9 22

9_21

9_23

Visit 9 22's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 22's page at Knotilus! Visit 9 22's page at the original Knot Atlas!

9 22 Quick Notes

9 22 Further Notes and Views

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]

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]=

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

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

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 22]]
Out[2]=	9
In[3]:=	PD[Knot[9, 22]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[16, 12, 17, 11], X[14, 7, 15, 8], X[6, 15, 7, 16], X[18, 14, 1, 13], X[12, 18, 13, 17]]
In[4]:=	GaussCode[Knot[9, 22]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -9, 8, -6, 7, -5, 9, -8]
In[5]:=	BR[Knot[9, 22]]
Out[5]=	BR[4, {-1, 2, -1, 2, -3, 2, 2, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 22]][t]
Out[6]=	-3 5 10 2 3 -11 + t - -- + -- + 10 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[9, 22]][z]
Out[7]=	2 4 6 1 - z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 22], Knot[11, NonAlternating, 128]}
In[9]:=	{KnotDet[Knot[9, 22]], KnotSignature[Knot[9, 22]]}
Out[9]=	{43, 2}
In[10]:=	J=Jones[Knot[9, 22]][q]
Out[10]=	-3 2 4 2 3 4 5 6 -6 + q - -- + - + 7 q - 7 q + 7 q - 5 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 22], Knot[11, NonAlternating, 3]}
In[12]:=	A2Invariant[Knot[9, 22]][q]
Out[12]=	-10 -8 -4 2 4 6 10 14 16 18 -1 + q + q + q - -- - q + 3 q + 2 q - q + q - q 2 q
In[13]:=	Kauffman[Knot[9, 22]][a, z]
Out[13]=	2 2 -4 4 2 z z 2 z 2 z 5 z -4 - a - -- - 2 a + -- + -- - --- - 2 a z + 16 z - -- + ---- + 2 5 3 a 6 4 a a a a a 2 3 3 3 3 4 17 z 2 2 z 4 z 2 z 10 z 3 4 3 z ----- + 5 a z + -- - ---- - ---- + ----- + 7 a z - 15 z + ---- - 2 7 5 3 a 6 a a a a a 4 4 5 5 5 6 9 z 23 z 2 4 5 z 4 z 16 z 5 6 6 z ---- - ----- - 4 a z + ---- - ---- - ----- - 7 a z + 2 z + ---- + 4 2 5 3 a 4 a a a a a 6 7 7 8 7 z 2 6 4 z 6 z 7 8 z ---- + a z + ---- + ---- + 2 a z + z + -- 2 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[9, 22]], Vassiliev[3][Knot[9, 22]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[9, 22]][q, t]
Out[15]=	3 1 1 1 3 1 3 3 q 4 q + 4 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 4 q t + 3 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 11 4 13 5 2 q t + q t