9 20

9_19

9_21

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9 20 Quick Notes

From knotilus

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,14,6,15 X7,16,8,17 X11,1,12,18 X15,6,16,7 X17,13,18,12 X13,8,14,9 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 6, -4, 8, -9, 2, -5, 7, -8, 3, -6, 4, -7, 5
Dowker-Thistlethwaite code	4 10 14 16 2 18 8 6 12
Conway Notation	[31212]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 9.6443 A-Polynomial See Data:9 20/A-polynomial

[edit Notes for 9 20's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 20's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-9t+11-9t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, -4 }
Jones polynomial	$1-2q^{-1}+4q^{-2}-5q^{-3}+7q^{-4}-7q^{-5}+6q^{-6}-5q^{-7}+3q^{-8}-q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{8}-a^{8}+2z^{4}a^{6}+5z^{2}a^{6}+2a^{6}-z^{6}a^{4}-4z^{4}a^{4}-5z^{2}a^{4}-2a^{4}+z^{4}a^{2}+3z^{2}a^{2}+2a^{2}$
Kauffman polynomial (db, data sources)	$z^{3}a^{11}+3z^{4}a^{10}-z^{2}a^{10}+5z^{5}a^{9}-5z^{3}a^{9}+2za^{9}+5z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-a^{8}+3z^{7}a^{7}-7z^{3}a^{7}+2za^{7}+z^{8}a^{6}+5z^{6}a^{6}-16z^{4}a^{6}+10z^{2}a^{6}-2a^{6}+5z^{7}a^{5}-12z^{5}a^{5}+5z^{3}a^{5}+z^{8}a^{4}+z^{6}a^{4}-11z^{4}a^{4}+11z^{2}a^{4}-2a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+6z^{3}a^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$
The A2 invariant	$-q^{28}+q^{24}-q^{22}+q^{20}-q^{18}+q^{14}-q^{12}+2q^{10}-q^{8}+q^{6}+q^{4}+1$
The G2 invariant	$q^{148}-2q^{146}+3q^{144}-4q^{142}+2q^{140}-q^{138}-2q^{136}+9q^{134}-12q^{132}+15q^{130}-13q^{128}+5q^{126}+2q^{124}-13q^{122}+23q^{120}-27q^{118}+23q^{116}-13q^{114}-q^{112}+15q^{110}-23q^{108}+25q^{106}-21q^{104}+5q^{102}+7q^{100}-17q^{98}+16q^{96}-5q^{94}-9q^{92}+23q^{90}-24q^{88}+14q^{86}+3q^{84}-26q^{82}+44q^{80}-44q^{78}+30q^{76}-4q^{74}-21q^{72}+43q^{70}-45q^{68}+33q^{66}-17q^{64}-6q^{62}+22q^{60}-28q^{58}+22q^{56}-6q^{54}-9q^{52}+19q^{50}-19q^{48}+7q^{46}+8q^{44}-23q^{42}+31q^{40}-28q^{38}+12q^{36}+10q^{34}-26q^{32}+36q^{30}-30q^{28}+17q^{26}-q^{24}-13q^{22}+21q^{20}-19q^{18}+14q^{16}-3q^{14}-2q^{12}+5q^{10}-5q^{8}+4q^{6}-q^{4}+q^{2}$

Further Quantum Invariants

Further quantum knot invariants for 9_20.

A1 Invariants.

Weight	Invariant
1	$-q^{19}+2q^{17}-2q^{15}+q^{13}-q^{11}+2q^{7}-q^{5}+2q^{3}-q+q^{-1}$
2	$q^{52}-2q^{50}+5q^{46}-6q^{44}-q^{42}+8q^{40}-8q^{38}-q^{36}+9q^{34}-4q^{32}-4q^{30}+4q^{28}+2q^{26}-5q^{24}-3q^{22}+6q^{20}-q^{18}-7q^{16}+8q^{14}+3q^{12}-8q^{10}+5q^{8}+5q^{6}-6q^{4}+q^{2}+4-2q^{-2}-q^{-4}+q^{-6}$
3	$-q^{99}+2q^{97}-3q^{93}+5q^{89}+2q^{87}-10q^{85}+13q^{81}-2q^{79}-18q^{77}+2q^{75}+25q^{73}-2q^{71}-28q^{69}-2q^{67}+30q^{65}+7q^{63}-26q^{61}-13q^{59}+16q^{57}+19q^{55}-6q^{53}-20q^{51}-5q^{49}+21q^{47}+13q^{45}-16q^{43}-22q^{41}+13q^{39}+23q^{37}-9q^{35}-27q^{33}+2q^{31}+28q^{29}+4q^{27}-27q^{25}-11q^{23}+26q^{21}+17q^{19}-19q^{17}-20q^{15}+13q^{13}+23q^{11}-3q^{9}-20q^{7}-2q^{5}+14q^{3}+7q-8q^{-1}-7q^{-3}+3q^{-5}+5q^{-7}-2q^{-11}-q^{-13}+q^{-15}$
4	$q^{160}-2q^{158}+3q^{154}-2q^{152}+q^{150}-6q^{148}+3q^{146}+9q^{144}-8q^{142}+2q^{140}-10q^{138}+11q^{136}+18q^{134}-25q^{132}-10q^{130}-9q^{128}+40q^{126}+37q^{124}-53q^{122}-46q^{120}-15q^{118}+79q^{116}+79q^{114}-60q^{112}-93q^{110}-51q^{108}+86q^{106}+122q^{104}-15q^{102}-92q^{100}-97q^{98}+30q^{96}+111q^{94}+48q^{92}-28q^{90}-96q^{88}-45q^{86}+39q^{84}+78q^{82}+50q^{80}-51q^{78}-87q^{76}-28q^{74}+73q^{72}+85q^{70}-5q^{68}-93q^{66}-68q^{64}+60q^{62}+92q^{60}+24q^{58}-90q^{56}-92q^{54}+44q^{52}+92q^{50}+58q^{48}-70q^{46}-114q^{44}+3q^{42}+76q^{40}+98q^{38}-20q^{36}-112q^{34}-52q^{32}+26q^{30}+114q^{28}+46q^{26}-63q^{24}-76q^{22}-40q^{20}+70q^{18}+73q^{16}+6q^{14}-44q^{12}-68q^{10}+7q^{8}+44q^{6}+35q^{4}+4q^{2}-40-19q^{-2}+3q^{-4}+19q^{-6}+18q^{-8}-8q^{-10}-9q^{-12}-7q^{-14}+q^{-16}+7q^{-18}+q^{-20}-2q^{-24}-q^{-26}+q^{-28}$
5	$-q^{235}+2q^{233}-3q^{229}+2q^{227}+q^{225}+q^{221}-2q^{219}-4q^{217}+2q^{215}+6q^{213}-q^{211}-8q^{209}-5q^{207}+10q^{205}+15q^{203}+8q^{201}-18q^{199}-46q^{197}-15q^{195}+50q^{193}+81q^{191}+32q^{189}-77q^{187}-147q^{185}-73q^{183}+114q^{181}+236q^{179}+133q^{177}-137q^{175}-326q^{173}-232q^{171}+119q^{169}+420q^{167}+350q^{165}-68q^{163}-460q^{161}-464q^{159}-46q^{157}+443q^{155}+556q^{153}+176q^{151}-353q^{149}-573q^{147}-306q^{145}+198q^{143}+514q^{141}+400q^{139}-23q^{137}-388q^{135}-422q^{133}-140q^{131}+213q^{129}+390q^{127}+262q^{125}-44q^{123}-304q^{121}-338q^{119}-100q^{117}+215q^{115}+354q^{113}+205q^{111}-132q^{109}-361q^{107}-261q^{105}+87q^{103}+349q^{101}+288q^{99}-49q^{97}-356q^{95}-311q^{93}+44q^{91}+362q^{89}+335q^{87}-17q^{85}-376q^{83}-381q^{81}-16q^{79}+378q^{77}+426q^{75}+83q^{73}-346q^{71}-472q^{69}-180q^{67}+280q^{65}+491q^{63}+277q^{61}-165q^{59}-467q^{57}-379q^{55}+21q^{53}+390q^{51}+433q^{49}+129q^{47}-259q^{45}-429q^{43}-260q^{41}+102q^{39}+361q^{37}+336q^{35}+55q^{33}-241q^{31}-332q^{29}-176q^{27}+97q^{25}+275q^{23}+234q^{21}+28q^{19}-167q^{17}-220q^{15}-117q^{13}+61q^{11}+167q^{9}+140q^{7}+22q^{5}-87q^{3}-119q-65q^{-1}+23q^{-3}+75q^{-5}+67q^{-7}+15q^{-9}-32q^{-11}-45q^{-13}-29q^{-15}+4q^{-17}+25q^{-19}+22q^{-21}+5q^{-23}-6q^{-25}-11q^{-27}-9q^{-29}+q^{-31}+5q^{-33}+3q^{-35}+q^{-37}-2q^{-41}-q^{-43}+q^{-45}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{28}+q^{24}-q^{22}+q^{20}-q^{18}+q^{14}-q^{12}+2q^{10}-q^{8}+q^{6}+q^{4}+1$
1,1	$q^{76}-4q^{74}+8q^{72}-12q^{70}+22q^{68}-36q^{66}+46q^{64}-56q^{62}+72q^{60}-84q^{58}+84q^{56}-82q^{54}+77q^{52}-62q^{50}+32q^{48}+4q^{46}-43q^{44}+90q^{42}-132q^{40}+168q^{38}-192q^{36}+198q^{34}-190q^{32}+158q^{30}-126q^{28}+82q^{26}-30q^{24}-18q^{22}+57q^{20}-82q^{18}+106q^{16}-110q^{14}+104q^{12}-84q^{10}+70q^{8}-48q^{6}+29q^{4}-16q^{2}+8-2q^{-2}+q^{-4}$
2,0	$q^{70}-q^{66}+q^{62}-3q^{58}+q^{56}+2q^{54}-3q^{52}-q^{50}+4q^{48}+3q^{46}-4q^{44}-q^{42}+3q^{40}-2q^{38}-4q^{36}+2q^{34}+q^{32}-4q^{30}+q^{28}+2q^{26}-2q^{24}-2q^{22}+4q^{20}+3q^{18}-2q^{16}+q^{14}+5q^{12}-2q^{8}+q^{6}+2q^{4}-1+q^{-4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{62}-2q^{60}-q^{58}+5q^{56}-3q^{54}-3q^{52}+8q^{50}-3q^{48}-6q^{46}+6q^{44}-2q^{42}-6q^{40}+4q^{38}+q^{36}-2q^{34}-q^{32}+3q^{30}+3q^{28}-6q^{26}+3q^{24}+5q^{22}-7q^{20}+5q^{16}-5q^{14}+2q^{12}+5q^{10}-2q^{8}+2q^{6}+2q^{4}-q^{2}+1$
1,0,0	$-q^{37}-q^{33}+q^{31}-q^{29}+2q^{27}-q^{25}+q^{23}-q^{15}+q^{13}-q^{11}+2q^{9}+2q^{5}+q$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{80}-2q^{76}-q^{74}+2q^{72}+2q^{70}-2q^{68}+5q^{64}-6q^{60}+2q^{56}-6q^{54}-3q^{52}+3q^{50}+q^{48}-4q^{46}+3q^{44}+5q^{42}-4q^{40}-q^{38}+7q^{36}-7q^{32}+4q^{30}+3q^{28}-4q^{26}-3q^{24}+3q^{22}+q^{20}-2q^{18}+2q^{16}+4q^{14}+q^{12}+q^{10}+3q^{8}+q^{6}+q^{2}$
1,0,0,0	$-q^{46}-q^{42}-q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}+q^{28}-q^{22}-2q^{18}+q^{16}-q^{14}+2q^{12}+q^{10}+q^{8}+2q^{6}+q^{2}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{62}+2q^{60}-3q^{58}+5q^{56}-7q^{54}+7q^{52}-8q^{50}+7q^{48}-6q^{46}+4q^{44}-4q^{40}+8q^{38}-11q^{36}+14q^{34}-15q^{32}+15q^{30}-13q^{28}+10q^{26}-7q^{24}+3q^{22}+q^{20}-4q^{18}+7q^{16}-7q^{14}+8q^{12}-7q^{10}+6q^{8}-4q^{6}+4q^{4}-q^{2}+1$
1,0	$q^{100}-2q^{96}-2q^{94}+q^{92}+5q^{90}+2q^{88}-5q^{86}-5q^{84}+2q^{82}+8q^{80}+2q^{78}-7q^{76}-6q^{74}+3q^{72}+7q^{70}-q^{68}-7q^{66}-2q^{64}+6q^{62}+4q^{60}-5q^{58}-5q^{56}+3q^{54}+5q^{52}-2q^{50}-5q^{48}+q^{46}+5q^{44}-6q^{40}-q^{38}+7q^{36}+5q^{34}-5q^{32}-8q^{30}+q^{28}+9q^{26}+3q^{24}-6q^{22}-6q^{20}+4q^{18}+7q^{16}+q^{14}-4q^{12}-2q^{10}+3q^{8}+3q^{6}-q^{4}-q^{2}+q^{-2}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{86}-2q^{84}+q^{82}-2q^{80}+5q^{78}-5q^{76}+4q^{74}-5q^{72}+8q^{70}-5q^{68}+4q^{66}-6q^{64}+3q^{62}-2q^{60}-2q^{58}-6q^{54}+7q^{52}-8q^{50}+11q^{48}-11q^{46}+13q^{44}-9q^{42}+12q^{40}-9q^{38}+8q^{36}-5q^{34}+3q^{32}-3q^{30}-2q^{28}+2q^{26}-5q^{24}+4q^{22}-6q^{20}+8q^{18}-4q^{16}+7q^{14}-3q^{12}+6q^{10}-2q^{8}+3q^{6}-q^{4}+q^{2}$

G2 Invariants.

Weight

Invariant

1,0

$q^{148}-2q^{146}+3q^{144}-4q^{142}+2q^{140}-q^{138}-2q^{136}+9q^{134}-12q^{132}+15q^{130}-13q^{128}+5q^{126}+2q^{124}-13q^{122}+23q^{120}-27q^{118}+23q^{116}-13q^{114}-q^{112}+15q^{110}-23q^{108}+25q^{106}-21q^{104}+5q^{102}+7q^{100}-17q^{98}+16q^{96}-5q^{94}-9q^{92}+23q^{90}-24q^{88}+14q^{86}+3q^{84}-26q^{82}+44q^{80}-44q^{78}+30q^{76}-4q^{74}-21q^{72}+43q^{70}-45q^{68}+33q^{66}-17q^{64}-6q^{62}+22q^{60}-28q^{58}+22q^{56}-6q^{54}-9q^{52}+19q^{50}-19q^{48}+7q^{46}+8q^{44}-23q^{42}+31q^{40}-28q^{38}+12q^{36}+10q^{34}-26q^{32}+36q^{30}-30q^{28}+17q^{26}-q^{24}-13q^{22}+21q^{20}-19q^{18}+14q^{16}-3q^{14}-2q^{12}+5q^{10}-5q^{8}+4q^{6}-q^{4}+q^{2}$

.

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 20"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-9t+11-9t^{-1}+5t^{-2}-t^{-3}$

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]=

{ 41, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{8}-a^{8}+2z^{4}a^{6}+5z^{2}a^{6}+2a^{6}-z^{6}a^{4}-4z^{4}a^{4}-5z^{2}a^{4}-2a^{4}+z^{4}a^{2}+3z^{2}a^{2}+2a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{11}+3z^{4}a^{10}-z^{2}a^{10}+5z^{5}a^{9}-5z^{3}a^{9}+2za^{9}+5z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-a^{8}+3z^{7}a^{7}-7z^{3}a^{7}+2za^{7}+z^{8}a^{6}+5z^{6}a^{6}-16z^{4}a^{6}+10z^{2}a^{6}-2a^{6}+5z^{7}a^{5}-12z^{5}a^{5}+5z^{3}a^{5}+z^{8}a^{4}+z^{6}a^{4}-11z^{4}a^{4}+11z^{2}a^{4}-2a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+6z^{3}a^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$

Vassiliev invariants

V2 and V3:

(2, -4)

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 $8$ $-32$ $32$ ${\frac {412}{3}}$ ${\frac {68}{3}}$ $-256$ $-{\frac {1856}{3}}$ $-{\frac {224}{3}}$ $-128$ ${\frac {256}{3}}$ $512$ ${\frac {3296}{3}}$ ${\frac {544}{3}}$ ${\frac {42751}{15}}$ $-{\frac {4124}{15}}$ ${\frac {64924}{45}}$ ${\frac {833}{9}}$ ${\frac {2911}{15}}$

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 $s=$-4 is the signature of 9 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-7	-6	-5	-4	-3	-2	-1	0	1	2	χ
1										1	1
-1									1		-1
-3								3	1		2
-5							3	2			-1
-7						4	2				2
-9					3	3					0
-11				3	4						-1
-13			2	3							1
-15		1	3								-2
-17		2									2
-19	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, 20]]
Out[2]=	9
In[3]:=	PD[Knot[9, 20]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 14, 6, 15], X[7, 16, 8, 17], X[11, 1, 12, 18], X[15, 6, 16, 7], X[17, 13, 18, 12], X[13, 8, 14, 9], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[9, 20]]
Out[4]=	GaussCode[-1, 9, -2, 1, -3, 6, -4, 8, -9, 2, -5, 7, -8, 3, -6, 4, -7, 5]
In[5]:=	BR[Knot[9, 20]]
Out[5]=	BR[4, {-1, -1, -1, 2, -1, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[9, 20]][t]
Out[6]=	-3 5 9 2 3 11 - t + -- - - - 9 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[9, 20]][z]
Out[7]=	2 4 6 1 + 2 z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 20], Knot[10, 149], Knot[11, NonAlternating, 26]}
In[9]:=	{KnotDet[Knot[9, 20]], KnotSignature[Knot[9, 20]]}
Out[9]=	{41, -4}
In[10]:=	J=Jones[Knot[9, 20]][q]
Out[10]=	-9 3 5 6 7 7 5 4 2 1 - q + -- - -- + -- - -- + -- - -- + -- - - 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 20], Knot[11, NonAlternating, 90]}
In[12]:=	A2Invariant[Knot[9, 20]][q]
Out[12]=	-28 -24 -22 -20 -18 -14 -12 2 -8 -6 1 - q + q - q + q - q + q - q + --- - q + q + 10 q -4 q
In[13]:=	Kauffman[Knot[9, 20]][a, z]
Out[13]=	2 4 6 8 7 9 2 2 4 2 -2 a - 2 a - 2 a - a + 2 a z + 2 a z + 5 a z + 11 a z + 6 2 8 2 10 2 3 3 5 3 7 3 9 3 10 a z + 3 a z - a z + 6 a z + 5 a z - 7 a z - 5 a z + 11 3 2 4 4 4 6 4 8 4 10 4 a z - 4 a z - 11 a z - 16 a z - 6 a z + 3 a z - 3 5 5 5 9 5 2 6 4 6 6 6 8 6 7 a z - 12 a z + 5 a z + a z + a z + 5 a z + 5 a z + 3 7 5 7 7 7 4 8 6 8 2 a z + 5 a z + 3 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 20]], Vassiliev[3][Knot[9, 20]]}
Out[14]=	{0, -4}
In[15]:=	Kh[Knot[9, 20]][q, t]
Out[15]=	2 3 1 2 1 3 2 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 q q q t q t q t q t q t q t 3 4 3 3 4 2 3 t t 2 ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + q t 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q