9 29

9_28

9_30

Visit 9 29's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 29's page at Knotilus! Visit 9 29's page at the original Knot Atlas!

9 29 Quick Notes

9 29 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,11,17,12 X10,4,11,3 X2,15,3,16 X14,5,15,6 X18,8,1,7 X4,10,5,9 X12,17,13,18 X8,13,9,14
Gauss code	1, -4, 3, -7, 5, -1, 6, -9, 7, -3, 2, -8, 9, -5, 4, -2, 8, -6
Dowker-Thistlethwaite code	6 10 14 18 4 16 8 2 12
Conway Notation	[.2.20.2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-3] Hyperbolic Volume 12.2059 A-Polynomial See Data:9 29/A-polynomial

[edit Notes for 9 29'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 29's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, -2 }
Jones polynomial	$q^{3}-3q^{2}+5q-7+9q^{-1}-8q^{-2}+8q^{-3}-6q^{-4}+3q^{-5}-q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{2}z^{6}-a^{4}z^{4}+4a^{2}z^{4}-2z^{4}-2a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}-5z^{2}-2a^{4}+5a^{2}+a^{-2}-3$
Kauffman polynomial (db, data sources)	$2a^{2}z^{8}+2z^{8}+6a^{3}z^{7}+9az^{7}+3z^{7}a^{-1}+8a^{4}z^{6}+6a^{2}z^{6}+z^{6}a^{-2}-z^{6}+6a^{5}z^{5}-8a^{3}z^{5}-24az^{5}-10z^{5}a^{-1}+3a^{6}z^{4}-13a^{4}z^{4}-24a^{2}z^{4}-3z^{4}a^{-2}-11z^{4}+a^{7}z^{3}-5a^{5}z^{3}-a^{3}z^{3}+14az^{3}+9z^{3}a^{-1}+8a^{4}z^{2}+17a^{2}z^{2}+3z^{2}a^{-2}+12z^{2}+2a^{5}z+2a^{3}z-az-za^{-1}-2a^{4}-5a^{2}-a^{-2}-3$
The A2 invariant	$-q^{18}+q^{16}-2q^{14}-q^{12}+2q^{10}+4q^{6}+q^{2}-2q^{-2}+q^{-4}-q^{-6}+q^{-10}$
The G2 invariant	$q^{100}-2q^{98}+3q^{96}-4q^{94}+3q^{92}-2q^{90}-q^{88}+8q^{86}-12q^{84}+16q^{82}-19q^{80}+13q^{78}-6q^{76}-9q^{74}+28q^{72}-39q^{70}+44q^{68}-37q^{66}+16q^{64}+13q^{62}-46q^{60}+63q^{58}-61q^{56}+31q^{54}+5q^{52}-41q^{50}+57q^{48}-42q^{46}+9q^{44}+30q^{42}-59q^{40}+56q^{38}-21q^{36}-30q^{34}+78q^{32}-91q^{30}+77q^{28}-24q^{26}-29q^{24}+77q^{22}-96q^{20}+88q^{18}-50q^{16}+47q^{12}-69q^{10}+69q^{8}-36q^{6}-5q^{4}+36q^{2}-55+41q^{-2}-7q^{-4}-39q^{-6}+68q^{-8}-70q^{-10}+39q^{-12}+10q^{-14}-56q^{-16}+77q^{-18}-70q^{-20}+40q^{-22}-3q^{-24}-32q^{-26}+47q^{-28}-40q^{-30}+27q^{-32}-6q^{-34}-7q^{-36}+11q^{-38}-10q^{-40}+6q^{-42}-2q^{-44}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 9_29.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+2q^{11}-3q^{9}+2q^{7}+q^{3}+2q-2q^{-1}+2q^{-3}-2q^{-5}+q^{-7}$
2	$q^{36}-2q^{34}+q^{32}+4q^{30}-9q^{28}+3q^{26}+11q^{24}-15q^{22}-q^{20}+14q^{18}-9q^{16}-6q^{14}+10q^{12}+4q^{10}-8q^{8}+11q^{4}-5q^{2}-10+13q^{-2}+q^{-4}-15q^{-6}+9q^{-8}+8q^{-10}-11q^{-12}+q^{-14}+7q^{-16}-3q^{-18}-2q^{-20}+q^{-22}$
3	$-q^{69}+2q^{67}-q^{65}-2q^{63}+3q^{61}+3q^{59}-6q^{57}-8q^{55}+17q^{53}+16q^{51}-30q^{49}-29q^{47}+36q^{45}+50q^{43}-36q^{41}-71q^{39}+25q^{37}+78q^{35}-76q^{31}-23q^{29}+55q^{27}+45q^{25}-32q^{23}-53q^{21}+6q^{19}+58q^{17}+17q^{15}-56q^{13}-32q^{11}+52q^{9}+46q^{7}-46q^{5}-56q^{3}+32q+70q^{-1}-19q^{-3}-74q^{-5}-4q^{-7}+74q^{-9}+28q^{-11}-61q^{-13}-45q^{-15}+40q^{-17}+54q^{-19}-14q^{-21}-50q^{-23}-7q^{-25}+34q^{-27}+17q^{-29}-16q^{-31}-18q^{-33}+4q^{-35}+10q^{-37}+2q^{-39}-3q^{-41}-2q^{-43}+q^{-45}$
4	$q^{112}-2q^{110}+q^{108}+2q^{106}-5q^{104}+3q^{102}+6q^{98}-3q^{96}-26q^{94}+13q^{92}+27q^{90}+31q^{88}-28q^{86}-108q^{84}+116q^{80}+152q^{78}-38q^{76}-292q^{74}-141q^{72}+179q^{70}+390q^{68}+125q^{66}-400q^{64}-409q^{62}+9q^{60}+492q^{58}+406q^{56}-200q^{54}-488q^{52}-290q^{50}+251q^{48}+476q^{46}+141q^{44}-244q^{42}-395q^{40}-96q^{38}+269q^{36}+317q^{34}+61q^{32}-296q^{30}-286q^{28}+46q^{26}+330q^{24}+214q^{22}-185q^{20}-349q^{18}-75q^{16}+332q^{14}+294q^{12}-117q^{10}-407q^{8}-186q^{6}+320q^{4}+392q^{2}+25-417q^{-2}-365q^{-4}+167q^{-6}+440q^{-8}+273q^{-10}-247q^{-12}-476q^{-14}-129q^{-16}+271q^{-18}+433q^{-20}+77q^{-22}-324q^{-24}-318q^{-26}-57q^{-28}+298q^{-30}+267q^{-32}-10q^{-34}-206q^{-36}-224q^{-38}+24q^{-40}+162q^{-42}+130q^{-44}+8q^{-46}-122q^{-48}-76q^{-50}+57q^{-54}+56q^{-56}-8q^{-58}-24q^{-60}-23q^{-62}-3q^{-64}+13q^{-66}+5q^{-68}+2q^{-70}-3q^{-72}-2q^{-74}+q^{-76}$
5	$-q^{165}+2q^{163}-q^{161}-2q^{159}+5q^{157}-q^{155}-6q^{153}+5q^{149}+9q^{147}+6q^{145}-16q^{143}-38q^{141}-10q^{139}+59q^{137}+89q^{135}+14q^{133}-130q^{131}-200q^{129}-67q^{127}+254q^{125}+441q^{123}+189q^{121}-403q^{119}-816q^{117}-496q^{115}+483q^{113}+1329q^{111}+1073q^{109}-382q^{107}-1849q^{105}-1866q^{103}-78q^{101}+2147q^{99}+2768q^{97}+911q^{95}-2033q^{93}-3470q^{91}-1957q^{89}+1380q^{87}+3686q^{85}+2946q^{83}-303q^{81}-3318q^{79}-3528q^{77}-872q^{75}+2349q^{73}+3541q^{71}+1880q^{69}-1126q^{67}-3004q^{65}-2413q^{63}-87q^{61}+2092q^{59}+2513q^{57}+1006q^{55}-1127q^{53}-2239q^{51}-1560q^{49}+319q^{47}+1843q^{45}+1785q^{43}+217q^{41}-1502q^{39}-1837q^{37}-484q^{35}+1323q^{33}+1861q^{31}+586q^{29}-1311q^{27}-1969q^{25}-675q^{23}+1392q^{21}+2211q^{19}+855q^{17}-1456q^{15}-2516q^{13}-1226q^{11}+1355q^{9}+2841q^{7}+1764q^{5}-1010q^{3}-2984q-2389q^{-1}+331q^{-3}+2866q^{-5}+2967q^{-7}+552q^{-9}-2323q^{-11}-3273q^{-13}-1548q^{-15}+1403q^{-17}+3166q^{-19}+2365q^{-21}-246q^{-23}-2544q^{-25}-2779q^{-27}-907q^{-29}+1522q^{-31}+2650q^{-33}+1737q^{-35}-353q^{-37}-1991q^{-39}-2039q^{-41}-653q^{-43}+1031q^{-45}+1796q^{-47}+1222q^{-49}-105q^{-51}-1161q^{-53}-1270q^{-55}-531q^{-57}+437q^{-59}+947q^{-61}+741q^{-63}+97q^{-65}-465q^{-67}-608q^{-69}-348q^{-71}+74q^{-73}+346q^{-75}+327q^{-77}+114q^{-79}-100q^{-81}-192q^{-83}-147q^{-85}-20q^{-87}+72q^{-89}+85q^{-91}+44q^{-93}-2q^{-95}-30q^{-97}-31q^{-99}-8q^{-101}+6q^{-103}+8q^{-105}+5q^{-107}+2q^{-109}-3q^{-111}-2q^{-113}+q^{-115}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{18}+q^{16}-2q^{14}-q^{12}+2q^{10}+4q^{6}+q^{2}-2q^{-2}+q^{-4}-q^{-6}+q^{-10}$
1,1	$q^{52}-4q^{50}+8q^{48}-12q^{46}+22q^{44}-36q^{42}+48q^{40}-64q^{38}+93q^{36}-118q^{34}+142q^{32}-176q^{30}+194q^{28}-204q^{26}+174q^{24}-128q^{22}+52q^{20}+62q^{18}-170q^{16}+292q^{14}-375q^{12}+438q^{10}-454q^{8}+428q^{6}-372q^{4}+276q^{2}-162+38q^{-2}+70q^{-4}-164q^{-6}+234q^{-8}-258q^{-10}+250q^{-12}-212q^{-14}+168q^{-16}-114q^{-18}+66q^{-20}-36q^{-22}+14q^{-24}-4q^{-26}+q^{-28}$
2,0	$q^{46}-q^{44}+q^{42}+2q^{40}-2q^{38}-2q^{36}+3q^{34}+3q^{32}-8q^{30}-8q^{28}+3q^{26}+3q^{24}-7q^{22}+2q^{20}+11q^{18}+4q^{16}+q^{14}+3q^{12}+2q^{10}-4q^{8}-q^{6}+2q^{4}-6q^{2}-4+5q^{-2}+2q^{-4}-6q^{-6}+q^{-8}+7q^{-10}+2q^{-12}-4q^{-14}-q^{-16}+5q^{-18}+q^{-20}-3q^{-22}-2q^{-24}+q^{-28}$

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{42}+2q^{40}-3q^{38}+6q^{36}-9q^{34}+10q^{32}-14q^{30}+13q^{28}-12q^{26}+8q^{24}-3q^{22}-2q^{20}+10q^{18}-15q^{16}+22q^{14}-23q^{12}+27q^{10}-23q^{8}+20q^{6}-13q^{4}+7q^{2}-1-6q^{-2}+10q^{-4}-13q^{-6}+13q^{-8}-13q^{-10}+10q^{-12}-8q^{-14}+6q^{-16}-2q^{-18}+q^{-20}$
1,0	$q^{68}-2q^{64}-2q^{62}+q^{60}+6q^{58}+3q^{56}-6q^{54}-8q^{52}+12q^{48}+5q^{46}-10q^{44}-11q^{42}+4q^{40}+13q^{38}-13q^{34}-6q^{32}+10q^{30}+7q^{28}-7q^{26}-9q^{24}+5q^{22}+11q^{20}+q^{18}-8q^{16}+2q^{14}+10q^{12}+3q^{10}-9q^{8}-4q^{6}+9q^{4}+9q^{2}-7-15q^{-2}+14q^{-6}+6q^{-8}-11q^{-10}-11q^{-12}+6q^{-14}+12q^{-16}-8q^{-20}-4q^{-22}+5q^{-24}+4q^{-26}-2q^{-28}-2q^{-30}+q^{-34}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{58}-2q^{56}+q^{54}-2q^{52}+6q^{50}-6q^{48}+5q^{46}-8q^{44}+12q^{42}-10q^{40}+8q^{38}-11q^{36}+7q^{34}-6q^{32}-2q^{28}-8q^{26}+9q^{24}-11q^{22}+17q^{20}-15q^{18}+24q^{16}-14q^{14}+24q^{12}-14q^{10}+16q^{8}-10q^{6}+7q^{4}-7q^{2}-2+2q^{-2}-8q^{-4}+6q^{-6}-11q^{-8}+12q^{-10}-10q^{-12}+10q^{-14}-8q^{-16}+8q^{-18}-5q^{-20}+4q^{-22}-2q^{-24}+q^{-26}$

G2 Invariants.

Weight

Invariant

1,0

$q^{100}-2q^{98}+3q^{96}-4q^{94}+3q^{92}-2q^{90}-q^{88}+8q^{86}-12q^{84}+16q^{82}-19q^{80}+13q^{78}-6q^{76}-9q^{74}+28q^{72}-39q^{70}+44q^{68}-37q^{66}+16q^{64}+13q^{62}-46q^{60}+63q^{58}-61q^{56}+31q^{54}+5q^{52}-41q^{50}+57q^{48}-42q^{46}+9q^{44}+30q^{42}-59q^{40}+56q^{38}-21q^{36}-30q^{34}+78q^{32}-91q^{30}+77q^{28}-24q^{26}-29q^{24}+77q^{22}-96q^{20}+88q^{18}-50q^{16}+47q^{12}-69q^{10}+69q^{8}-36q^{6}-5q^{4}+36q^{2}-55+41q^{-2}-7q^{-4}-39q^{-6}+68q^{-8}-70q^{-10}+39q^{-12}+10q^{-14}-56q^{-16}+77q^{-18}-70q^{-20}+40q^{-22}-3q^{-24}-32q^{-26}+47q^{-28}-40q^{-30}+27q^{-32}-6q^{-34}-7q^{-36}+11q^{-38}-10q^{-40}+6q^{-42}-2q^{-44}+q^{-46}$

.

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 29"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$

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]=

{ 51, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-3q^{2}+5q-7+9q^{-1}-8q^{-2}+8q^{-3}-6q^{-4}+3q^{-5}-q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{2}z^{6}-a^{4}z^{4}+4a^{2}z^{4}-2z^{4}-2a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}-5z^{2}-2a^{4}+5a^{2}+a^{-2}-3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{8}+2z^{8}+6a^{3}z^{7}+9az^{7}+3z^{7}a^{-1}+8a^{4}z^{6}+6a^{2}z^{6}+z^{6}a^{-2}-z^{6}+6a^{5}z^{5}-8a^{3}z^{5}-24az^{5}-10z^{5}a^{-1}+3a^{6}z^{4}-13a^{4}z^{4}-24a^{2}z^{4}-3z^{4}a^{-2}-11z^{4}+a^{7}z^{3}-5a^{5}z^{3}-a^{3}z^{3}+14az^{3}+9z^{3}a^{-1}+8a^{4}z^{2}+17a^{2}z^{2}+3z^{2}a^{-2}+12z^{2}+2a^{5}z+2a^{3}z-az-za^{-1}-2a^{4}-5a^{2}-a^{-2}-3$

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 $4$ $-16$ $8$ ${\frac {62}{3}}$ $-{\frac {14}{3}}$ $-64$ $-{\frac {352}{3}}$ ${\frac {128}{3}}$ $-80$ ${\frac {32}{3}}$ $128$ ${\frac {248}{3}}$ $-{\frac {56}{3}}$ ${\frac {11311}{30}}$ $-{\frac {634}{5}}$ ${\frac {11822}{45}}$ ${\frac {209}{18}}$ ${\frac {751}{30}}$

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 29. 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									2		-2
3								3	1		2
1							4	2			-2
-1						5	3				2
-3					4	5					1
-5				4	4						0
-7			2	4							2
-9		1	4								-3
-11		2									2
-13	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, 29]]
Out[2]=	9
In[3]:=	PD[Knot[9, 29]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 11, 17, 12], X[10, 4, 11, 3], X[2, 15, 3, 16], X[14, 5, 15, 6], X[18, 8, 1, 7], X[4, 10, 5, 9], X[12, 17, 13, 18], X[8, 13, 9, 14]]
In[4]:=	GaussCode[Knot[9, 29]]
Out[4]=	GaussCode[1, -4, 3, -7, 5, -1, 6, -9, 7, -3, 2, -8, 9, -5, 4, -2, 8, -6]
In[5]:=	BR[Knot[9, 29]]
Out[5]=	BR[4, {1, -2, -2, 3, -2, 1, -2, 3, -2}]
In[6]:=	alex = Alexander[Knot[9, 29]][t]
Out[6]=	-3 5 12 2 3 -15 + t - -- + -- + 12 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[9, 29]][z]
Out[7]=	2 4 6 1 + z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 28], Knot[9, 29], Knot[10, 163], Knot[11, NonAlternating, 87]}
In[9]:=	{KnotDet[Knot[9, 29]], KnotSignature[Knot[9, 29]]}
Out[9]=	{51, -2}
In[10]:=	J=Jones[Knot[9, 29]][q]
Out[10]=	-6 3 6 8 8 9 2 3 -7 - q + -- - -- + -- - -- + - + 5 q - 3 q + q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 29]}
In[12]:=	A2Invariant[Knot[9, 29]][q]
Out[12]=	-18 -16 2 -12 2 4 -2 2 4 6 10 -q + q - --- - q + --- + -- + q - 2 q + q - q + q 14 10 6 q q q
In[13]:=	Kauffman[Knot[9, 29]][a, z]
Out[13]=	2 -2 2 4 z 3 5 2 3 z -3 - a - 5 a - 2 a - - - a z + 2 a z + 2 a z + 12 z + ---- + a 2 a 3 2 2 4 2 9 z 3 3 3 5 3 7 3 17 a z + 8 a z + ---- + 14 a z - a z - 5 a z + a z - a 4 5 4 3 z 2 4 4 4 6 4 10 z 5 11 z - ---- - 24 a z - 13 a z + 3 a z - ----- - 24 a z - 2 a a 6 7 3 5 5 5 6 z 2 6 4 6 3 z 7 8 a z + 6 a z - z + -- + 6 a z + 8 a z + ---- + 9 a z + 2 a a 3 7 8 2 8 6 a z + 2 z + 2 a z
In[14]:=	{Vassiliev[2][Knot[9, 29]], Vassiliev[3][Knot[9, 29]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[9, 29]][q, t]
Out[15]=	5 5 1 2 1 4 2 4 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t 4 4 3 t 2 3 2 3 3 5 3 7 4 ---- + ---- + --- + 4 q t + 2 q t + 3 q t + q t + 2 q t + q t 5 3 q q t q t