^{-14} -9 q^{-16} +12 q^{-18} -21 q^{-20} +25 q^{-22} -13 q^{-24} +9 q^{-28} -19 q^{-30} +23 q^{-32} -18 q^{-34} +4 q^{-36} +3 q^{-38} -14 q^{-40} +19 q^{-42} -11 q^{-44} -3 q^{-46} +15 q^{-48} -19 q^{-50} +12 q^{-52} + q^{-54} -17 q^{-56} +20 q^{-58} -17 q^{-60} +11 q^{-62} +2 q^{-64} -11 q^{-66} +14 q^{-68} -12 q^{-70} +8 q^{-72} -5 q^{-76} +2 q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/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 47"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-4 t^2+6 t-5+6 t^{-1} -4 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+2 z^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{ \{3,t+1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+4 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^7 a^{-1} +2 z^7 a^{-3} +6 z^6 a^{-2} +3 z^6 a^{-4} +3 z^6+a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} -16 z^4 a^{-2} -7 z^4 a^{-4} -9 z^4-2 a z^3+z^3 a^{-1} +6 z^3 a^{-3} +3 z^3 a^{-5} +11 z^2 a^{-2} +9 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1 }[/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 47"];

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

Out[5]=

{}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-1, -2)

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{ -4 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{110}{3} }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{440}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ \frac{10529}{30} }[/math] [math]\displaystyle{ -\frac{418}{15} }[/math] [math]\displaystyle{ \frac{10738}{45} }[/math] [math]\displaystyle{ -\frac{833}{18} }[/math] [math]\displaystyle{ \frac{1409}{30} }[/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 47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 χ 11               2 2 9             2   -2 7           2 2   0 5         3 2     -1 3       2 2       0 1     2 4         2 -1   1 1           0 -3   2             2 -5 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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^{15}-6 q^{13}+6 q^{12}+6 q^{11}-17 q^{10}+10 q^9+14 q^8-25 q^7+8 q^6+19 q^5-25 q^4+2 q^3+20 q^2-18 q-3+17 q^{-1} -8 q^{-2} -6 q^{-3} +9 q^{-4} - q^{-5} -3 q^{-6} + q^{-7} }[/math]
3	[math]\displaystyle{ 2 q^{29}-4 q^{28}-2 q^{27}+q^{26}+15 q^{25}-3 q^{24}-25 q^{23}-4 q^{22}+37 q^{21}+19 q^{20}-51 q^{19}-32 q^{18}+54 q^{17}+50 q^{16}-59 q^{15}-58 q^{14}+51 q^{13}+69 q^{12}-46 q^{11}-70 q^{10}+37 q^9+70 q^8-25 q^7-69 q^6+16 q^5+60 q^4+3 q^3-60 q^2-10 q+43+26 q^{-1} -35 q^{-2} -28 q^{-3} +17 q^{-4} +32 q^{-5} -6 q^{-6} -23 q^{-7} -5 q^{-8} +17 q^{-9} +6 q^{-10} -7 q^{-11} -5 q^{-12} + q^{-13} +3 q^{-14} - q^{-15} }[/math]
4	[math]\displaystyle{ q^{48}-6 q^{46}-3 q^{45}+12 q^{44}+14 q^{43}+5 q^{42}-34 q^{41}-46 q^{40}+23 q^{39}+68 q^{38}+73 q^{37}-58 q^{36}-161 q^{35}-30 q^{34}+118 q^{33}+215 q^{32}-7 q^{31}-278 q^{30}-150 q^{29}+100 q^{28}+344 q^{27}+100 q^{26}-319 q^{25}-248 q^{24}+28 q^{23}+392 q^{22}+185 q^{21}-297 q^{20}-276 q^{19}-38 q^{18}+374 q^{17}+217 q^{16}-245 q^{15}-254 q^{14}-92 q^{13}+321 q^{12}+228 q^{11}-173 q^{10}-214 q^9-145 q^8+236 q^7+227 q^6-72 q^5-149 q^4-193 q^3+115 q^2+190 q+32-47 q^{-1} -192 q^{-2} -7 q^{-3} +98 q^{-4} +77 q^{-5} +56 q^{-6} -115 q^{-7} -60 q^{-8} -3 q^{-9} +39 q^{-10} +88 q^{-11} -21 q^{-12} -33 q^{-13} -38 q^{-14} -12 q^{-15} +46 q^{-16} +11 q^{-17} +3 q^{-18} -15 q^{-19} -16 q^{-20} +7 q^{-21} +3 q^{-22} +5 q^{-23} - q^{-24} -3 q^{-25} + q^{-26} }[/math]
5	[math]\displaystyle{ 2 q^{71}-4 q^{70}-2 q^{69}-2 q^{68}+3 q^{67}+21 q^{66}+19 q^{65}-21 q^{64}-51 q^{63}-49 q^{62}-8 q^{61}+111 q^{60}+154 q^{59}+42 q^{58}-151 q^{57}-283 q^{56}-192 q^{55}+154 q^{54}+472 q^{53}+404 q^{52}-83 q^{51}-626 q^{50}-691 q^{49}-95 q^{48}+727 q^{47}+1010 q^{46}+338 q^{45}-755 q^{44}-1255 q^{43}-637 q^{42}+675 q^{41}+1466 q^{40}+904 q^{39}-563 q^{38}-1549 q^{37}-1124 q^{36}+397 q^{35}+1590 q^{34}+1268 q^{33}-270 q^{32}-1542 q^{31}-1351 q^{30}+145 q^{29}+1495 q^{28}+1373 q^{27}-65 q^{26}-1406 q^{25}-1369 q^{24}-15 q^{23}+1323 q^{22}+1349 q^{21}+84 q^{20}-1221 q^{19}-1314 q^{18}-173 q^{17}+1083 q^{16}+1287 q^{15}+290 q^{14}-944 q^{13}-1226 q^{12}-404 q^{11}+721 q^{10}+1161 q^9+546 q^8-516 q^7-1028 q^6-630 q^5+231 q^4+858 q^3+714 q^2-15 q-619-674 q^{-1} -224 q^{-2} +367 q^{-3} +593 q^{-4} +336 q^{-5} -114 q^{-6} -410 q^{-7} -392 q^{-8} -78 q^{-9} +224 q^{-10} +318 q^{-11} +192 q^{-12} -34 q^{-13} -213 q^{-14} -209 q^{-15} -71 q^{-16} +71 q^{-17} +153 q^{-18} +132 q^{-19} +17 q^{-20} -82 q^{-21} -101 q^{-22} -61 q^{-23} +5 q^{-24} +66 q^{-25} +61 q^{-26} +17 q^{-27} -21 q^{-28} -33 q^{-29} -24 q^{-30} -5 q^{-31} +18 q^{-32} +14 q^{-33} +3 q^{-34} -3 q^{-35} -3 q^{-36} -5 q^{-37} + q^{-38} +3 q^{-39} - q^{-40} }[/math]
6	[math]\displaystyle{ q^{99}-6 q^{97}-3 q^{96}+6 q^{95}+11 q^{94}+14 q^{93}+12 q^{92}-7 q^{91}-64 q^{90}-83 q^{89}-18 q^{88}+78 q^{87}+162 q^{86}+204 q^{85}+92 q^{84}-220 q^{83}-477 q^{82}-434 q^{81}-65 q^{80}+445 q^{79}+935 q^{78}+897 q^{77}+32 q^{76}-1075 q^{75}-1656 q^{74}-1225 q^{73}+77 q^{72}+1873 q^{71}+2748 q^{70}+1631 q^{69}-829 q^{68}-3059 q^{67}-3524 q^{66}-1819 q^{65}+1792 q^{64}+4634 q^{63}+4325 q^{62}+970 q^{61}-3303 q^{60}-5637 q^{59}-4594 q^{58}+244 q^{57}+5213 q^{56}+6574 q^{55}+3386 q^{54}-2186 q^{53}-6367 q^{52}-6680 q^{51}-1701 q^{50}+4511 q^{49}+7418 q^{48}+5032 q^{47}-767 q^{46}-5949 q^{45}-7440 q^{44}-2940 q^{43}+3523 q^{42}+7244 q^{41}+5580 q^{40}+139 q^{39}-5253 q^{38}-7358 q^{37}-3392 q^{36}+2818 q^{35}+6777 q^{34}+5549 q^{33}+612 q^{32}-4638 q^{31}-7027 q^{30}-3574 q^{29}+2217 q^{28}+6240 q^{27}+5439 q^{26}+1127 q^{25}-3892 q^{24}-6593 q^{23}-3894 q^{22}+1292 q^{21}+5430 q^{20}+5347 q^{19}+1991 q^{18}-2664 q^{17}-5850 q^{16}-4346 q^{15}-152 q^{14}+4030 q^{13}+4979 q^{12}+3060 q^{11}-844 q^{10}-4434 q^9-4462 q^8-1809 q^7+1944 q^6+3828 q^5+3677 q^4+1147 q^3-2236 q^2-3594 q-2831-290 q^{-1} +1775 q^{-2} +3078 q^{-3} +2319 q^{-4} +68 q^{-5} -1695 q^{-6} -2423 q^{-7} -1547 q^{-8} -320 q^{-9} +1362 q^{-10} +1925 q^{-11} +1247 q^{-12} +161 q^{-13} -917 q^{-14} -1199 q^{-15} -1184 q^{-16} -195 q^{-17} +578 q^{-18} +881 q^{-19} +759 q^{-20} +271 q^{-21} -126 q^{-22} -684 q^{-23} -536 q^{-24} -284 q^{-25} +48 q^{-26} +284 q^{-27} +368 q^{-28} +360 q^{-29} -15 q^{-30} -124 q^{-31} -236 q^{-32} -196 q^{-33} -118 q^{-34} +28 q^{-35} +186 q^{-36} +103 q^{-37} +89 q^{-38} -2 q^{-39} -48 q^{-40} -94 q^{-41} -62 q^{-42} +12 q^{-43} +10 q^{-44} +39 q^{-45} +25 q^{-46} +17 q^{-47} -16 q^{-48} -17 q^{-49} - q^{-50} -7 q^{-51} +3 q^{-52} +3 q^{-53} +5 q^{-54} - q^{-55} -3 q^{-56} + q^{-57} }[/math]
7	[math]\displaystyle{ 2 q^{131}-4 q^{130}-2 q^{129}-2 q^{128}+15 q^{126}+19 q^{125}+15 q^{124}-13 q^{123}-47 q^{122}-70 q^{121}-68 q^{120}-28 q^{119}+109 q^{118}+246 q^{117}+278 q^{116}+144 q^{115}-160 q^{114}-489 q^{113}-725 q^{112}-649 q^{111}-75 q^{110}+846 q^{109}+1567 q^{108}+1653 q^{107}+791 q^{106}-812 q^{105}-2516 q^{104}-3461 q^{103}-2611 q^{102}+49 q^{101}+3396 q^{100}+5762 q^{99}+5505 q^{98}+2098 q^{97}-3244 q^{96}-8154 q^{95}-9574 q^{94}-5881 q^{93}+1670 q^{92}+9855 q^{91}+13991 q^{90}+11087 q^{89}+1803 q^{88}-9990 q^{87}-17999 q^{86}-17161 q^{85}-6983 q^{84}+8363 q^{83}+20776 q^{82}+22927 q^{81}+13091 q^{80}-4888 q^{79}-21648 q^{78}-27727 q^{77}-19347 q^{76}+351 q^{75}+20885 q^{74}+30785 q^{73}+24610 q^{72}+4578 q^{71}-18666 q^{70}-32143 q^{69}-28605 q^{68}-8988 q^{67}+15921 q^{66}+32023 q^{65}+30941 q^{64}+12461 q^{63}-13064 q^{62}-31045 q^{61}-32019 q^{60}-14775 q^{59}+10730 q^{58}+29642 q^{57}+32091 q^{56}+16131 q^{55}-8934 q^{54}-28287 q^{53}-31718 q^{52}-16723 q^{51}+7756 q^{50}+27068 q^{49}+31078 q^{48}+16987 q^{47}-6870 q^{46}-26075 q^{45}-30510 q^{44}-17103 q^{43}+6142 q^{42}+25133 q^{41}+29948 q^{40}+17342 q^{39}-5240 q^{38}-24091 q^{37}-29454 q^{36}-17828 q^{35}+4011 q^{34}+22811 q^{33}+28917 q^{32}+18523 q^{31}-2320 q^{30}-20988 q^{29}-28182 q^{28}-19533 q^{27}+19 q^{26}+18678 q^{25}+27138 q^{24}+20527 q^{23}+2746 q^{22}-15509 q^{21}-25433 q^{20}-21539 q^{19}-6000 q^{18}+11708 q^{17}+23066 q^{16}+21943 q^{15}+9245 q^{14}-7100 q^{13}-19589 q^{12}-21648 q^{11}-12361 q^{10}+2229 q^9+15299 q^8+20080 q^7+14458 q^6+2675 q^5-10035 q^4-17269 q^3-15407 q^2-6824 q+4661+13133 q^{-1} +14552 q^{-2} +9688 q^{-3} +496 q^{-4} -8266 q^{-5} -12190 q^{-6} -10696 q^{-7} -4404 q^{-8} +3309 q^{-9} +8477 q^{-10} +9890 q^{-11} +6668 q^{-12} +794 q^{-13} -4385 q^{-14} -7487 q^{-15} -6944 q^{-16} -3494 q^{-17} +670 q^{-18} +4387 q^{-19} +5691 q^{-20} +4371 q^{-21} +1803 q^{-22} -1381 q^{-23} -3472 q^{-24} -3781 q^{-25} -2888 q^{-26} -721 q^{-27} +1299 q^{-28} +2355 q^{-29} +2579 q^{-30} +1629 q^{-31} +318 q^{-32} -772 q^{-33} -1648 q^{-34} -1589 q^{-35} -970 q^{-36} -256 q^{-37} +579 q^{-38} +907 q^{-39} +926 q^{-40} +734 q^{-41} +138 q^{-42} -303 q^{-43} -529 q^{-44} -605 q^{-45} -368 q^{-46} -146 q^{-47} +94 q^{-48} +360 q^{-49} +332 q^{-50} +236 q^{-51} +88 q^{-52} -96 q^{-53} -140 q^{-54} -175 q^{-55} -152 q^{-56} -30 q^{-57} +42 q^{-58} +84 q^{-59} +90 q^{-60} +32 q^{-61} +21 q^{-62} -12 q^{-63} -42 q^{-64} -30 q^{-65} -18 q^{-66} +4 q^{-67} +15 q^{-68} +4 q^{-69} +5 q^{-70} +7 q^{-71} -3 q^{-72} -3 q^{-73} -5 q^{-74} + q^{-75} +3 q^{-76} - q^{-77} }[/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.

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