9 24

9_23

9_25

Visit 9 24's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 24's page at Knotilus! Visit 9 24's page at the original Knot Atlas!

9 24 Quick Notes

9 24 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X5,14,6,15 X9,17,10,16 X11,1,12,18 X17,11,18,10 X15,13,16,12 X13,6,14,7 X7283
Gauss code	-1, 9, -2, 1, -3, 8, -9, 2, -4, 6, -5, 7, -8, 3, -7, 4, -6, 5
Dowker-Thistlethwaite code	4 8 14 2 16 18 6 12 10
Conway Notation	[3,21,2+]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.8337 A-Polynomial See Data:9 24/A-polynomial

[edit Notes for 9 24's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 24's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-10t+13-10t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 45, 0 }
Jones polynomial	$q^{4}-3q^{3}+5q^{2}-7q+8-7q^{-1}+7q^{-2}-4q^{-3}+2q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-4z^{4}-a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}-6z^{2}-2a^{4}+5a^{2}+a^{-2}-3$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+2a^{3}z^{7}+5az^{7}+3z^{7}a^{-1}+2a^{4}z^{6}+3a^{2}z^{6}+4z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-z^{5}a^{-1}+3z^{5}a^{-3}-5a^{4}z^{4}-10a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-11z^{4}-3a^{5}z^{3}-3a^{3}z^{3}+az^{3}-3z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+10a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+9z^{2}+2a^{5}z+3a^{3}z+2az+2za^{-1}+za^{-3}-2a^{4}-5a^{2}-a^{-2}-3$
The A2 invariant	$-q^{16}-q^{14}-q^{10}+3q^{8}+2q^{6}+q^{4}+2q^{2}-2+q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}$
The G2 invariant	$q^{80}-q^{78}+3q^{76}-4q^{74}+3q^{72}-3q^{70}-3q^{68}+9q^{66}-16q^{64}+19q^{62}-19q^{60}+8q^{58}+7q^{56}-26q^{54}+41q^{52}-47q^{50}+34q^{48}-11q^{46}-23q^{44}+45q^{42}-53q^{40}+46q^{38}-16q^{36}-11q^{34}+34q^{32}-36q^{30}+22q^{28}+10q^{26}-32q^{24}+44q^{22}-27q^{20}+2q^{18}+37q^{16}-60q^{14}+71q^{12}-55q^{10}+21q^{8}+20q^{6}-60q^{4}+77q^{2}-69+40q^{-2}-4q^{-4}-32q^{-6}+46q^{-8}-43q^{-10}+18q^{-12}+8q^{-14}-31q^{-16}+33q^{-18}-15q^{-20}-14q^{-22}+41q^{-24}-50q^{-26}+44q^{-28}-20q^{-30}-11q^{-32}+35q^{-34}-48q^{-36}+47q^{-38}-29q^{-40}+9q^{-42}+9q^{-44}-21q^{-46}+24q^{-48}-19q^{-50}+13q^{-52}-4q^{-54}-2q^{-56}+4q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_24.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{9}-2q^{7}+3q^{5}+q+q^{-1}-2q^{-3}+2q^{-5}-2q^{-7}+q^{-9}$
2	$q^{32}-q^{30}-q^{28}+4q^{26}-3q^{24}-6q^{22}+9q^{20}-q^{18}-12q^{16}+11q^{14}+5q^{12}-12q^{10}+5q^{8}+7q^{6}-5q^{4}-2q^{2}+5+5q^{-2}-10q^{-4}+12q^{-8}-11q^{-10}-4q^{-12}+13q^{-14}-5q^{-16}-5q^{-18}+6q^{-20}-q^{-22}-2q^{-24}+q^{-26}$
3	$-q^{63}+q^{61}+q^{59}-q^{57}-3q^{55}+3q^{53}+7q^{51}-3q^{49}-13q^{47}+q^{45}+21q^{43}+5q^{41}-30q^{39}-17q^{37}+34q^{35}+30q^{33}-30q^{31}-48q^{29}+26q^{27}+54q^{25}-11q^{23}-59q^{21}-q^{19}+55q^{17}+12q^{15}-46q^{13}-18q^{11}+34q^{9}+25q^{7}-16q^{5}-30q^{3}+3q+32q^{-1}+17q^{-3}-36q^{-5}-32q^{-7}+32q^{-9}+49q^{-11}-27q^{-13}-56q^{-15}+16q^{-17}+57q^{-19}-3q^{-21}-53q^{-23}-7q^{-25}+41q^{-27}+13q^{-29}-26q^{-31}-14q^{-33}+14q^{-35}+11q^{-37}-8q^{-39}-5q^{-41}+3q^{-43}+3q^{-45}-q^{-47}-2q^{-49}+q^{-51}$
4	$q^{104}-q^{102}-q^{100}+q^{98}+3q^{94}-5q^{92}-5q^{90}+5q^{88}+5q^{86}+13q^{84}-13q^{82}-24q^{80}+18q^{76}+51q^{74}-7q^{72}-62q^{70}-48q^{68}+6q^{66}+121q^{64}+61q^{62}-67q^{60}-138q^{58}-92q^{56}+148q^{54}+182q^{52}+34q^{50}-177q^{48}-244q^{46}+62q^{44}+244q^{42}+183q^{40}-104q^{38}-320q^{36}-78q^{34}+190q^{32}+264q^{30}+10q^{28}-275q^{26}-155q^{24}+89q^{22}+236q^{20}+81q^{18}-164q^{16}-163q^{14}-5q^{12}+160q^{10}+121q^{8}-41q^{6}-149q^{4}-95q^{2}+70+162q^{-2}+100q^{-4}-130q^{-6}-197q^{-8}-46q^{-10}+180q^{-12}+245q^{-14}-61q^{-16}-257q^{-18}-180q^{-20}+124q^{-22}+327q^{-24}+56q^{-26}-203q^{-28}-251q^{-30}-3q^{-32}+269q^{-34}+135q^{-36}-65q^{-38}-202q^{-40}-92q^{-42}+129q^{-44}+108q^{-46}+32q^{-48}-86q^{-50}-80q^{-52}+28q^{-54}+36q^{-56}+36q^{-58}-18q^{-60}-31q^{-62}+6q^{-64}+3q^{-66}+12q^{-68}-3q^{-70}-8q^{-72}+3q^{-74}+3q^{-78}-q^{-80}-2q^{-82}+q^{-84}$
5	$-q^{155}+q^{153}+q^{151}-q^{149}-q^{143}+3q^{141}+4q^{139}-5q^{137}-8q^{135}-3q^{133}+3q^{131}+15q^{129}+18q^{127}-4q^{125}-33q^{123}-37q^{121}-7q^{119}+45q^{117}+80q^{115}+44q^{113}-51q^{111}-137q^{109}-117q^{107}+22q^{105}+191q^{103}+236q^{101}+75q^{99}-208q^{97}-385q^{95}-252q^{93}+141q^{91}+507q^{89}+504q^{87}+49q^{85}-542q^{83}-782q^{81}-365q^{79}+452q^{77}+991q^{75}+743q^{73}-182q^{71}-1079q^{69}-1137q^{67}-171q^{65}+1004q^{63}+1392q^{61}+604q^{59}-770q^{57}-1536q^{55}-963q^{53}+466q^{51}+1481q^{49}+1212q^{47}-122q^{45}-1326q^{43}-1310q^{41}-155q^{39}+1078q^{37}+1280q^{35}+345q^{33}-813q^{31}-1161q^{29}-460q^{27}+570q^{25}+999q^{23}+517q^{21}-356q^{19}-836q^{17}-557q^{15}+171q^{13}+691q^{11}+615q^{9}+34q^{7}-577q^{5}-700q^{3}-242q+443q^{-1}+820q^{-3}+521q^{-5}-300q^{-7}-942q^{-9}-808q^{-11}+70q^{-13}+1011q^{-15}+1140q^{-17}+203q^{-19}-1002q^{-21}-1395q^{-23}-550q^{-25}+857q^{-27}+1563q^{-29}+895q^{-31}-596q^{-33}-1567q^{-35}-1165q^{-37}+236q^{-39}+1391q^{-41}+1310q^{-43}+137q^{-45}-1074q^{-47}-1286q^{-49}-423q^{-51}+677q^{-53}+1100q^{-55}+588q^{-57}-303q^{-59}-818q^{-61}-603q^{-63}+32q^{-65}+508q^{-67}+498q^{-69}+118q^{-71}-248q^{-73}-349q^{-75}-160q^{-77}+94q^{-79}+196q^{-81}+126q^{-83}-7q^{-85}-90q^{-87}-85q^{-89}-12q^{-91}+40q^{-93}+36q^{-95}+11q^{-97}-11q^{-99}-15q^{-101}-8q^{-103}+6q^{-105}+8q^{-107}-2q^{-109}-3q^{-111}+3q^{-119}-q^{-121}-2q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}-q^{14}-q^{10}+3q^{8}+2q^{6}+q^{4}+2q^{2}-2+q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}$
1,1	$q^{44}-2q^{42}+6q^{40}-12q^{38}+25q^{36}-42q^{34}+66q^{32}-100q^{30}+130q^{28}-168q^{26}+186q^{24}-196q^{22}+177q^{20}-132q^{18}+72q^{16}+26q^{14}-114q^{12}+216q^{10}-294q^{8}+352q^{6}-385q^{4}+376q^{2}-338+268q^{-2}-177q^{-4}+80q^{-6}+16q^{-8}-96q^{-10}+156q^{-12}-186q^{-14}+192q^{-16}-178q^{-18}+152q^{-20}-120q^{-22}+86q^{-24}-58q^{-26}+36q^{-28}-20q^{-30}+10q^{-32}-4q^{-34}+q^{-36}$
2,0	$q^{42}+q^{40}-q^{36}+q^{34}+q^{32}-4q^{30}-6q^{28}+q^{24}-4q^{22}+8q^{18}+7q^{16}-3q^{14}+2q^{12}+5q^{10}-3q^{8}-3q^{6}+3q^{4}-4+2q^{-2}+4q^{-4}-4q^{-6}-3q^{-8}+6q^{-10}+2q^{-12}-6q^{-14}+q^{-16}+5q^{-18}-q^{-20}-3q^{-22}+2q^{-26}-q^{-28}-q^{-30}+q^{-32}$

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+q^{32}-3q^{30}+4q^{28}-7q^{26}+8q^{24}-10q^{22}+10q^{20}-9q^{18}+8q^{16}-2q^{14}+9q^{10}-11q^{8}+17q^{6}-18q^{4}+20q^{2}-20+15q^{-2}-11q^{-4}+5q^{-6}-q^{-8}-4q^{-10}+8q^{-12}-10q^{-14}+11q^{-16}-10q^{-18}+8q^{-20}-6q^{-22}+4q^{-24}-2q^{-26}+q^{-28}$
1,0	$q^{56}-q^{52}-q^{50}+2q^{48}+3q^{46}-q^{44}-6q^{42}-3q^{40}+5q^{38}+6q^{36}-5q^{34}-12q^{32}-3q^{30}+10q^{28}+8q^{26}-7q^{24}-8q^{22}+4q^{20}+13q^{18}+3q^{16}-5q^{14}-q^{12}+8q^{10}+4q^{8}-5q^{6}-6q^{4}+3q^{2}+6-4q^{-2}-9q^{-4}+8q^{-8}+q^{-10}-9q^{-12}-5q^{-14}+9q^{-16}+10q^{-18}-4q^{-20}-11q^{-22}-q^{-24}+10q^{-26}+6q^{-28}-5q^{-30}-7q^{-32}+5q^{-36}+2q^{-38}-2q^{-40}-2q^{-42}+q^{-46}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-q^{44}+2q^{42}-2q^{40}+4q^{38}-6q^{36}+4q^{34}-9q^{32}+5q^{30}-12q^{28}+4q^{26}-8q^{24}+7q^{22}-q^{20}+4q^{18}+9q^{16}+3q^{14}+15q^{12}-8q^{10}+14q^{8}-13q^{6}+14q^{4}-19q^{2}+9-17q^{-2}+9q^{-4}-8q^{-6}+3q^{-8}-3q^{-10}+q^{-12}+7q^{-14}-4q^{-16}+7q^{-18}-8q^{-20}+10q^{-22}-7q^{-24}+6q^{-26}-7q^{-28}+5q^{-30}-3q^{-32}+2q^{-34}-2q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-q^{78}+3q^{76}-4q^{74}+3q^{72}-3q^{70}-3q^{68}+9q^{66}-16q^{64}+19q^{62}-19q^{60}+8q^{58}+7q^{56}-26q^{54}+41q^{52}-47q^{50}+34q^{48}-11q^{46}-23q^{44}+45q^{42}-53q^{40}+46q^{38}-16q^{36}-11q^{34}+34q^{32}-36q^{30}+22q^{28}+10q^{26}-32q^{24}+44q^{22}-27q^{20}+2q^{18}+37q^{16}-60q^{14}+71q^{12}-55q^{10}+21q^{8}+20q^{6}-60q^{4}+77q^{2}-69+40q^{-2}-4q^{-4}-32q^{-6}+46q^{-8}-43q^{-10}+18q^{-12}+8q^{-14}-31q^{-16}+33q^{-18}-15q^{-20}-14q^{-22}+41q^{-24}-50q^{-26}+44q^{-28}-20q^{-30}-11q^{-32}+35q^{-34}-48q^{-36}+47q^{-38}-29q^{-40}+9q^{-42}+9q^{-44}-21q^{-46}+24q^{-48}-19q^{-50}+13q^{-52}-4q^{-54}-2q^{-56}+4q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

.

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 24"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-10t+13-10t^{-1}+5t^{-2}-t^{-3}$

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]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{4}-3q^{3}+5q^{2}-7q+8-7q^{-1}+7q^{-2}-4q^{-3}+2q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+2a^{3}z^{7}+5az^{7}+3z^{7}a^{-1}+2a^{4}z^{6}+3a^{2}z^{6}+4z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-z^{5}a^{-1}+3z^{5}a^{-3}-5a^{4}z^{4}-10a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-11z^{4}-3a^{5}z^{3}-3a^{3}z^{3}+az^{3}-3z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+10a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+9z^{2}+2a^{5}z+3a^{3}z+2az+2za^{-1}+za^{-3}-2a^{4}-5a^{2}-a^{-2}-3$

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 $4$ $-16$ $8$ ${\frac {110}{3}}$ ${\frac {34}{3}}$ $-64$ $-{\frac {352}{3}}$ ${\frac {32}{3}}$ $-48$ ${\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ ${\frac {136}{3}}$ ${\frac {13471}{30}}$ $-{\frac {782}{15}}$ ${\frac {10862}{45}}$ ${\frac {545}{18}}$ ${\frac {991}{30}}$

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=$0 is the signature of 9 24. 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									2		-2
5								3	1		2
3							4	2			-2
1						4	3				1
-1					4	5					1
-3				3	3						0
-5			1	4							3
-7		1	3								-2
-9		1									1
-11	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, 24]]
Out[2]=	9
In[3]:=	PD[Knot[9, 24]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[9, 17, 10, 16], X[11, 1, 12, 18], X[17, 11, 18, 10], X[15, 13, 16, 12], X[13, 6, 14, 7], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[9, 24]]
Out[4]=	GaussCode[-1, 9, -2, 1, -3, 8, -9, 2, -4, 6, -5, 7, -8, 3, -7, 4, -6, 5]
In[5]:=	BR[Knot[9, 24]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, 2, 2, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 24]][t]
Out[6]=	-3 5 10 2 3 13 - t + -- - -- - 10 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[9, 24]][z]
Out[7]=	2 4 6 1 + z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 18], Knot[9, 24], Knot[11, NonAlternating, 85], Knot[11, NonAlternating, 164]}
In[9]:=	{KnotDet[Knot[9, 24]], KnotSignature[Knot[9, 24]]}
Out[9]=	{45, 0}
In[10]:=	J=Jones[Knot[9, 24]][q]
Out[10]=	-5 2 4 7 7 2 3 4 8 - q + -- - -- + -- - - - 7 q + 5 q - 3 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 24]}
In[12]:=	A2Invariant[Knot[9, 24]][q]
Out[12]=	-16 -14 -10 3 2 -4 2 2 4 8 10 -2 - q - q - q + -- + -- + q + -- + q - 2 q + q - q + 8 6 2 q q q 12 q
In[13]:=	Kauffman[Knot[9, 24]][a, z]
Out[13]=	-2 2 4 z 2 z 3 5 2 -3 - a - 5 a - 2 a + -- + --- + 2 a z + 3 a z + 2 a z + 9 z - 3 a a 2 2 3 3 z 2 z 2 2 4 2 4 z 3 z 3 3 3 -- + ---- + 10 a z + 4 a z - ---- - ---- + a z - 3 a z - 4 2 3 a a a a 4 4 5 5 5 3 4 z 5 z 2 4 4 4 3 z z 3 a z - 11 z + -- - ---- - 10 a z - 5 a z + ---- - -- - 4 2 3 a a a a 6 7 5 3 5 5 5 6 4 z 2 6 4 6 3 z 7 a z - 2 a z + a z + 5 z + ---- + 3 a z + 2 a z + ---- + 2 a a 7 3 7 8 2 8 5 a z + 2 a z + z + a z
In[14]:=	{Vassiliev[2][Knot[9, 24]], Vassiliev[3][Knot[9, 24]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[9, 24]][q, t]
Out[15]=	5 1 1 1 3 1 4 3 - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 3 4 3 3 2 5 2 5 3 7 3 ---- + --- + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 3 q t q t 9 4 q t