9 22

9_21

9_23

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 22's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 22 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17
Gauss code	1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -9, 8, -6, 7, -5, 9, -8
Dowker-Thistlethwaite code	4 8 10 14 2 16 18 6 12
Conway Notation	[211,3,2]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{6, 12}, {1, 9}, {11, 4}, {12, 10}, {8, 3}, {9, 7}, {5, 8}, {7, 11}, {4, 2}, {3, 6}, {2, 5}, {10, 1}]

[edit Notes on presentations of 9 22]

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 22"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 2 16 18 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[211,3,2]

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]=

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

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[{6, 12}, {1, 9}, {11, 4}, {12, 10}, {8, 3}, {9, 7}, {5, 8}, {7, 11}, {4, 2}, {3, 6}, {2, 5}, {10, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-8] Hyperbolic Volume 10.6207 A-Polynomial See Data:9 22/A-polynomial

[edit Notes for 9 22'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 22's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+10t-11+10t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, 2 }
Jones polynomial	$-q^{6}+3q^{5}-5q^{4}+7q^{3}-7q^{2}+7q-6+4q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+6z^{2}a^{-2}-2z^{2}a^{-4}-6z^{2}+2a^{2}+4a^{-2}-a^{-4}-4$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}+2az^{7}+6z^{7}a^{-1}+4z^{7}a^{-3}+a^{2}z^{6}+7z^{6}a^{-2}+6z^{6}a^{-4}+2z^{6}-7az^{5}-16z^{5}a^{-1}-4z^{5}a^{-3}+5z^{5}a^{-5}-4a^{2}z^{4}-23z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}-15z^{4}+7az^{3}+10z^{3}a^{-1}-2z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+5a^{2}z^{2}+17z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+16z^{2}-2az-2za^{-1}+za^{-3}+za^{-5}-2a^{2}-4a^{-2}-a^{-4}-4$
The A2 invariant	$q^{10}+q^{8}+q^{4}-2q^{2}-1-q^{-4}+3q^{-6}+2q^{-10}-q^{-14}+q^{-16}-q^{-18}$
The G2 invariant	$q^{46}-q^{44}+4q^{42}-5q^{40}+5q^{38}-3q^{36}-3q^{34}+14q^{32}-20q^{30}+25q^{28}-17q^{26}+2q^{24}+19q^{22}-35q^{20}+43q^{18}-35q^{16}+14q^{14}+11q^{12}-35q^{10}+41q^{8}-32q^{6}+10q^{4}+12q^{2}-28+24q^{-2}-14q^{-4}-11q^{-6}+29q^{-8}-38q^{-10}+31q^{-12}-9q^{-14}-20q^{-16}+46q^{-18}-56q^{-20}+50q^{-22}-25q^{-24}-5q^{-26}+37q^{-28}-52q^{-30}+54q^{-32}-30q^{-34}+5q^{-36}+23q^{-38}-34q^{-40}+28q^{-42}-9q^{-44}-11q^{-46}+25q^{-48}-25q^{-50}+12q^{-52}+8q^{-54}-28q^{-56}+36q^{-58}-33q^{-60}+17q^{-62}+2q^{-64}-21q^{-66}+28q^{-68}-29q^{-70}+24q^{-72}-11q^{-74}+8q^{-78}-14q^{-80}+14q^{-82}-11q^{-84}+8q^{-86}-2q^{-88}-q^{-90}+2q^{-92}-4q^{-94}+3q^{-96}-2q^{-98}+q^{-100}$

Further Quantum Invariants

Further quantum knot invariants for 9_22.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+2q^{3}-2q+q^{-1}+2q^{-7}-2q^{-9}+2q^{-11}-q^{-13}$
2	$q^{22}-q^{20}-2q^{18}+4q^{16}-7q^{12}+6q^{10}+5q^{8}-10q^{6}+3q^{4}+9q^{2}-8-2q^{-2}+8q^{-4}-2q^{-6}-5q^{-8}+3q^{-10}+5q^{-12}-5q^{-14}-5q^{-16}+10q^{-18}-2q^{-20}-9q^{-22}+10q^{-24}+q^{-26}-7q^{-28}+4q^{-30}-2q^{-34}+q^{-36}$
3	$q^{45}-q^{43}-2q^{41}+5q^{37}+2q^{35}-8q^{33}-7q^{31}+10q^{29}+15q^{27}-7q^{25}-24q^{23}+29q^{19}+12q^{17}-29q^{15}-25q^{13}+25q^{11}+33q^{9}-14q^{7}-39q^{5}+4q^{3}+41q+6q^{-1}-37q^{-3}-14q^{-5}+33q^{-7}+21q^{-9}-26q^{-11}-25q^{-13}+19q^{-15}+27q^{-17}-6q^{-19}-29q^{-21}-7q^{-23}+26q^{-25}+21q^{-27}-21q^{-29}-34q^{-31}+12q^{-33}+41q^{-35}-q^{-37}-42q^{-39}-5q^{-41}+35q^{-43}+10q^{-45}-25q^{-47}-10q^{-49}+16q^{-51}+7q^{-53}-10q^{-55}-2q^{-57}+3q^{-59}+q^{-61}-2q^{-63}+2q^{-67}-q^{-69}$
4	$q^{76}-q^{74}-2q^{72}+q^{68}+7q^{66}-8q^{62}-8q^{60}-5q^{58}+22q^{56}+18q^{54}-5q^{52}-28q^{50}-41q^{48}+19q^{46}+51q^{44}+44q^{42}-13q^{40}-95q^{38}-46q^{36}+32q^{34}+108q^{32}+78q^{30}-79q^{28}-121q^{26}-74q^{24}+89q^{22}+172q^{20}+30q^{18}-106q^{16}-177q^{14}-20q^{12}+172q^{10}+138q^{8}-13q^{6}-194q^{4}-119q^{2}+99+172q^{-2}+70q^{-4}-149q^{-6}-156q^{-8}+29q^{-10}+160q^{-12}+102q^{-14}-102q^{-16}-152q^{-18}-16q^{-20}+133q^{-22}+114q^{-24}-44q^{-26}-137q^{-28}-75q^{-30}+81q^{-32}+126q^{-34}+53q^{-36}-87q^{-38}-148q^{-40}-30q^{-42}+100q^{-44}+171q^{-46}+25q^{-48}-171q^{-50}-152q^{-52}+4q^{-54}+212q^{-56}+142q^{-58}-100q^{-60}-184q^{-62}-95q^{-64}+142q^{-66}+159q^{-68}-8q^{-70}-106q^{-72}-107q^{-74}+48q^{-76}+89q^{-78}+18q^{-80}-28q^{-82}-56q^{-84}+10q^{-86}+26q^{-88}+4q^{-90}+2q^{-92}-17q^{-94}+3q^{-96}+5q^{-98}-2q^{-100}+3q^{-102}-3q^{-104}+2q^{-106}-2q^{-110}+q^{-112}$
5	$q^{115}-q^{113}-2q^{111}+q^{107}+3q^{105}+5q^{103}-10q^{99}-10q^{97}-3q^{95}+9q^{93}+24q^{91}+21q^{89}-8q^{87}-41q^{85}-47q^{83}-16q^{81}+44q^{79}+90q^{77}+69q^{75}-20q^{73}-120q^{71}-146q^{69}-57q^{67}+105q^{65}+222q^{63}+186q^{61}-16q^{59}-251q^{57}-324q^{55}-155q^{53}+173q^{51}+423q^{49}+375q^{47}+12q^{45}-413q^{43}-563q^{41}-286q^{39}+257q^{37}+662q^{35}+579q^{33}+12q^{31}-616q^{29}-795q^{27}-352q^{25}+419q^{23}+903q^{21}+668q^{19}-136q^{17}-856q^{15}-894q^{13}-189q^{11}+700q^{9}+1010q^{7}+464q^{5}-482q^{3}-1003q-655q^{-1}+255q^{-3}+925q^{-5}+753q^{-7}-78q^{-9}-803q^{-11}-765q^{-13}-37q^{-15}+679q^{-17}+733q^{-19}+97q^{-21}-587q^{-23}-679q^{-25}-126q^{-27}+520q^{-29}+642q^{-31}+153q^{-33}-461q^{-35}-629q^{-37}-222q^{-39}+383q^{-41}+638q^{-43}+332q^{-45}-242q^{-47}-624q^{-49}-507q^{-51}+28q^{-53}+568q^{-55}+687q^{-57}+264q^{-59}-413q^{-61}-829q^{-63}-605q^{-65}+162q^{-67}+876q^{-69}+912q^{-71}+168q^{-73}-782q^{-75}-1123q^{-77}-518q^{-79}+571q^{-81}+1183q^{-83}+780q^{-85}-275q^{-87}-1073q^{-89}-926q^{-91}-8q^{-93}+845q^{-95}+911q^{-97}+213q^{-99}-565q^{-101}-768q^{-103}-311q^{-105}+314q^{-107}+568q^{-109}+309q^{-111}-146q^{-113}-362q^{-115}-237q^{-117}+40q^{-119}+203q^{-121}+162q^{-123}-2q^{-125}-110q^{-127}-82q^{-129}-9q^{-131}+43q^{-133}+45q^{-135}+8q^{-137}-21q^{-139}-19q^{-141}+8q^{-145}+4q^{-147}+2q^{-149}-q^{-151}-4q^{-153}-q^{-155}+3q^{-157}-2q^{-159}+2q^{-163}-q^{-165}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}+q^{8}+q^{4}-2q^{2}-1-q^{-4}+3q^{-6}+2q^{-10}-q^{-14}+q^{-16}-q^{-18}$
1,1	$q^{28}-2q^{26}+8q^{24}-16q^{22}+31q^{20}-54q^{18}+78q^{16}-106q^{14}+126q^{12}-140q^{10}+138q^{8}-110q^{6}+76q^{4}-16q^{2}-44+112q^{-2}-171q^{-4}+214q^{-6}-248q^{-8}+250q^{-10}-236q^{-12}+200q^{-14}-148q^{-16}+90q^{-18}-24q^{-20}-26q^{-22}+70q^{-24}-96q^{-26}+108q^{-28}-108q^{-30}+98q^{-32}-86q^{-34}+70q^{-36}-54q^{-38}+42q^{-40}-32q^{-42}+20q^{-44}-12q^{-46}+8q^{-48}-4q^{-50}+q^{-52}$
2,0	$q^{28}+q^{26}-2q^{22}+2q^{18}-q^{16}-5q^{14}-q^{12}+4q^{10}+2q^{8}-q^{6}+4q^{4}+7q^{2}-1-3q^{-2}+q^{-4}-2q^{-6}-5q^{-8}+q^{-12}-2q^{-14}+6q^{-18}+q^{-20}-5q^{-22}+3q^{-24}+5q^{-26}-2q^{-28}-3q^{-30}+3q^{-32}+q^{-34}-2q^{-36}-2q^{-38}+q^{-40}-q^{-44}+q^{-46}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-q^{18}+2q^{16}+2q^{14}-3q^{12}+4q^{10}+q^{8}-7q^{6}+5q^{4}-8+6q^{-2}+2q^{-4}-6q^{-6}+3q^{-8}+4q^{-10}-q^{-14}+2q^{-16}+5q^{-18}-5q^{-20}-2q^{-22}+8q^{-24}-7q^{-26}-3q^{-28}+8q^{-30}-4q^{-32}-3q^{-34}+5q^{-36}-q^{-38}-2q^{-40}+q^{-42}$
1,0,0	$q^{13}+q^{11}+2q^{9}+q^{5}-3q^{3}-q-3q^{-1}+2q^{-7}+3q^{-9}+q^{-11}+2q^{-13}-q^{-15}+q^{-17}-2q^{-19}+q^{-21}-q^{-23}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-q^{18}+4q^{16}-4q^{14}+7q^{12}-8q^{10}+9q^{8}-9q^{6}+7q^{4}-6q^{2}+2q^{-2}-8q^{-4}+12q^{-6}-15q^{-8}+18q^{-10}-16q^{-12}+17q^{-14}-12q^{-16}+9q^{-18}-3q^{-20}+4q^{-24}-7q^{-26}+9q^{-28}-10q^{-30}+8q^{-32}-7q^{-34}+5q^{-36}-3q^{-38}+2q^{-40}-q^{-42}$
1,0	$q^{34}-q^{30}-q^{28}+3q^{26}+3q^{24}-2q^{22}-5q^{20}+7q^{16}+4q^{14}-7q^{12}-8q^{10}+3q^{8}+10q^{6}+2q^{4}-10q^{2}-5+6q^{-2}+8q^{-4}-2q^{-6}-7q^{-8}+6q^{-12}+q^{-14}-6q^{-16}-q^{-18}+6q^{-20}+4q^{-22}-5q^{-24}-5q^{-26}+4q^{-28}+8q^{-30}-2q^{-32}-8q^{-34}-q^{-36}+9q^{-38}+4q^{-40}-7q^{-42}-8q^{-44}+3q^{-46}+9q^{-48}+q^{-50}-6q^{-52}-5q^{-54}+2q^{-56}+5q^{-58}+q^{-60}-2q^{-62}-2q^{-64}+q^{-68}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{26}-q^{24}+3q^{22}-2q^{20}+6q^{18}-4q^{16}+7q^{14}-6q^{12}+8q^{10}-8q^{8}+4q^{6}-7q^{4}+2q^{2}-3-4q^{-2}+3q^{-4}-6q^{-6}+9q^{-8}-10q^{-10}+15q^{-12}-10q^{-14}+16q^{-16}-11q^{-18}+13q^{-20}-8q^{-22}+9q^{-24}-6q^{-26}+q^{-28}-q^{-30}-2q^{-32}+4q^{-34}-7q^{-36}+5q^{-38}-7q^{-40}+9q^{-42}-6q^{-44}+4q^{-46}-5q^{-48}+5q^{-50}-2q^{-52}+q^{-54}-2q^{-56}+q^{-58}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-q^{44}+4q^{42}-5q^{40}+5q^{38}-3q^{36}-3q^{34}+14q^{32}-20q^{30}+25q^{28}-17q^{26}+2q^{24}+19q^{22}-35q^{20}+43q^{18}-35q^{16}+14q^{14}+11q^{12}-35q^{10}+41q^{8}-32q^{6}+10q^{4}+12q^{2}-28+24q^{-2}-14q^{-4}-11q^{-6}+29q^{-8}-38q^{-10}+31q^{-12}-9q^{-14}-20q^{-16}+46q^{-18}-56q^{-20}+50q^{-22}-25q^{-24}-5q^{-26}+37q^{-28}-52q^{-30}+54q^{-32}-30q^{-34}+5q^{-36}+23q^{-38}-34q^{-40}+28q^{-42}-9q^{-44}-11q^{-46}+25q^{-48}-25q^{-50}+12q^{-52}+8q^{-54}-28q^{-56}+36q^{-58}-33q^{-60}+17q^{-62}+2q^{-64}-21q^{-66}+28q^{-68}-29q^{-70}+24q^{-72}-11q^{-74}+8q^{-78}-14q^{-80}+14q^{-82}-11q^{-84}+8q^{-86}-2q^{-88}-q^{-90}+2q^{-92}-4q^{-94}+3q^{-96}-2q^{-98}+q^{-100}$

.

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 22"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+10t-11+10t^{-1}-5t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 43, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-2}+z^{8}+2az^{7}+6z^{7}a^{-1}+4z^{7}a^{-3}+a^{2}z^{6}+7z^{6}a^{-2}+6z^{6}a^{-4}+2z^{6}-7az^{5}-16z^{5}a^{-1}-4z^{5}a^{-3}+5z^{5}a^{-5}-4a^{2}z^{4}-23z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}-15z^{4}+7az^{3}+10z^{3}a^{-1}-2z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+5a^{2}z^{2}+17z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+16z^{2}-2az-2za^{-1}+za^{-3}+za^{-5}-2a^{2}-4a^{-2}-a^{-4}-4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n128,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n3,}

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 22"];

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]=

{ $t^{3}-5t^{2}+10t-11+10t^{-1}-5t^{-2}+t^{-3}$, $-q^{6}+3q^{5}-5q^{4}+7q^{3}-7q^{2}+7q-6+4q^{-1}-2q^{-2}+q^{-3}$ }

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]=

{K11n128,}

In[6]:=

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

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

Out[6]=

{K11n3,}

Vassiliev invariants

V2 and V3:

(-1, 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 $-4$ $8$ $8$ ${\frac {34}{3}}$ $-{\frac {10}{3}}$ $-32$ $-{\frac {208}{3}}$ $-{\frac {160}{3}}$ $8$ $-{\frac {32}{3}}$ $32$ $-{\frac {136}{3}}$ ${\frac {40}{3}}$ $-{\frac {751}{30}}$ $-{\frac {766}{5}}$ ${\frac {7018}{45}}$ $-{\frac {305}{18}}$ ${\frac {1169}{30}}$

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 22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   1 -1 11                 2   2 9               3 1   -2 7             4 2     2 5           3 3       0 3         4 4         0 1       3 4           1 -1     1 3             -2 -3   1 3               2 -5   1                 -1 -7 1                   1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{17}-3q^{16}+2q^{15}+5q^{14}-14q^{13}+10q^{12}+14q^{11}-33q^{10}+17q^{9}+26q^{8}-48q^{7}+17q^{6}+36q^{5}-50q^{4}+9q^{3}+39q^{2}-40q-1+33q^{-1}-23q^{-2}-7q^{-3}+20q^{-4}-8q^{-5}-6q^{-6}+7q^{-7}-q^{-8}-2q^{-9}+q^{-10}$
3	$-q^{33}+3q^{32}-2q^{31}-2q^{30}+2q^{29}+5q^{28}-7q^{27}-10q^{26}+19q^{25}+14q^{24}-33q^{23}-25q^{22}+54q^{21}+39q^{20}-73q^{19}-62q^{18}+95q^{17}+81q^{16}-102q^{15}-108q^{14}+108q^{13}+123q^{12}-97q^{11}-141q^{10}+86q^{9}+146q^{8}-64q^{7}-149q^{6}+42q^{5}+145q^{4}-17q^{3}-137q^{2}-5q+122+26q^{-1}-102q^{-2}-42q^{-3}+79q^{-4}+51q^{-5}-55q^{-6}-50q^{-7}+29q^{-8}+47q^{-9}-14q^{-10}-33q^{-11}+23q^{-13}+3q^{-14}-11q^{-15}-5q^{-16}+6q^{-17}+2q^{-18}-q^{-19}-2q^{-20}+q^{-21}$
4	$q^{54}-3q^{53}+2q^{52}+2q^{51}-5q^{50}+7q^{49}-8q^{48}+9q^{47}-25q^{45}+26q^{44}-6q^{43}+31q^{42}-16q^{41}-91q^{40}+54q^{39}+40q^{38}+102q^{37}-57q^{36}-246q^{35}+55q^{34}+138q^{33}+269q^{32}-74q^{31}-483q^{30}-34q^{29}+222q^{28}+511q^{27}-4q^{26}-691q^{25}-190q^{24}+203q^{23}+707q^{22}+142q^{21}-762q^{20}-320q^{19}+85q^{18}+768q^{17}+282q^{16}-689q^{15}-365q^{14}-71q^{13}+706q^{12}+375q^{11}-531q^{10}-346q^{9}-220q^{8}+570q^{7}+425q^{6}-327q^{5}-288q^{4}-351q^{3}+385q^{2}+432q-108-186q^{-1}-424q^{-2}+167q^{-3}+357q^{-4}+73q^{-5}-35q^{-6}-390q^{-7}-25q^{-8}+200q^{-9}+144q^{-10}+101q^{-11}-248q^{-12}-108q^{-13}+37q^{-14}+97q^{-15}+143q^{-16}-91q^{-17}-78q^{-18}-39q^{-19}+19q^{-20}+94q^{-21}-9q^{-22}-21q^{-23}-32q^{-24}-13q^{-25}+34q^{-26}+4q^{-27}+2q^{-28}-9q^{-29}-9q^{-30}+7q^{-31}+q^{-32}+2q^{-33}-q^{-34}-2q^{-35}+q^{-36}$
5	$-q^{80}+3q^{79}-2q^{78}-2q^{77}+5q^{76}-4q^{75}-4q^{74}+6q^{73}+q^{72}+9q^{70}-12q^{69}-23q^{68}+4q^{67}+30q^{66}+37q^{65}+7q^{64}-64q^{63}-96q^{62}-24q^{61}+138q^{60}+201q^{59}+48q^{58}-227q^{57}-373q^{56}-149q^{55}+354q^{54}+656q^{53}+307q^{52}-481q^{51}-998q^{50}-606q^{49}+557q^{48}+1434q^{47}+1005q^{46}-547q^{45}-1851q^{44}-1524q^{43}+410q^{42}+2232q^{41}+2060q^{40}-144q^{39}-2463q^{38}-2613q^{37}-195q^{36}+2573q^{35}+3010q^{34}+600q^{33}-2499q^{32}-3327q^{31}-962q^{30}+2349q^{29}+3426q^{28}+1277q^{27}-2076q^{26}-3446q^{25}-1502q^{24}+1814q^{23}+3309q^{22}+1659q^{21}-1502q^{20}-3140q^{19}-1757q^{18}+1209q^{17}+2902q^{16}+1827q^{15}-888q^{14}-2651q^{13}-1879q^{12}+563q^{11}+2349q^{10}+1919q^{9}-204q^{8}-2015q^{7}-1933q^{6}-153q^{5}+1621q^{4}+1881q^{3}+521q^{2}-1184q-1761-823q^{-1}+711q^{-2}+1533q^{-3}+1042q^{-4}-238q^{-5}-1215q^{-6}-1133q^{-7}-178q^{-8}+828q^{-9}+1080q^{-10}+482q^{-11}-411q^{-12}-898q^{-13}-662q^{-14}+57q^{-15}+637q^{-16}+661q^{-17}+217q^{-18}-331q^{-19}-579q^{-20}-348q^{-21}+94q^{-22}+384q^{-23}+367q^{-24}+94q^{-25}-216q^{-26}-300q^{-27}-156q^{-28}+56q^{-29}+198q^{-30}+167q^{-31}+19q^{-32}-98q^{-33}-120q^{-34}-61q^{-35}+36q^{-36}+78q^{-37}+45q^{-38}+2q^{-39}-31q^{-40}-40q^{-41}-10q^{-42}+18q^{-43}+14q^{-44}+8q^{-45}+2q^{-46}-11q^{-47}-7q^{-48}+3q^{-49}+2q^{-50}+q^{-51}+2q^{-52}-q^{-53}-2q^{-54}+q^{-55}$
6	$q^{111}-3q^{110}+2q^{109}+2q^{108}-5q^{107}+4q^{106}+q^{105}+6q^{104}-16q^{103}-q^{102}+16q^{101}-17q^{100}+17q^{99}+13q^{98}+5q^{97}-59q^{96}-21q^{95}+51q^{94}-5q^{93}+79q^{92}+55q^{91}-43q^{90}-234q^{89}-130q^{88}+138q^{87}+148q^{86}+370q^{85}+216q^{84}-239q^{83}-806q^{82}-626q^{81}+211q^{80}+689q^{79}+1358q^{78}+895q^{77}-572q^{76}-2210q^{75}-2180q^{74}-236q^{73}+1653q^{72}+3649q^{71}+2956q^{70}-391q^{69}-4448q^{68}-5493q^{67}-2268q^{66}+2177q^{65}+7078q^{64}+7070q^{63}+1559q^{62}-6289q^{61}-10064q^{60}-6488q^{59}+787q^{58}+9980q^{57}+12336q^{56}+5755q^{55}-6027q^{54}-13807q^{53}-11655q^{52}-2849q^{51}+10506q^{50}+16389q^{49}+10695q^{48}-3445q^{47}-14891q^{46}-15386q^{45}-7086q^{44}+8624q^{43}+17616q^{42}+14128q^{41}-185q^{40}-13507q^{39}-16448q^{38}-9964q^{37}+5922q^{36}+16471q^{35}+15219q^{34}+2167q^{33}-11149q^{32}-15521q^{31}-11055q^{30}+3634q^{29}+14349q^{28}+14794q^{27}+3509q^{26}-8770q^{25}-13869q^{24}-11249q^{23}+1687q^{22}+11987q^{21}+13921q^{20}+4662q^{19}-6221q^{18}-11990q^{17}-11359q^{16}-578q^{15}+9173q^{14}+12821q^{13}+6111q^{12}-3022q^{11}-9519q^{10}-11259q^{9}-3325q^{8}+5476q^{7}+10899q^{6}+7376q^{5}+755q^{4}-5958q^{3}-10084q^{2}-5760q+1096+7547q^{-1}+7321q^{-2}+4068q^{-3}-1572q^{-4}-7131q^{-5}-6524q^{-6}-2724q^{-7}+3124q^{-8}+5174q^{-9}+5400q^{-10}+2215q^{-11}-2922q^{-12}-4862q^{-13}-4343q^{-14}-724q^{-15}+1641q^{-16}+4096q^{-17}+3710q^{-18}+667q^{-19}-1735q^{-20}-3270q^{-21}-2272q^{-22}-1239q^{-23}+1390q^{-24}+2652q^{-25}+1967q^{-26}+699q^{-27}-980q^{-28}-1493q^{-29}-1986q^{-30}-556q^{-31}+696q^{-32}+1233q^{-33}+1215q^{-34}+485q^{-35}-97q^{-36}-1132q^{-37}-848q^{-38}-376q^{-39}+151q^{-40}+549q^{-41}+589q^{-42}+486q^{-43}-222q^{-44}-331q^{-45}-378q^{-46}-224q^{-47}-10q^{-48}+195q^{-49}+340q^{-50}+65q^{-51}+9q^{-52}-108q^{-53}-126q^{-54}-109q^{-55}-10q^{-56}+110q^{-57}+36q^{-58}+47q^{-59}+2q^{-60}-19q^{-61}-48q^{-62}-25q^{-63}+22q^{-64}+q^{-65}+15q^{-66}+7q^{-67}+5q^{-68}-11q^{-69}-9q^{-70}+5q^{-71}-2q^{-72}+2q^{-73}+q^{-74}+2q^{-75}-q^{-76}-2q^{-77}+q^{-78}$
7	$-q^{147}+3q^{146}-2q^{145}-2q^{144}+5q^{143}-4q^{142}-q^{141}-3q^{140}+4q^{139}+16q^{138}-15q^{137}-8q^{136}+12q^{135}-13q^{134}+2q^{133}-q^{132}+20q^{131}+46q^{130}-49q^{129}-51q^{128}-8q^{127}-28q^{126}+62q^{125}+73q^{124}+91q^{123}+92q^{122}-189q^{121}-268q^{120}-197q^{119}-63q^{118}+376q^{117}+542q^{116}+472q^{115}+158q^{114}-737q^{113}-1208q^{112}-1078q^{111}-286q^{110}+1401q^{109}+2345q^{108}+2165q^{107}+688q^{106}-2240q^{105}-4243q^{104}-4291q^{103}-1697q^{102}+3385q^{101}+7198q^{100}+7661q^{99}+3690q^{98}-4322q^{97}-11066q^{96}-12903q^{95}-7506q^{94}+4628q^{93}+15832q^{92}+20120q^{91}+13474q^{90}-3445q^{89}-20589q^{88}-29035q^{87}-22218q^{86}-134q^{85}+24548q^{84}+39027q^{83}+33387q^{82}+6506q^{81}-26403q^{80}-48697q^{79}-46300q^{78}-15913q^{77}+25355q^{76}+56781q^{75}+59556q^{74}+27610q^{73}-21019q^{72}-61992q^{71}-71665q^{70}-40304q^{69}+13813q^{68}+63582q^{67}+81127q^{66}+52649q^{65}-4618q^{64}-61892q^{63}-87234q^{62}-62947q^{61}-4978q^{60}+57312q^{59}+89586q^{58}+70609q^{57}+13971q^{56}-51320q^{55}-89026q^{54}-75006q^{53}-21070q^{52}+44775q^{51}+86065q^{50}+76696q^{49}+26316q^{48}-38744q^{47}-82094q^{46}-76257q^{45}-29468q^{44}+33541q^{43}+77532q^{42}+74601q^{41}+31433q^{40}-29178q^{39}-73186q^{38}-72451q^{37}-32606q^{36}+25279q^{35}+68923q^{34}+70322q^{33}+33847q^{32}-21376q^{31}-64741q^{30}-68373q^{29}-35479q^{28}+16944q^{27}+60183q^{26}+66589q^{25}+37742q^{24}-11653q^{23}-54911q^{22}-64609q^{21}-40506q^{20}+5296q^{19}+48518q^{18}+62011q^{17}+43492q^{16}+2025q^{15}-40817q^{14}-58308q^{13}-46079q^{12}-9919q^{11}+31642q^{10}+53001q^{9}+47681q^{8}+17915q^{7}-21368q^{6}-45901q^{5}-47466q^{4}-25050q^{3}+10375q^{2}+36815q+45006+30590q^{-1}+395q^{-2}-26306q^{-3}-39906q^{-4}-33501q^{-5}-9964q^{-6}+14992q^{-7}+32305q^{-8}+33373q^{-9}+17267q^{-10}-4072q^{-11}-22958q^{-12}-30013q^{-13}-21349q^{-14}-5224q^{-15}+12850q^{-16}+23874q^{-17}+21982q^{-18}+11861q^{-19}-3461q^{-20}-16144q^{-21}-19327q^{-22}-15020q^{-23}-3962q^{-24}+8009q^{-25}+14268q^{-26}+14978q^{-27}+8606q^{-28}-1119q^{-29}-8295q^{-30}-12166q^{-31}-10059q^{-32}-3738q^{-33}+2545q^{-34}+7993q^{-35}+9113q^{-36}+5977q^{-37}+1579q^{-38}-3635q^{-39}-6417q^{-40}-5987q^{-41}-3895q^{-42}+182q^{-43}+3414q^{-44}+4533q^{-45}+4272q^{-46}+1836q^{-47}-820q^{-48}-2521q^{-49}-3480q^{-50}-2469q^{-51}-758q^{-52}+744q^{-53}+2154q^{-54}+2148q^{-55}+1365q^{-56}+360q^{-57}-939q^{-58}-1363q^{-59}-1246q^{-60}-849q^{-61}+103q^{-62}+659q^{-63}+856q^{-64}+795q^{-65}+232q^{-66}-125q^{-67}-387q^{-68}-584q^{-69}-341q^{-70}-90q^{-71}+141q^{-72}+321q^{-73}+210q^{-74}+143q^{-75}+47q^{-76}-137q^{-77}-145q^{-78}-124q^{-79}-47q^{-80}+59q^{-81}+40q^{-82}+53q^{-83}+62q^{-84}+2q^{-85}-20q^{-86}-40q^{-87}-30q^{-88}+10q^{-89}-2q^{-90}+3q^{-91}+17q^{-92}+7q^{-93}+4q^{-94}-8q^{-95}-9q^{-96}+3q^{-97}-2q^{-99}+2q^{-100}+q^{-101}+2q^{-102}-q^{-103}-2q^{-104}+q^{-105}$

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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