9 19

9_18

9_20

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 19's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 19 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X11,18,12,1 X17,12,18,13
Gauss code	-1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 10 14 2 18 16 6 12
Conway Notation	[23112]

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

[edit Notes on presentations of 9 19]

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 19"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 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]=

[23112]

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]=

$\textrm{BR}(5,\{1,-2,1,-2,-2,-3,2,4,-3,4\})$

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.0325 A-Polynomial See Data:9 19/A-polynomial

[edit Notes for 9 19's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 9 19's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2 t^2-10 t+17-10 t^{-1} +2 t^{-2}$
Conway polynomial	$2 z^4-2 z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, 0 }
Jones polynomial	$q^4-2 q^3+4 q^2-6 q+7-7 q^{-1} +6 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^2 a^4+z^4 a^2+z^2 a^2+a^2+z^4-2 z^2 a^{-2} - a^{-2} + a^{-4}$
Kauffman polynomial (db, data sources)	$a^2 z^8+z^8+3 a^3 z^7+5 a z^7+2 z^7 a^{-1} +3 a^4 z^6+3 a^2 z^6+2 z^6 a^{-2} +2 z^6+a^5 z^5-7 a^3 z^5-11 a z^5-z^5 a^{-1} +2 z^5 a^{-3} -8 a^4 z^4-11 a^2 z^4+z^4 a^{-4} -4 z^4-2 a^5 z^3+4 a^3 z^3+10 a z^3+z^3 a^{-1} -3 z^3 a^{-3} +4 a^4 z^2+8 a^2 z^2-3 z^2 a^{-2} -2 z^2 a^{-4} +3 z^2-a^3 z-3 a z-z a^{-1} +z a^{-3} -a^2+ a^{-2} + a^{-4}$
The A2 invariant	$-q^{16}+q^{14}+q^{12}-q^{10}+2 q^8+q^2-1+ q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14}$
The G2 invariant	$q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-3 q^{70}-5 q^{68}+16 q^{66}-21 q^{64}+24 q^{62}-18 q^{60}+2 q^{58}+18 q^{56}-34 q^{54}+40 q^{52}-31 q^{50}+13 q^{48}+11 q^{46}-28 q^{44}+33 q^{42}-24 q^{40}+8 q^{38}+10 q^{36}-23 q^{34}+20 q^{32}-6 q^{30}-13 q^{28}+30 q^{26}-34 q^{24}+27 q^{22}-6 q^{20}-20 q^{18}+41 q^{16}-51 q^{14}+48 q^{12}-25 q^{10}-3 q^8+30 q^6-44 q^4+44 q^2-27+4 q^{-2} +14 q^{-4} -24 q^{-6} +19 q^{-8} -4 q^{-10} -13 q^{-12} +23 q^{-14} -22 q^{-16} +7 q^{-18} +9 q^{-20} -26 q^{-22} +33 q^{-24} -29 q^{-26} +17 q^{-28} -2 q^{-30} -14 q^{-32} +22 q^{-34} -24 q^{-36} +21 q^{-38} -12 q^{-40} +4 q^{-42} +4 q^{-44} -9 q^{-46} +11 q^{-48} -9 q^{-50} +8 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_19.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2 q^9-q^7+2 q^5-q^3+ q^{-1} -2 q^{-3} +2 q^{-5} - q^{-7} + q^{-9}$
2	$q^{32}-2 q^{30}-2 q^{28}+6 q^{26}-q^{24}-7 q^{22}+7 q^{20}+3 q^{18}-10 q^{16}+5 q^{14}+6 q^{12}-8 q^{10}+q^8+5 q^6-2 q^4-4 q^2+2+7 q^{-2} -7 q^{-4} -3 q^{-6} +11 q^{-8} -5 q^{-10} -5 q^{-12} +8 q^{-14} -2 q^{-16} -3 q^{-18} +3 q^{-20} - q^{-22} - q^{-24} + q^{-26}$
3	$-q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-6 q^{55}+q^{53}+13 q^{51}+2 q^{49}-16 q^{47}-10 q^{45}+17 q^{43}+20 q^{41}-13 q^{39}-30 q^{37}+6 q^{35}+33 q^{33}+5 q^{31}-35 q^{29}-13 q^{27}+34 q^{25}+20 q^{23}-27 q^{21}-24 q^{19}+21 q^{17}+23 q^{15}-13 q^{13}-24 q^{11}+5 q^9+21 q^7+6 q^5-17 q^3-17 q+13 q^{-1} +29 q^{-3} -5 q^{-5} -34 q^{-7} -4 q^{-9} +37 q^{-11} +13 q^{-13} -35 q^{-15} -17 q^{-17} +25 q^{-19} +19 q^{-21} -17 q^{-23} -16 q^{-25} +10 q^{-27} +11 q^{-29} -5 q^{-31} -6 q^{-33} +3 q^{-35} +3 q^{-37} -2 q^{-39} - q^{-41} +2 q^{-43} - q^{-47} - q^{-49} + q^{-51}$
4	$q^{104}-2 q^{102}-2 q^{100}+3 q^{98}+3 q^{96}+6 q^{94}-8 q^{92}-13 q^{90}-q^{88}+7 q^{86}+31 q^{84}+q^{82}-30 q^{80}-28 q^{78}-13 q^{76}+57 q^{74}+44 q^{72}-7 q^{70}-56 q^{68}-81 q^{66}+31 q^{64}+85 q^{62}+70 q^{60}-25 q^{58}-138 q^{56}-53 q^{54}+60 q^{52}+142 q^{50}+61 q^{48}-130 q^{46}-126 q^{44}-14 q^{42}+147 q^{40}+128 q^{38}-73 q^{36}-141 q^{34}-73 q^{32}+105 q^{30}+139 q^{28}-17 q^{26}-110 q^{24}-87 q^{22}+54 q^{20}+112 q^{18}+27 q^{16}-69 q^{14}-85 q^{12}+q^{10}+74 q^8+76 q^6-15 q^4-84 q^2-75+20 q^{-2} +131 q^{-4} +61 q^{-6} -60 q^{-8} -145 q^{-10} -61 q^{-12} +138 q^{-14} +132 q^{-16} +10 q^{-18} -155 q^{-20} -132 q^{-22} +79 q^{-24} +134 q^{-26} +76 q^{-28} -88 q^{-30} -131 q^{-32} +6 q^{-34} +69 q^{-36} +82 q^{-38} -16 q^{-40} -73 q^{-42} -17 q^{-44} +9 q^{-46} +43 q^{-48} +9 q^{-50} -23 q^{-52} -6 q^{-54} -9 q^{-56} +12 q^{-58} +5 q^{-60} -4 q^{-62} +4 q^{-64} -6 q^{-66} + q^{-68} - q^{-72} +4 q^{-74} - q^{-76} - q^{-80} - q^{-82} + q^{-84}$
5	$-q^{155}+2 q^{153}+2 q^{151}-3 q^{149}-3 q^{147}-3 q^{145}+q^{143}+8 q^{141}+13 q^{139}+q^{137}-17 q^{135}-22 q^{133}-13 q^{131}+13 q^{129}+40 q^{127}+44 q^{125}-3 q^{123}-57 q^{121}-72 q^{119}-39 q^{117}+43 q^{115}+114 q^{113}+106 q^{111}-123 q^{107}-173 q^{105}-104 q^{103}+78 q^{101}+233 q^{99}+226 q^{97}+30 q^{95}-227 q^{93}-348 q^{91}-200 q^{89}+147 q^{87}+423 q^{85}+394 q^{83}+14 q^{81}-429 q^{79}-553 q^{77}-225 q^{75}+338 q^{73}+664 q^{71}+439 q^{69}-191 q^{67}-687 q^{65}-607 q^{63}+4 q^{61}+633 q^{59}+715 q^{57}+168 q^{55}-531 q^{53}-746 q^{51}-292 q^{49}+397 q^{47}+709 q^{45}+376 q^{43}-277 q^{41}-635 q^{39}-394 q^{37}+172 q^{35}+533 q^{33}+392 q^{31}-91 q^{29}-442 q^{27}-362 q^{25}+22 q^{23}+351 q^{21}+348 q^{19}+48 q^{17}-267 q^{15}-349 q^{13}-139 q^{11}+184 q^9+362 q^7+257 q^5-75 q^3-379 q-403 q^{-1} -65 q^{-3} +381 q^{-5} +543 q^{-7} +243 q^{-9} -324 q^{-11} -667 q^{-13} -442 q^{-15} +217 q^{-17} +733 q^{-19} +619 q^{-21} -50 q^{-23} -705 q^{-25} -752 q^{-27} -143 q^{-29} +598 q^{-31} +799 q^{-33} +301 q^{-35} -412 q^{-37} -740 q^{-39} -423 q^{-41} +218 q^{-43} +610 q^{-45} +453 q^{-47} -48 q^{-49} -430 q^{-51} -413 q^{-53} -72 q^{-55} +261 q^{-57} +325 q^{-59} +124 q^{-61} -130 q^{-63} -221 q^{-65} -125 q^{-67} +41 q^{-69} +133 q^{-71} +103 q^{-73} +2 q^{-75} -71 q^{-77} -68 q^{-79} -17 q^{-81} +30 q^{-83} +40 q^{-85} +20 q^{-87} -9 q^{-89} -23 q^{-91} -14 q^{-93} + q^{-95} +7 q^{-97} +9 q^{-99} +6 q^{-101} -4 q^{-103} -6 q^{-105} - q^{-107} -2 q^{-109} + q^{-111} +4 q^{-113} + q^{-115} - q^{-117} - q^{-121} - q^{-123} + q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+q^{12}-q^{10}+2 q^8+q^2-1+ q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14}$
1,1	$q^{44}-4 q^{42}+10 q^{40}-22 q^{38}+40 q^{36}-62 q^{34}+86 q^{32}-112 q^{30}+129 q^{28}-130 q^{26}+120 q^{24}-90 q^{22}+48 q^{20}+8 q^{18}-70 q^{16}+128 q^{14}-177 q^{12}+216 q^{10}-234 q^8+234 q^6-214 q^4+172 q^2-122+64 q^{-2} -9 q^{-4} -38 q^{-6} +78 q^{-8} -94 q^{-10} +104 q^{-12} -100 q^{-14} +92 q^{-16} -78 q^{-18} +62 q^{-20} -50 q^{-22} +36 q^{-24} -26 q^{-26} +17 q^{-28} -10 q^{-30} +6 q^{-32} -2 q^{-34} + q^{-36}$
2,0	$q^{42}-q^{40}-2 q^{38}+3 q^{34}+2 q^{32}-5 q^{30}-q^{28}+5 q^{26}+3 q^{24}-5 q^{22}-3 q^{20}+6 q^{18}+3 q^{16}-5 q^{14}-q^{12}+5 q^{10}-q^8-2 q^6+q^4-q^2-1+2 q^{-2} +2 q^{-4} -5 q^{-6} -2 q^{-8} +8 q^{-10} +2 q^{-12} -6 q^{-14} + q^{-16} +7 q^{-18} + q^{-20} -5 q^{-22} -2 q^{-24} +2 q^{-26} -2 q^{-30} + q^{-34} + q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2 q^{32}+2 q^{28}-5 q^{26}+4 q^{24}+4 q^{22}-6 q^{20}+5 q^{18}+5 q^{16}-8 q^{14}+q^{12}+4 q^{10}-5 q^8-q^6+3 q^4+3 q^2-1- q^{-2} +7 q^{-4} -4 q^{-6} -6 q^{-8} +8 q^{-10} -3 q^{-12} -6 q^{-14} +6 q^{-16} -3 q^{-20} +3 q^{-22} + q^{-24} - q^{-26} + q^{-28}$
1,0,0	$-q^{21}+q^{19}+q^{15}-q^{13}+2 q^{11}+q^7+q^3- q^{-1} + q^{-3} -2 q^{-5} - q^{-9} + q^{-11} - q^{-13} + q^{-15} + q^{-17} + q^{-19}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2 q^{32}-4 q^{30}+6 q^{28}-7 q^{26}+8 q^{24}-8 q^{22}+8 q^{20}-5 q^{18}+3 q^{16}+2 q^{14}-5 q^{12}+10 q^{10}-13 q^8+15 q^6-15 q^4+15 q^2-13+9 q^{-2} -5 q^{-4} +2 q^{-8} -6 q^{-10} +7 q^{-12} -8 q^{-14} +8 q^{-16} -6 q^{-18} +5 q^{-20} -3 q^{-22} +3 q^{-24} - q^{-26} + q^{-28}$
1,0	$q^{56}-2 q^{52}-2 q^{50}+2 q^{48}+4 q^{46}-2 q^{44}-6 q^{42}-q^{40}+8 q^{38}+6 q^{36}-5 q^{34}-8 q^{32}+2 q^{30}+9 q^{28}+4 q^{26}-7 q^{24}-6 q^{22}+3 q^{20}+6 q^{18}-2 q^{16}-6 q^{14}+5 q^{10}+q^8-5 q^6-2 q^4+6 q^2+5-4 q^{-2} -5 q^{-4} +4 q^{-6} +7 q^{-8} -2 q^{-10} -8 q^{-12} -2 q^{-14} +8 q^{-16} +5 q^{-18} -5 q^{-20} -8 q^{-22} +7 q^{-26} +3 q^{-28} -3 q^{-30} -4 q^{-32} +3 q^{-36} +2 q^{-38} - q^{-40} - q^{-42} + q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-3 q^{70}-5 q^{68}+16 q^{66}-21 q^{64}+24 q^{62}-18 q^{60}+2 q^{58}+18 q^{56}-34 q^{54}+40 q^{52}-31 q^{50}+13 q^{48}+11 q^{46}-28 q^{44}+33 q^{42}-24 q^{40}+8 q^{38}+10 q^{36}-23 q^{34}+20 q^{32}-6 q^{30}-13 q^{28}+30 q^{26}-34 q^{24}+27 q^{22}-6 q^{20}-20 q^{18}+41 q^{16}-51 q^{14}+48 q^{12}-25 q^{10}-3 q^8+30 q^6-44 q^4+44 q^2-27+4 q^{-2} +14 q^{-4} -24 q^{-6} +19 q^{-8} -4 q^{-10} -13 q^{-12} +23 q^{-14} -22 q^{-16} +7 q^{-18} +9 q^{-20} -26 q^{-22} +33 q^{-24} -29 q^{-26} +17 q^{-28} -2 q^{-30} -14 q^{-32} +22 q^{-34} -24 q^{-36} +21 q^{-38} -12 q^{-40} +4 q^{-42} +4 q^{-44} -9 q^{-46} +11 q^{-48} -9 q^{-50} +8 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66}$

. </div></div>

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 19"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2 t^2-10 t+17-10 t^{-1} +2 t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2 z^4-2 z^2+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 41, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^4-2 q^3+4 q^2-6 q+7-7 q^{-1} +6 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^2 a^4+z^4 a^2+z^2 a^2+a^2+z^4-2 z^2 a^{-2} - a^{-2} + a^{-4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^2 z^8+z^8+3 a^3 z^7+5 a z^7+2 z^7 a^{-1} +3 a^4 z^6+3 a^2 z^6+2 z^6 a^{-2} +2 z^6+a^5 z^5-7 a^3 z^5-11 a z^5-z^5 a^{-1} +2 z^5 a^{-3} -8 a^4 z^4-11 a^2 z^4+z^4 a^{-4} -4 z^4-2 a^5 z^3+4 a^3 z^3+10 a z^3+z^3 a^{-1} -3 z^3 a^{-3} +4 a^4 z^2+8 a^2 z^2-3 z^2 a^{-2} -2 z^2 a^{-4} +3 z^2-a^3 z-3 a z-z a^{-1} +z a^{-3} -a^2+ a^{-2} + a^{-4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

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 19"];

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]=

{ $2 t^2-10 t+17-10 t^{-1} +2 t^{-2}$, $q^4-2 q^3+4 q^2-6 q+7-7 q^{-1} +6 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5}$ }

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:

(-2, -1)

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 $-8$ $-8$ $32$ $\frac{68}{3}$ $\frac{4}{3}$ $64$ $\frac{304}{3}$ $\frac{160}{3}$ $-8$ $-\frac{256}{3}$ $32$ $-\frac{544}{3}$ $-\frac{32}{3}$ $-\frac{991}{15}$ $-\frac{1676}{15}$ $\frac{5276}{45}$ $-\frac{113}{9}$ $\frac{449}{15}$

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 $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 9 19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 1   -1 5               3 1   2 3             3 1     -2 1           4 3       1 -1         4 4         0 -3       2 3           -1 -5     2 4             2 -7   1 2               -1 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

<p>

$n$	$J_n$
2	$q^{12}-2 q^{11}+5 q^9-8 q^8+q^7+15 q^6-21 q^5+q^4+31 q^3-35 q^2-3 q+45-40 q^{-1} -9 q^{-2} +47 q^{-3} -33 q^{-4} -13 q^{-5} +38 q^{-6} -19 q^{-7} -14 q^{-8} +23 q^{-9} -6 q^{-10} -10 q^{-11} +9 q^{-12} -3 q^{-14} + q^{-15}$
3	$q^{24}-2 q^{23}+q^{21}+3 q^{20}-5 q^{19}-q^{18}+6 q^{17}+3 q^{16}-14 q^{15}+22 q^{13}+2 q^{12}-40 q^{11}-q^{10}+58 q^9+8 q^8-82 q^7-19 q^6+106 q^5+32 q^4-123 q^3-49 q^2+135 q+66-139 q^{-1} -79 q^{-2} +135 q^{-3} +89 q^{-4} -124 q^{-5} -95 q^{-6} +106 q^{-7} +100 q^{-8} -88 q^{-9} -97 q^{-10} +61 q^{-11} +97 q^{-12} -41 q^{-13} -83 q^{-14} +14 q^{-15} +75 q^{-16} - q^{-17} -55 q^{-18} -13 q^{-19} +39 q^{-20} +16 q^{-21} -22 q^{-22} -16 q^{-23} +12 q^{-24} +10 q^{-25} -4 q^{-26} -5 q^{-27} +3 q^{-29} - q^{-30}$
4	$q^{40}-2 q^{39}+q^{37}-q^{36}+6 q^{35}-7 q^{34}+q^{33}+2 q^{32}-8 q^{31}+16 q^{30}-15 q^{29}+10 q^{28}+9 q^{27}-29 q^{26}+19 q^{25}-32 q^{24}+42 q^{23}+43 q^{22}-63 q^{21}-7 q^{20}-88 q^{19}+99 q^{18}+141 q^{17}-76 q^{16}-70 q^{15}-225 q^{14}+142 q^{13}+305 q^{12}-18 q^{11}-125 q^{10}-436 q^9+119 q^8+470 q^7+104 q^6-119 q^5-635 q^4+35 q^3+555 q^2+225 q-49-746 q^{-1} -60 q^{-2} +546 q^{-3} +294 q^{-4} +42 q^{-5} -748 q^{-6} -133 q^{-7} +460 q^{-8} +310 q^{-9} +138 q^{-10} -663 q^{-11} -191 q^{-12} +319 q^{-13} +287 q^{-14} +231 q^{-15} -507 q^{-16} -225 q^{-17} +141 q^{-18} +219 q^{-19} +299 q^{-20} -306 q^{-21} -206 q^{-22} -20 q^{-23} +107 q^{-24} +295 q^{-25} -115 q^{-26} -125 q^{-27} -102 q^{-28} -6 q^{-29} +210 q^{-30} -2 q^{-31} -30 q^{-32} -87 q^{-33} -60 q^{-34} +98 q^{-35} +23 q^{-36} +19 q^{-37} -36 q^{-38} -47 q^{-39} +28 q^{-40} +8 q^{-41} +17 q^{-42} -5 q^{-43} -17 q^{-44} +4 q^{-45} +5 q^{-47} -3 q^{-49} + q^{-50}$
5	$q^{60}-2 q^{59}+q^{57}-q^{56}+2 q^{55}+4 q^{54}-5 q^{53}-3 q^{52}+2 q^{51}-6 q^{50}+4 q^{49}+14 q^{48}-2 q^{47}-5 q^{46}-4 q^{45}-21 q^{44}-5 q^{43}+28 q^{42}+27 q^{41}+15 q^{40}-14 q^{39}-68 q^{38}-56 q^{37}+25 q^{36}+100 q^{35}+116 q^{34}+16 q^{33}-160 q^{32}-222 q^{31}-71 q^{30}+191 q^{29}+370 q^{28}+217 q^{27}-224 q^{26}-555 q^{25}-412 q^{24}+174 q^{23}+752 q^{22}+718 q^{21}-67 q^{20}-947 q^{19}-1053 q^{18}-143 q^{17}+1080 q^{16}+1431 q^{15}+431 q^{14}-1148 q^{13}-1794 q^{12}-752 q^{11}+1127 q^{10}+2086 q^9+1100 q^8-1034 q^7-2310 q^6-1411 q^5+902 q^4+2429 q^3+1667 q^2-734 q-2472-1857 q^{-1} +564 q^{-2} +2453 q^{-3} +1971 q^{-4} -402 q^{-5} -2367 q^{-6} -2035 q^{-7} +241 q^{-8} +2243 q^{-9} +2053 q^{-10} -87 q^{-11} -2067 q^{-12} -2032 q^{-13} -88 q^{-14} +1859 q^{-15} +1973 q^{-16} +264 q^{-17} -1584 q^{-18} -1891 q^{-19} -449 q^{-20} +1293 q^{-21} +1732 q^{-22} +622 q^{-23} -931 q^{-24} -1558 q^{-25} -761 q^{-26} +604 q^{-27} +1278 q^{-28} +837 q^{-29} -232 q^{-30} -1011 q^{-31} -843 q^{-32} -25 q^{-33} +667 q^{-34} +757 q^{-35} +264 q^{-36} -381 q^{-37} -618 q^{-38} -351 q^{-39} +104 q^{-40} +429 q^{-41} +388 q^{-42} +62 q^{-43} -238 q^{-44} -322 q^{-45} -172 q^{-46} +82 q^{-47} +240 q^{-48} +183 q^{-49} +19 q^{-50} -126 q^{-51} -165 q^{-52} -73 q^{-53} +58 q^{-54} +114 q^{-55} +69 q^{-56} -3 q^{-57} -59 q^{-58} -65 q^{-59} -13 q^{-60} +32 q^{-61} +36 q^{-62} +12 q^{-63} -5 q^{-64} -18 q^{-65} -17 q^{-66} +5 q^{-67} +10 q^{-68} +3 q^{-69} -5 q^{-72} +3 q^{-74} - q^{-75}$
6	$q^{84}-2 q^{83}+q^{81}-q^{80}+2 q^{79}+6 q^{77}-9 q^{76}-3 q^{75}+4 q^{74}-7 q^{73}+5 q^{72}+5 q^{71}+24 q^{70}-21 q^{69}-12 q^{68}+5 q^{67}-27 q^{66}+q^{65}+20 q^{64}+74 q^{63}-27 q^{62}-25 q^{61}-7 q^{60}-89 q^{59}-31 q^{58}+49 q^{57}+196 q^{56}+17 q^{55}-23 q^{54}-51 q^{53}-269 q^{52}-168 q^{51}+67 q^{50}+454 q^{49}+246 q^{48}+118 q^{47}-103 q^{46}-697 q^{45}-637 q^{44}-120 q^{43}+844 q^{42}+874 q^{41}+752 q^{40}+116 q^{39}-1384 q^{38}-1763 q^{37}-1019 q^{36}+999 q^{35}+1922 q^{34}+2314 q^{33}+1243 q^{32}-1859 q^{31}-3543 q^{30}-3125 q^{29}+165 q^{28}+2796 q^{27}+4727 q^{26}+3756 q^{25}-1278 q^{24}-5230 q^{23}-6201 q^{22}-2076 q^{21}+2590 q^{20}+7062 q^{19}+7201 q^{18}+693 q^{17}-5842 q^{16}-9126 q^{15}-5123 q^{14}+1061 q^{13}+8292 q^{12}+10321 q^{11}+3355 q^{10}-5164 q^9-10853 q^8-7759 q^7-1079 q^6+8234 q^5+12163 q^4+5616 q^3-3831 q^2-11252 q-9251-2914 q^{-1} +7437 q^{-2} +12695 q^{-3} +6961 q^{-4} -2533 q^{-5} -10799 q^{-6} -9712 q^{-7} -4130 q^{-8} +6408 q^{-9} +12358 q^{-10} +7589 q^{-11} -1401 q^{-12} -9859 q^{-13} -9555 q^{-14} -4989 q^{-15} +5152 q^{-16} +11426 q^{-17} +7868 q^{-18} -143 q^{-19} -8380 q^{-20} -8972 q^{-21} -5790 q^{-22} +3411 q^{-23} +9815 q^{-24} +7871 q^{-25} +1419 q^{-26} -6164 q^{-27} -7790 q^{-28} -6448 q^{-29} +1157 q^{-30} +7355 q^{-31} +7270 q^{-32} +2976 q^{-33} -3310 q^{-34} -5756 q^{-35} -6442 q^{-36} -1100 q^{-37} +4217 q^{-38} +5691 q^{-39} +3790 q^{-40} -491 q^{-41} -3019 q^{-42} -5286 q^{-43} -2470 q^{-44} +1183 q^{-45} +3279 q^{-46} +3298 q^{-47} +1297 q^{-48} -405 q^{-49} -3171 q^{-50} -2394 q^{-51} -712 q^{-52} +933 q^{-53} +1773 q^{-54} +1543 q^{-55} +1079 q^{-56} -1078 q^{-57} -1268 q^{-58} -1073 q^{-59} -366 q^{-60} +262 q^{-61} +767 q^{-62} +1182 q^{-63} +60 q^{-64} -157 q^{-65} -525 q^{-66} -498 q^{-67} -415 q^{-68} +6 q^{-69} +601 q^{-70} +228 q^{-71} +268 q^{-72} -15 q^{-73} -152 q^{-74} -365 q^{-75} -224 q^{-76} +143 q^{-77} +49 q^{-78} +196 q^{-79} +107 q^{-80} +58 q^{-81} -139 q^{-82} -140 q^{-83} +7 q^{-84} -38 q^{-85} +56 q^{-86} +52 q^{-87} +64 q^{-88} -28 q^{-89} -43 q^{-90} - q^{-91} -26 q^{-92} +6 q^{-93} +9 q^{-94} +26 q^{-95} -5 q^{-96} -10 q^{-97} +4 q^{-98} -7 q^{-99} +5 q^{-102} -3 q^{-104} + q^{-105}$
7	$q^{112}-2 q^{111}+q^{109}-q^{108}+2 q^{107}+2 q^{105}+2 q^{104}-9 q^{103}-q^{102}+3 q^{101}-6 q^{100}+7 q^{99}+3 q^{98}+11 q^{97}+11 q^{96}-28 q^{95}-7 q^{94}+2 q^{93}-19 q^{92}+12 q^{91}+9 q^{90}+38 q^{89}+41 q^{88}-56 q^{87}-29 q^{86}-16 q^{85}-51 q^{84}+25 q^{83}+27 q^{82}+96 q^{81}+115 q^{80}-91 q^{79}-96 q^{78}-111 q^{77}-139 q^{76}+62 q^{75}+122 q^{74}+276 q^{73}+308 q^{72}-112 q^{71}-286 q^{70}-463 q^{69}-495 q^{68}+34 q^{67}+402 q^{66}+869 q^{65}+980 q^{64}+139 q^{63}-599 q^{62}-1430 q^{61}-1736 q^{60}-661 q^{59}+632 q^{58}+2222 q^{57}+2998 q^{56}+1670 q^{55}-351 q^{54}-3062 q^{53}-4786 q^{52}-3517 q^{51}-588 q^{50}+3816 q^{49}+7082 q^{48}+6256 q^{47}+2521 q^{46}-3931 q^{45}-9630 q^{44}-10112 q^{43}-5829 q^{42}+3173 q^{41}+12096 q^{40}+14679 q^{39}+10528 q^{38}-900 q^{37}-13795 q^{36}-19766 q^{35}-16639 q^{34}-2931 q^{33}+14414 q^{32}+24606 q^{31}+23543 q^{30}+8369 q^{29}-13428 q^{28}-28609 q^{27}-30788 q^{26}-15023 q^{25}+10894 q^{24}+31380 q^{23}+37534 q^{22}+22153 q^{21}-6984 q^{20}-32519 q^{19}-43182 q^{18}-29277 q^{17}+2194 q^{16}+32266 q^{15}+47435 q^{14}+35554 q^{13}+2811 q^{12}-30755 q^{11}-50085 q^{10}-40694 q^9-7636 q^8+28531 q^7+51410 q^6+44492 q^5+11758 q^4-26022 q^3-51586 q^2-46997 q-15071+23460 q^{-1} +51043 q^{-2} +48493 q^{-3} +17596 q^{-4} -21149 q^{-5} -50051 q^{-6} -49152 q^{-7} -19460 q^{-8} +18995 q^{-9} +48723 q^{-10} +49321 q^{-11} +20961 q^{-12} -16936 q^{-13} -47183 q^{-14} -49153 q^{-15} -22238 q^{-16} +14779 q^{-17} +45278 q^{-18} +48678 q^{-19} +23585 q^{-20} -12258 q^{-21} -42961 q^{-22} -47959 q^{-23} -25002 q^{-24} +9292 q^{-25} +39963 q^{-26} +46799 q^{-27} +26559 q^{-28} -5721 q^{-29} -36200 q^{-30} -45063 q^{-31} -28062 q^{-32} +1588 q^{-33} +31519 q^{-34} +42587 q^{-35} +29294 q^{-36} +2862 q^{-37} -25974 q^{-38} -39041 q^{-39} -29920 q^{-40} -7486 q^{-41} +19703 q^{-42} +34560 q^{-43} +29637 q^{-44} +11556 q^{-45} -13068 q^{-46} -28889 q^{-47} -28087 q^{-48} -14950 q^{-49} +6456 q^{-50} +22647 q^{-51} +25292 q^{-52} +16844 q^{-53} -625 q^{-54} -15869 q^{-55} -21155 q^{-56} -17348 q^{-57} -4126 q^{-58} +9515 q^{-59} +16312 q^{-60} +16060 q^{-61} +7132 q^{-62} -3882 q^{-63} -11017 q^{-64} -13576 q^{-65} -8487 q^{-66} -263 q^{-67} +6155 q^{-68} +10143 q^{-69} +8141 q^{-70} +2916 q^{-71} -2143 q^{-72} -6607 q^{-73} -6670 q^{-74} -3880 q^{-75} -595 q^{-76} +3349 q^{-77} +4603 q^{-78} +3714 q^{-79} +2037 q^{-80} -1038 q^{-81} -2529 q^{-82} -2688 q^{-83} -2354 q^{-84} -432 q^{-85} +844 q^{-86} +1549 q^{-87} +1994 q^{-88} +958 q^{-89} +142 q^{-90} -484 q^{-91} -1255 q^{-92} -945 q^{-93} -644 q^{-94} -187 q^{-95} +659 q^{-96} +643 q^{-97} +618 q^{-98} +447 q^{-99} -136 q^{-100} -264 q^{-101} -485 q^{-102} -517 q^{-103} -64 q^{-104} +77 q^{-105} +242 q^{-106} +349 q^{-107} +137 q^{-108} +105 q^{-109} -88 q^{-110} -260 q^{-111} -122 q^{-112} -86 q^{-113} +21 q^{-114} +110 q^{-115} +55 q^{-116} +91 q^{-117} +41 q^{-118} -65 q^{-119} -47 q^{-120} -50 q^{-121} -12 q^{-122} +28 q^{-123} -3 q^{-124} +26 q^{-125} +25 q^{-126} -6 q^{-127} -9 q^{-128} -17 q^{-129} -4 q^{-130} +10 q^{-131} -4 q^{-132} +7 q^{-134} -5 q^{-137} +3 q^{-139} - q^{-140}$

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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