9 17

9_16

9_18

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 17's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 17 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code	-1, 4, -3, 1, -2, 6, -5, 9, -8, 3, -4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code	4 10 12 14 16 2 6 18 8
Conway Notation	[21312]

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

[edit Notes on presentations of 9 17]

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 17"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 12 14 16 2 6 18 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[21312]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-3] Hyperbolic Volume 9.47458 A-Polynomial See Data:9 17/A-polynomial

[edit Notes for 9 17's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for 9 17's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+9t-9+9t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 39, -2 }
Jones polynomial	$q^{3}-2q^{2}+4q-5+6q^{-1}-7q^{-2}+6q^{-3}-4q^{-4}+3q^{-5}-q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{2}z^{6}-a^{4}z^{4}+4a^{2}z^{4}-2z^{4}-2a^{4}z^{2}+5a^{2}z^{2}+z^{2}a^{-2}-6z^{2}+2a^{2}+2a^{-2}-3$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+5az^{7}+2z^{7}a^{-1}+4a^{4}z^{6}+4a^{2}z^{6}+z^{6}a^{-2}+z^{6}+4a^{5}z^{5}-2a^{3}z^{5}-13az^{5}-7z^{5}a^{-1}+3a^{6}z^{4}-3a^{4}z^{4}-14a^{2}z^{4}-4z^{4}a^{-2}-12z^{4}+a^{7}z^{3}-3a^{5}z^{3}-4a^{3}z^{3}+6az^{3}+6z^{3}a^{-1}-2a^{6}z^{2}-a^{4}z^{2}+9a^{2}z^{2}+5z^{2}a^{-2}+13z^{2}+a^{5}z+3a^{3}z+az-za^{-1}-2a^{2}-2a^{-2}-3$
The A2 invariant	$-q^{18}+q^{16}+q^{12}+2q^{10}-q^{8}+q^{6}-2q^{4}-q^{-2}+q^{-4}+q^{-8}+q^{-10}$
The G2 invariant	$q^{100}-2q^{98}+3q^{96}-4q^{94}+q^{92}-3q^{88}+8q^{86}-9q^{84}+12q^{82}-9q^{80}+3q^{78}+4q^{76}-9q^{74}+14q^{72}-20q^{70}+17q^{68}-12q^{66}+q^{64}+10q^{62}-18q^{60}+22q^{58}-16q^{56}+8q^{54}+q^{52}-14q^{50}+16q^{48}-7q^{46}-2q^{44}+15q^{42}-18q^{40}+15q^{38}+3q^{36}-17q^{34}+30q^{32}-35q^{30}+26q^{28}-6q^{26}-14q^{24}+33q^{22}-38q^{20}+32q^{18}-16q^{16}-4q^{14}+15q^{12}-26q^{10}+22q^{8}-12q^{6}-4q^{4}+14q^{2}-17+10q^{-2}+3q^{-4}-17q^{-6}+23q^{-8}-24q^{-10}+11q^{-12}+5q^{-14}-21q^{-16}+32q^{-18}-27q^{-20}+17q^{-22}-q^{-24}-11q^{-26}+19q^{-28}-18q^{-30}+14q^{-32}-4q^{-34}-2q^{-36}+5q^{-38}-5q^{-40}+4q^{-42}-q^{-44}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 9_17.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+2q^{11}-q^{9}+2q^{7}-q^{5}-q^{3}+q-q^{-1}+2q^{-3}-q^{-5}+q^{-7}$
2	$q^{36}-2q^{34}-q^{32}+4q^{30}-4q^{28}+q^{26}+7q^{24}-7q^{22}-q^{20}+7q^{18}-5q^{16}-4q^{14}+4q^{12}+2q^{10}-3q^{8}-q^{6}+6q^{4}-q^{2}-5+7q^{-2}+q^{-4}-8q^{-6}+4q^{-8}+4q^{-10}-6q^{-12}+q^{-14}+4q^{-16}-2q^{-18}-q^{-20}+q^{-22}$
3	$-q^{69}+2q^{67}+q^{65}-2q^{63}-2q^{61}+q^{59}+2q^{57}-5q^{55}+3q^{53}+7q^{51}-3q^{49}-12q^{47}+7q^{45}+17q^{43}-8q^{41}-22q^{39}+4q^{37}+24q^{35}-22q^{31}-8q^{29}+18q^{27}+12q^{25}-6q^{23}-17q^{21}+18q^{17}+7q^{15}-16q^{13}-13q^{11}+15q^{9}+16q^{7}-11q^{5}-21q^{3}+9q+22q^{-1}-2q^{-3}-23q^{-5}-4q^{-7}+23q^{-9}+11q^{-11}-19q^{-13}-15q^{-15}+12q^{-17}+18q^{-19}-4q^{-21}-18q^{-23}-q^{-25}+13q^{-27}+6q^{-29}-8q^{-31}-7q^{-33}+3q^{-35}+5q^{-37}-2q^{-41}-q^{-43}+q^{-45}$
4	$q^{112}-2q^{110}-q^{108}+2q^{106}+5q^{102}-4q^{100}-q^{98}-9q^{94}+9q^{92}+9q^{88}+q^{86}-28q^{84}+q^{82}+12q^{80}+36q^{78}+9q^{76}-60q^{74}-30q^{72}+17q^{70}+76q^{68}+39q^{66}-80q^{64}-74q^{62}-6q^{60}+96q^{58}+80q^{56}-50q^{54}-86q^{52}-55q^{50}+56q^{48}+93q^{46}+14q^{44}-44q^{42}-75q^{40}-14q^{38}+51q^{36}+57q^{34}+19q^{32}-56q^{30}-64q^{28}+q^{26}+70q^{24}+53q^{22}-30q^{20}-80q^{18}-29q^{16}+71q^{14}+66q^{12}-12q^{10}-85q^{8}-51q^{6}+64q^{4}+77q^{2}+18-78q^{-2}-78q^{-4}+34q^{-6}+76q^{-8}+58q^{-10}-45q^{-12}-91q^{-14}-18q^{-16}+40q^{-18}+85q^{-20}+13q^{-22}-64q^{-24}-52q^{-26}-16q^{-28}+64q^{-30}+50q^{-32}-7q^{-34}-38q^{-36}-52q^{-38}+12q^{-40}+38q^{-42}+27q^{-44}+q^{-46}-37q^{-48}-16q^{-50}+4q^{-52}+18q^{-54}+17q^{-56}-8q^{-58}-9q^{-60}-7q^{-62}+q^{-64}+7q^{-66}+q^{-68}-2q^{-72}-q^{-74}+q^{-76}$
5	$-q^{165}+2q^{163}+q^{161}-2q^{159}-3q^{155}-2q^{153}+3q^{151}+6q^{149}+3q^{147}+2q^{145}-6q^{143}-13q^{141}-9q^{139}+8q^{137}+27q^{135}+16q^{133}-9q^{131}-40q^{129}-46q^{127}+10q^{125}+78q^{123}+71q^{121}-7q^{119}-103q^{117}-128q^{115}-13q^{113}+159q^{111}+194q^{109}+37q^{107}-194q^{105}-279q^{103}-97q^{101}+218q^{99}+372q^{97}+176q^{95}-211q^{93}-442q^{91}-274q^{89}+152q^{87}+468q^{85}+377q^{83}-56q^{81}-433q^{79}-435q^{77}-72q^{75}+338q^{73}+439q^{71}+185q^{69}-189q^{67}-386q^{65}-271q^{63}+43q^{61}+283q^{59}+291q^{57}+103q^{55}-157q^{53}-285q^{51}-193q^{49}+43q^{47}+239q^{45}+251q^{43}+50q^{41}-205q^{39}-281q^{37}-101q^{35}+174q^{33}+292q^{31}+133q^{29}-160q^{27}-311q^{25}-151q^{23}+168q^{21}+325q^{19}+176q^{17}-153q^{15}-356q^{13}-214q^{11}+143q^{9}+372q^{7}+260q^{5}-95q^{3}-375q-322q^{-1}+27q^{-3}+348q^{-5}+369q^{-7}+69q^{-9}-286q^{-11}-391q^{-13}-169q^{-15}+186q^{-17}+376q^{-19}+257q^{-21}-64q^{-23}-313q^{-25}-300q^{-27}-65q^{-29}+206q^{-31}+300q^{-33}+166q^{-35}-81q^{-37}-244q^{-39}-212q^{-41}-42q^{-43}+146q^{-45}+211q^{-47}+122q^{-49}-43q^{-51}-154q^{-53}-147q^{-55}-50q^{-57}+76q^{-59}+132q^{-61}+90q^{-63}-3q^{-65}-79q^{-67}-92q^{-69}-44q^{-71}+27q^{-73}+66q^{-75}+55q^{-77}+9q^{-79}-31q^{-81}-41q^{-83}-26q^{-85}+5q^{-87}+24q^{-89}+21q^{-91}+5q^{-93}-6q^{-95}-11q^{-97}-9q^{-99}+q^{-101}+5q^{-103}+3q^{-105}+q^{-107}-2q^{-111}-q^{-113}+q^{-115}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{18}+q^{16}+q^{12}+2q^{10}-q^{8}+q^{6}-2q^{4}-q^{-2}+q^{-4}+q^{-8}+q^{-10}$
1,1	$q^{52}-4q^{50}+8q^{48}-12q^{46}+18q^{44}-28q^{42}+36q^{40}-40q^{38}+45q^{36}-56q^{34}+60q^{32}-56q^{30}+60q^{28}-50q^{26}+40q^{24}-16q^{22}-17q^{20}+44q^{18}-88q^{16}+114q^{14}-141q^{12}+160q^{10}-156q^{8}+148q^{6}-117q^{4}+92q^{2}-50+6q^{-2}+31q^{-4}-62q^{-6}+80q^{-8}-94q^{-10}+92q^{-12}-78q^{-14}+64q^{-16}-46q^{-18}+29q^{-20}-16q^{-22}+8q^{-24}-2q^{-26}+q^{-28}$
2,0	$q^{46}-q^{44}-q^{42}-q^{38}+2q^{34}+3q^{32}-2q^{30}+4q^{26}+2q^{24}-6q^{22}-q^{20}+3q^{18}-3q^{16}-3q^{14}+2q^{10}-2q^{8}+2q^{6}+4q^{4}+q^{2}+1+4q^{-2}-5q^{-6}+2q^{-10}-q^{-12}-3q^{-14}+q^{-16}+3q^{-18}-2q^{-22}+q^{-26}+q^{-28}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{42}-2q^{40}-q^{38}+4q^{36}-3q^{34}-2q^{32}+7q^{30}-q^{28}-5q^{26}+6q^{24}-2q^{22}-6q^{20}+3q^{18}+q^{16}+q^{12}+5q^{10}+3q^{8}-5q^{6}+q^{4}+3q^{2}-7-q^{-2}+4q^{-4}-5q^{-6}+q^{-8}+4q^{-10}-2q^{-12}+2q^{-14}+2q^{-16}-q^{-18}+q^{-20}$
1,0,0	$-q^{23}+q^{21}-q^{19}+2q^{17}+2q^{13}+q^{9}-q^{5}-2q-2q^{-3}+q^{-5}+2q^{-9}+q^{-11}+q^{-13}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{42}+2q^{40}-3q^{38}+4q^{36}-5q^{34}+6q^{32}-7q^{30}+7q^{28}-5q^{26}+4q^{24}-2q^{20}+7q^{18}-9q^{16}+12q^{14}-13q^{12}+13q^{10}-13q^{8}+9q^{6}-7q^{4}+3q^{2}-1-3q^{-2}+6q^{-4}-7q^{-6}+7q^{-8}-6q^{-10}+6q^{-12}-4q^{-14}+4q^{-16}-q^{-18}+q^{-20}$
1,0	$q^{68}-2q^{64}-2q^{62}+q^{60}+4q^{58}+q^{56}-4q^{54}-4q^{52}+2q^{50}+7q^{48}+3q^{46}-5q^{44}-5q^{42}+2q^{40}+7q^{38}-q^{36}-7q^{34}-3q^{32}+5q^{30}+3q^{28}-4q^{26}-4q^{24}+3q^{22}+6q^{20}-3q^{16}+q^{14}+5q^{12}-5q^{8}-2q^{6}+5q^{4}+4q^{2}-4-8q^{-2}+8q^{-6}+3q^{-8}-6q^{-10}-6q^{-12}+3q^{-14}+6q^{-16}+q^{-18}-4q^{-20}-2q^{-22}+3q^{-24}+3q^{-26}-q^{-28}-q^{-30}+q^{-34}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{58}-2q^{56}+q^{54}-2q^{52}+4q^{50}-4q^{48}+3q^{46}-4q^{44}+7q^{42}-4q^{40}+4q^{38}-5q^{36}+4q^{34}-2q^{32}-q^{30}-5q^{26}+6q^{24}-6q^{22}+10q^{20}-8q^{18}+13q^{16}-7q^{14}+12q^{12}-8q^{10}+7q^{8}-6q^{6}+2q^{4}-5q^{2}-2-5q^{-4}+3q^{-6}-6q^{-8}+7q^{-10}-4q^{-12}+6q^{-14}-3q^{-16}+6q^{-18}-2q^{-20}+3q^{-22}-q^{-24}+q^{-26}$

G2 Invariants.

Weight

Invariant

1,0

$q^{100}-2q^{98}+3q^{96}-4q^{94}+q^{92}-3q^{88}+8q^{86}-9q^{84}+12q^{82}-9q^{80}+3q^{78}+4q^{76}-9q^{74}+14q^{72}-20q^{70}+17q^{68}-12q^{66}+q^{64}+10q^{62}-18q^{60}+22q^{58}-16q^{56}+8q^{54}+q^{52}-14q^{50}+16q^{48}-7q^{46}-2q^{44}+15q^{42}-18q^{40}+15q^{38}+3q^{36}-17q^{34}+30q^{32}-35q^{30}+26q^{28}-6q^{26}-14q^{24}+33q^{22}-38q^{20}+32q^{18}-16q^{16}-4q^{14}+15q^{12}-26q^{10}+22q^{8}-12q^{6}-4q^{4}+14q^{2}-17+10q^{-2}+3q^{-4}-17q^{-6}+23q^{-8}-24q^{-10}+11q^{-12}+5q^{-14}-21q^{-16}+32q^{-18}-27q^{-20}+17q^{-22}-q^{-24}-11q^{-26}+19q^{-28}-18q^{-30}+14q^{-32}-4q^{-34}-2q^{-36}+5q^{-38}-5q^{-40}+4q^{-42}-q^{-44}+q^{-46}$

.

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 17"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+9t-9+9t^{-1}-5t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+z^{4}-2z^{2}+1$

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]=

{ 39, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{2}z^{6}-a^{4}z^{4}+4a^{2}z^{4}-2z^{4}-2a^{4}z^{2}+5a^{2}z^{2}+z^{2}a^{-2}-6z^{2}+2a^{2}+2a^{-2}-3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+5az^{7}+2z^{7}a^{-1}+4a^{4}z^{6}+4a^{2}z^{6}+z^{6}a^{-2}+z^{6}+4a^{5}z^{5}-2a^{3}z^{5}-13az^{5}-7z^{5}a^{-1}+3a^{6}z^{4}-3a^{4}z^{4}-14a^{2}z^{4}-4z^{4}a^{-2}-12z^{4}+a^{7}z^{3}-3a^{5}z^{3}-4a^{3}z^{3}+6az^{3}+6z^{3}a^{-1}-2a^{6}z^{2}-a^{4}z^{2}+9a^{2}z^{2}+5z^{2}a^{-2}+13z^{2}+a^{5}z+3a^{3}z+az-za^{-1}-2a^{2}-2a^{-2}-3$

"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 17"];

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]=

{ $t^{3}-5t^{2}+9t-9+9t^{-1}-5t^{-2}+t^{-3}$, $q^{3}-2q^{2}+4q-5+6q^{-1}-7q^{-2}+6q^{-3}-4q^{-4}+3q^{-5}-q^{-6}$ }

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, 0)

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 $-8$ $0$ $32$ ${\frac {116}{3}}$ ${\frac {28}{3}}$ $0$ $32$ $32$ $0$ $-{\frac {256}{3}}$ $0$ $-{\frac {928}{3}}$ $-{\frac {224}{3}}$ $-{\frac {4951}{15}}$ $-{\frac {1796}{15}}$ $-{\frac {2764}{45}}$ ${\frac {7}{9}}$ ${\frac {89}{15}}$

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 $t^{r}q^{j}$ 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 $s=$-2 is the signature of 9 17. 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 7                   1 1 5                 1   -1 3               3 1   2 1             2 1     -1 -1           4 3       1 -3         4 3         -1 -5       2 3           -1 -7     2 4             2 -9   1 2               -1 -11   2                 2 -13 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{10}-2q^{9}-q^{8}+7q^{7}-5q^{6}-8q^{5}+17q^{4}-5q^{3}-20q^{2}+26q+1-32q^{-1}+30q^{-2}+8q^{-3}-39q^{-4}+28q^{-5}+13q^{-6}-37q^{-7}+20q^{-8}+12q^{-9}-25q^{-10}+12q^{-11}+6q^{-12}-11q^{-13}+6q^{-14}+q^{-15}-3q^{-16}+q^{-17}$
3	$q^{21}-2q^{20}-q^{19}+2q^{18}+6q^{17}-4q^{16}-11q^{15}+q^{14}+20q^{13}+3q^{12}-25q^{11}-16q^{10}+34q^{9}+25q^{8}-31q^{7}-43q^{6}+30q^{5}+55q^{4}-19q^{3}-70q^{2}+11q+76+5q^{-1}-83q^{-2}-19q^{-3}+86q^{-4}+32q^{-5}-84q^{-6}-47q^{-7}+83q^{-8}+55q^{-9}-73q^{-10}-65q^{-11}+66q^{-12}+66q^{-13}-55q^{-14}-59q^{-15}+40q^{-16}+52q^{-17}-33q^{-18}-35q^{-19}+20q^{-20}+26q^{-21}-19q^{-22}-10q^{-23}+10q^{-24}+7q^{-25}-10q^{-26}+6q^{-28}-q^{-29}-3q^{-30}-q^{-31}+3q^{-32}-q^{-33}$
4	$q^{36}-2q^{35}-q^{34}+2q^{33}+q^{32}+7q^{31}-8q^{30}-9q^{29}+2q^{27}+32q^{26}-7q^{25}-23q^{24}-20q^{23}-19q^{22}+70q^{21}+19q^{20}-12q^{19}-46q^{18}-83q^{17}+84q^{16}+50q^{15}+45q^{14}-32q^{13}-163q^{12}+48q^{11}+38q^{10}+122q^{9}+40q^{8}-208q^{7}-10q^{6}-35q^{5}+168q^{4}+143q^{3}-190q^{2}-52q-147+168q^{-1}+239q^{-2}-131q^{-3}-65q^{-4}-262q^{-5}+134q^{-6}+312q^{-7}-53q^{-8}-60q^{-9}-362q^{-10}+83q^{-11}+362q^{-12}+30q^{-13}-43q^{-14}-431q^{-15}+18q^{-16}+370q^{-17}+105q^{-18}-5q^{-19}-437q^{-20}-47q^{-21}+309q^{-22}+136q^{-23}+53q^{-24}-358q^{-25}-84q^{-26}+198q^{-27}+105q^{-28}+89q^{-29}-228q^{-30}-68q^{-31}+96q^{-32}+37q^{-33}+83q^{-34}-109q^{-35}-31q^{-36}+37q^{-37}-10q^{-38}+53q^{-39}-40q^{-40}-4q^{-41}+13q^{-42}-21q^{-43}+24q^{-44}-11q^{-45}+4q^{-46}+4q^{-47}-12q^{-48}+6q^{-49}-2q^{-50}+3q^{-51}+q^{-52}-3q^{-53}+q^{-54}$
5	$q^{55}-2q^{54}-q^{53}+2q^{52}+q^{51}+2q^{50}+3q^{49}-6q^{48}-11q^{47}+6q^{45}+13q^{44}+19q^{43}-3q^{42}-30q^{41}-31q^{40}-9q^{39}+23q^{38}+59q^{37}+43q^{36}-19q^{35}-70q^{34}-80q^{33}-25q^{32}+72q^{31}+119q^{30}+74q^{29}-28q^{28}-136q^{27}-151q^{26}-25q^{25}+112q^{24}+185q^{23}+137q^{22}-47q^{21}-216q^{20}-213q^{19}-58q^{18}+153q^{17}+300q^{16}+200q^{15}-82q^{14}-307q^{13}-329q^{12}-82q^{11}+287q^{10}+449q^{9}+239q^{8}-188q^{7}-519q^{6}-437q^{5}+65q^{4}+554q^{3}+594q^{2}+112q-540-758q^{-1}-284q^{-2}+501q^{-3}+874q^{-4}+467q^{-5}-428q^{-6}-987q^{-7}-641q^{-8}+359q^{-9}+1077q^{-10}+796q^{-11}-279q^{-12}-1144q^{-13}-960q^{-14}+199q^{-15}+1228q^{-16}+1089q^{-17}-120q^{-18}-1262q^{-19}-1235q^{-20}+19q^{-21}+1304q^{-22}+1344q^{-23}+81q^{-24}-1274q^{-25}-1431q^{-26}-217q^{-27}+1212q^{-28}+1472q^{-29}+341q^{-30}-1086q^{-31}-1439q^{-32}-457q^{-33}+898q^{-34}+1357q^{-35}+538q^{-36}-712q^{-37}-1185q^{-38}-558q^{-39}+488q^{-40}+994q^{-41}+540q^{-42}-335q^{-43}-752q^{-44}-467q^{-45}+172q^{-46}+568q^{-47}+372q^{-48}-104q^{-49}-365q^{-50}-271q^{-51}+18q^{-52}+253q^{-53}+190q^{-54}-19q^{-55}-134q^{-56}-114q^{-57}-17q^{-58}+81q^{-59}+75q^{-60}+6q^{-61}-38q^{-62}-36q^{-63}-10q^{-64}+13q^{-65}+19q^{-66}+12q^{-67}-7q^{-68}-11q^{-69}+q^{-70}-6q^{-71}+2q^{-72}+8q^{-73}-3q^{-75}+2q^{-76}-3q^{-77}-q^{-78}+3q^{-79}-q^{-80}$
6	$q^{78}-2q^{77}-q^{76}+2q^{75}+q^{74}+2q^{73}-2q^{72}+5q^{71}-8q^{70}-11q^{69}+3q^{68}+5q^{67}+14q^{66}+3q^{65}+24q^{64}-17q^{63}-39q^{62}-22q^{61}-11q^{60}+25q^{59}+19q^{58}+100q^{57}+17q^{56}-49q^{55}-73q^{54}-90q^{53}-39q^{52}-31q^{51}+197q^{50}+133q^{49}+68q^{48}-29q^{47}-136q^{46}-190q^{45}-267q^{44}+131q^{43}+158q^{42}+264q^{41}+216q^{40}+93q^{39}-162q^{38}-544q^{37}-172q^{36}-175q^{35}+177q^{34}+397q^{33}+591q^{32}+315q^{31}-406q^{30}-331q^{29}-750q^{28}-436q^{27}+5q^{26}+852q^{25}+1007q^{24}+335q^{23}+181q^{22}-959q^{21}-1230q^{20}-1058q^{19}+349q^{18}+1248q^{17}+1241q^{16}+1369q^{15}-319q^{14}-1523q^{13}-2269q^{12}-848q^{11}+615q^{10}+1642q^{9}+2680q^{8}+1030q^{7}-965q^{6}-2990q^{5}-2193q^{4}-702q^{3}+1264q^{2}+3551q+2555+213q^{-1}-3010q^{-2}-3198q^{-3}-2203q^{-4}+349q^{-5}+3838q^{-6}+3830q^{-7}+1550q^{-8}-2571q^{-9}-3766q^{-10}-3511q^{-11}-685q^{-12}+3770q^{-13}+4771q^{-14}+2724q^{-15}-2030q^{-16}-4101q^{-17}-4553q^{-18}-1573q^{-19}+3631q^{-20}+5526q^{-21}+3697q^{-22}-1559q^{-23}-4399q^{-24}-5450q^{-25}-2336q^{-26}+3485q^{-27}+6189q^{-28}+4596q^{-29}-1024q^{-30}-4589q^{-31}-6235q^{-32}-3156q^{-33}+3069q^{-34}+6557q^{-35}+5428q^{-36}-169q^{-37}-4292q^{-38}-6629q^{-39}-4019q^{-40}+2108q^{-41}+6172q^{-42}+5837q^{-43}+941q^{-44}-3233q^{-45}-6149q^{-46}-4501q^{-47}+771q^{-48}+4842q^{-49}+5341q^{-50}+1762q^{-51}-1720q^{-52}-4702q^{-53}-4120q^{-54}-303q^{-55}+3028q^{-56}+3959q^{-57}+1809q^{-58}-469q^{-59}-2869q^{-60}-2983q^{-61}-661q^{-62}+1502q^{-63}+2333q^{-64}+1232q^{-65}+114q^{-66}-1398q^{-67}-1719q^{-68}-483q^{-69}+630q^{-70}+1112q^{-71}+573q^{-72}+203q^{-73}-565q^{-74}-825q^{-75}-202q^{-76}+246q^{-77}+452q^{-78}+175q^{-79}+132q^{-80}-194q^{-81}-356q^{-82}-42q^{-83}+91q^{-84}+162q^{-85}+25q^{-86}+70q^{-87}-55q^{-88}-141q^{-89}+4q^{-90}+24q^{-91}+52q^{-92}-9q^{-93}+37q^{-94}-11q^{-95}-49q^{-96}+7q^{-97}+15q^{-99}-8q^{-100}+16q^{-101}-14q^{-103}+4q^{-104}-3q^{-105}+3q^{-106}-2q^{-107}+3q^{-108}+q^{-109}-3q^{-110}+q^{-111}$
7	$q^{105}-2q^{104}-q^{103}+2q^{102}+q^{101}+2q^{100}-2q^{99}+3q^{97}-8q^{96}-8q^{95}+2q^{94}+5q^{93}+16q^{92}+5q^{91}+q^{90}+13q^{89}-23q^{88}-34q^{87}-25q^{86}-13q^{85}+38q^{84}+38q^{83}+37q^{82}+70q^{81}-2q^{80}-59q^{79}-93q^{78}-132q^{77}-23q^{76}+24q^{75}+75q^{74}+209q^{73}+150q^{72}+74q^{71}-50q^{70}-267q^{69}-229q^{68}-208q^{67}-133q^{66}+198q^{65}+294q^{64}+399q^{63}+367q^{62}-25q^{61}-184q^{60}-453q^{59}-643q^{58}-324q^{57}-123q^{56}+322q^{55}+799q^{54}+676q^{53}+635q^{52}+114q^{51}-670q^{50}-874q^{49}-1205q^{48}-849q^{47}+115q^{46}+738q^{45}+1635q^{44}+1699q^{43}+788q^{42}-41q^{41}-1577q^{40}-2446q^{39}-2017q^{38}-1187q^{37}+992q^{36}+2731q^{35}+3090q^{34}+2807q^{33}+427q^{32}-2300q^{31}-3897q^{30}-4558q^{29}-2308q^{28}+1094q^{27}+3862q^{26}+5999q^{25}+4607q^{24}+949q^{23}-3046q^{22}-6888q^{21}-6771q^{20}-3467q^{19}+1263q^{18}+6854q^{17}+8590q^{16}+6257q^{15}+1220q^{14}-5947q^{13}-9686q^{12}-8868q^{11}-4215q^{10}+4155q^{9}+9981q^{8}+11106q^{7}+7335q^{6}-1728q^{5}-9431q^{4}-12709q^{3}-10385q^{2}-1134q+8236+13694q^{-1}+13062q^{-2}+4086q^{-3}-6474q^{-4}-14025q^{-5}-15370q^{-6}-7019q^{-7}+4495q^{-8}+13940q^{-9}+17159q^{-10}+9672q^{-11}-2401q^{-12}-13477q^{-13}-18579q^{-14}-12072q^{-15}+411q^{-16}+12923q^{-17}+19706q^{-18}+14102q^{-19}+1349q^{-20}-12339q^{-21}-20591q^{-22}-15887q^{-23}-2930q^{-24}+11890q^{-25}+21484q^{-26}+17463q^{-27}+4192q^{-28}-11593q^{-29}-22293q^{-30}-18922q^{-31}-5408q^{-32}+11376q^{-33}+23229q^{-34}+20426q^{-35}+6522q^{-36}-11219q^{-37}-24079q^{-38}-21880q^{-39}-7831q^{-40}+10781q^{-41}+24815q^{-42}+23452q^{-43}+9340q^{-44}-10065q^{-45}-25156q^{-46}-24782q^{-47}-11094q^{-48}+8682q^{-49}+24854q^{-50}+25852q^{-51}+13009q^{-52}-6760q^{-53}-23769q^{-54}-26211q^{-55}-14777q^{-56}+4299q^{-57}+21694q^{-58}+25700q^{-59}+16178q^{-60}-1574q^{-61}-18846q^{-62}-24203q^{-63}-16777q^{-64}-1005q^{-65}+15380q^{-66}+21658q^{-67}+16509q^{-68}+3183q^{-69}-11778q^{-70}-18449q^{-71}-15254q^{-72}-4551q^{-73}+8329q^{-74}+14823q^{-75}+13296q^{-76}+5144q^{-77}-5491q^{-78}-11273q^{-79}-10798q^{-80}-4997q^{-81}+3279q^{-82}+8091q^{-83}+8324q^{-84}+4310q^{-85}-1887q^{-86}-5491q^{-87}-5933q^{-88}-3371q^{-89}+951q^{-90}+3542q^{-91}+4074q^{-92}+2435q^{-93}-581q^{-94}-2220q^{-95}-2544q^{-96}-1575q^{-97}+327q^{-98}+1291q^{-99}+1583q^{-100}+1000q^{-101}-301q^{-102}-792q^{-103}-889q^{-104}-524q^{-105}+235q^{-106}+425q^{-107}+497q^{-108}+306q^{-109}-207q^{-110}-261q^{-111}-272q^{-112}-143q^{-113}+180q^{-114}+139q^{-115}+130q^{-116}+69q^{-117}-116q^{-118}-66q^{-119}-85q^{-120}-51q^{-121}+99q^{-122}+49q^{-123}+28q^{-124}+13q^{-125}-46q^{-126}-8q^{-127}-26q^{-128}-27q^{-129}+41q^{-130}+13q^{-131}+5q^{-132}+5q^{-133}-14q^{-134}+4q^{-135}-9q^{-136}-10q^{-137}+12q^{-138}+2q^{-139}-q^{-140}+3q^{-141}-3q^{-142}+2q^{-143}-3q^{-144}-q^{-145}+3q^{-146}-q^{-147}$

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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