10 98

10_97

10_99

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10 98 Quick Notes

10 98 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X3,10,4,11 X7,18,8,19 X17,8,18,9 X9,2,10,3 X11,16,12,17 X5,15,6,14 X15,5,16,4 X13,20,14,1 X19,12,20,13
Gauss code	-1, 5, -2, 8, -7, 1, -3, 4, -5, 2, -6, 10, -9, 7, -8, 6, -4, 3, -10, 9
Dowker-Thistlethwaite code	6 10 14 18 2 16 20 4 8 12
Conway Notation	[.2.2.2.20]

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-13][1] Hyperbolic Volume 14.4129 A-Polynomial See Data:10 98/A-polynomial

[edit Notes for 10 98's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 98's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+9t^{2}-18t+23-18t^{-1}+9t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 81, -4 }
Jones polynomial	$1-3q^{-1}+7q^{-2}-9q^{-3}+13q^{-4}-14q^{-5}+12q^{-6}-11q^{-7}+7q^{-8}-3q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+2z^{2}a^{8}+2a^{8}-z^{6}a^{6}-3z^{4}a^{6}-5z^{2}a^{6}-5a^{6}-z^{6}a^{4}-2z^{4}a^{4}+z^{2}a^{4}+3a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-2z^{3}a^{11}+6z^{6}a^{10}-7z^{4}a^{10}+4z^{2}a^{10}+8z^{7}a^{9}-14z^{5}a^{9}+14z^{3}a^{9}-6za^{9}+6z^{8}a^{8}-7z^{6}a^{8}+2z^{4}a^{8}+2a^{8}+2z^{9}a^{7}+8z^{7}a^{7}-26z^{5}a^{7}+25z^{3}a^{7}-12za^{7}+10z^{8}a^{6}-23z^{6}a^{6}+17z^{4}a^{6}-10z^{2}a^{6}+5a^{6}+2z^{9}a^{5}+3z^{7}a^{5}-17z^{5}a^{5}+14z^{3}a^{5}-6za^{5}+4z^{8}a^{4}-9z^{6}a^{4}+4z^{4}a^{4}-2z^{2}a^{4}+3a^{4}+3z^{7}a^{3}-8z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$
The A2 invariant	$q^{30}-q^{28}+2q^{26}+2q^{24}-3q^{22}-5q^{18}-q^{16}+q^{14}+5q^{10}-q^{8}+2q^{6}+q^{4}-q^{2}+1$
The G2 invariant	$q^{162}-2q^{160}+4q^{158}-6q^{156}+6q^{154}-5q^{152}+10q^{148}-21q^{146}+31q^{144}-39q^{142}+33q^{140}-18q^{138}-12q^{136}+58q^{134}-94q^{132}+117q^{130}-107q^{128}+56q^{126}+17q^{124}-112q^{122}+185q^{120}-203q^{118}+154q^{116}-41q^{114}-84q^{112}+183q^{110}-192q^{108}+134q^{106}-19q^{104}-110q^{102}+175q^{100}-143q^{98}+34q^{96}+116q^{94}-230q^{92}+256q^{90}-161q^{88}-7q^{86}+163q^{84}-297q^{82}+322q^{80}-242q^{78}+78q^{76}+92q^{74}-232q^{72}+288q^{70}-230q^{68}+90q^{66}+43q^{64}-158q^{62}+191q^{60}-129q^{58}+8q^{56}+128q^{54}-194q^{52}+181q^{50}-68q^{48}-83q^{46}+204q^{44}-244q^{42}+197q^{40}-81q^{38}-51q^{36}+157q^{34}-188q^{32}+162q^{30}-84q^{28}+3q^{26}+51q^{24}-78q^{22}+69q^{20}-44q^{18}+20q^{16}+2q^{14}-11q^{12}+13q^{10}-10q^{8}+6q^{6}-2q^{4}+q^{2}$

Further Quantum Invariants

Further quantum knot invariants for 10_98.

A1 Invariants.

Weight	Invariant
1	$q^{21}-2q^{19}+4q^{17}-4q^{15}+q^{13}-2q^{11}-q^{9}+4q^{7}-2q^{5}+4q^{3}-2q+q^{-1}$
2	$q^{58}-2q^{56}+q^{54}+5q^{52}-11q^{50}+2q^{48}+19q^{46}-24q^{44}-7q^{42}+34q^{40}-20q^{38}-16q^{36}+31q^{34}-20q^{30}+6q^{28}+16q^{26}-14q^{24}-20q^{22}+24q^{20}+4q^{18}-32q^{16}+21q^{14}+20q^{12}-29q^{10}+4q^{8}+21q^{6}-13q^{4}-5q^{2}+9-q^{-2}-2q^{-4}+q^{-6}$
3	$q^{111}-2q^{109}+q^{107}+2q^{105}-2q^{103}-5q^{101}+5q^{99}+13q^{97}-15q^{95}-30q^{93}+27q^{91}+60q^{89}-27q^{87}-110q^{85}+13q^{83}+166q^{81}+23q^{79}-198q^{77}-87q^{75}+213q^{73}+145q^{71}-179q^{69}-197q^{67}+115q^{65}+206q^{63}-39q^{61}-194q^{59}-40q^{57}+164q^{55}+103q^{53}-111q^{51}-152q^{49}+75q^{47}+186q^{45}-16q^{43}-218q^{41}-28q^{39}+217q^{37}+84q^{35}-213q^{33}-146q^{31}+174q^{29}+189q^{27}-111q^{25}-212q^{23}+41q^{21}+199q^{19}+31q^{17}-155q^{15}-71q^{13}+94q^{11}+86q^{9}-42q^{7}-66q^{5}+4q^{3}+42q+10q^{-1}-19q^{-3}-8q^{-5}+6q^{-7}+4q^{-9}-q^{-11}-2q^{-13}+q^{-15}$
4	$q^{180}-2q^{178}+q^{176}+2q^{174}-5q^{172}+4q^{170}-2q^{168}+5q^{166}+2q^{164}-27q^{162}+10q^{160}+16q^{158}+42q^{156}+6q^{154}-128q^{152}-54q^{150}+70q^{148}+250q^{146}+141q^{144}-353q^{142}-426q^{140}-73q^{138}+675q^{136}+776q^{134}-314q^{132}-1120q^{130}-884q^{128}+743q^{126}+1767q^{124}+546q^{122}-1303q^{120}-2018q^{118}-161q^{116}+2024q^{114}+1719q^{112}-374q^{110}-2249q^{108}-1343q^{106}+1040q^{104}+1980q^{102}+854q^{100}-1289q^{98}-1704q^{96}-275q^{94}+1237q^{92}+1411q^{90}-77q^{88}-1307q^{86}-1085q^{84}+381q^{82}+1433q^{80}+736q^{78}-864q^{76}-1504q^{74}-192q^{72}+1394q^{70}+1357q^{68}-486q^{66}-1871q^{64}-819q^{62}+1198q^{60}+1976q^{58}+222q^{56}-1893q^{54}-1621q^{52}+376q^{50}+2145q^{48}+1267q^{46}-1044q^{44}-1950q^{42}-877q^{40}+1316q^{38}+1772q^{36}+314q^{34}-1199q^{32}-1498q^{30}-13q^{28}+1127q^{26}+986q^{24}-8q^{22}-967q^{20}-634q^{18}+119q^{16}+606q^{14}+465q^{12}-173q^{10}-361q^{8}-235q^{6}+74q^{4}+245q^{2}+83-35q^{-2}-108q^{-4}-49q^{-6}+40q^{-8}+28q^{-10}+18q^{-12}-13q^{-14}-14q^{-16}+3q^{-18}+q^{-20}+4q^{-22}-q^{-24}-2q^{-26}+q^{-28}$

A2 Invariants.

Weight	Invariant
1,0	$q^{30}-q^{28}+2q^{26}+2q^{24}-3q^{22}-5q^{18}-q^{16}+q^{14}+5q^{10}-q^{8}+2q^{6}+q^{4}-q^{2}+1$
1,1	$q^{84}-4q^{82}+10q^{80}-20q^{78}+40q^{76}-72q^{74}+114q^{72}-174q^{70}+266q^{68}-376q^{66}+494q^{64}-632q^{62}+766q^{60}-860q^{58}+866q^{56}-786q^{54}+593q^{52}-272q^{50}-140q^{48}+616q^{46}-1070q^{44}+1472q^{42}-1744q^{40}+1876q^{38}-1845q^{36}+1644q^{34}-1308q^{32}+856q^{30}-396q^{28}-82q^{26}+494q^{24}-800q^{22}+968q^{20}-996q^{18}+934q^{16}-780q^{14}+598q^{12}-416q^{10}+268q^{8}-150q^{6}+80q^{4}-36q^{2}+14-4q^{-2}+q^{-4}$
2,0	$q^{76}-q^{74}+q^{72}+2q^{70}-q^{68}-4q^{66}+2q^{64}+7q^{62}-6q^{60}-13q^{58}+q^{56}+8q^{54}-11q^{52}-5q^{50}+19q^{48}+18q^{46}-2q^{44}+10q^{40}-8q^{38}-15q^{36}-q^{34}-6q^{32}-14q^{30}+3q^{28}+10q^{26}-6q^{24}-6q^{22}+14q^{20}+9q^{18}-10q^{16}-4q^{14}+13q^{12}+3q^{10}-7q^{8}-q^{6}+5q^{4}+3q^{2}-2-q^{-2}+q^{-4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{68}-2q^{66}+6q^{62}-7q^{60}-4q^{58}+17q^{56}-13q^{54}-13q^{52}+24q^{50}-14q^{48}-14q^{46}+27q^{44}-5q^{40}+16q^{38}+4q^{36}-8q^{34}-17q^{32}-q^{28}-26q^{26}+10q^{24}+19q^{22}-19q^{20}+11q^{18}+18q^{16}-17q^{14}+8q^{12}+8q^{10}-8q^{8}+4q^{6}+2q^{4}-2q^{2}+1$
1,0,0	$q^{39}-q^{37}+3q^{35}+3q^{31}-3q^{29}-6q^{25}-3q^{23}-2q^{21}+3q^{17}+q^{15}+5q^{13}-q^{11}+3q^{9}-q^{7}+2q^{5}-q^{3}+q$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{86}-q^{84}-2q^{82}+5q^{80}+4q^{78}-8q^{76}-2q^{74}+12q^{72}-q^{70}-20q^{68}-3q^{66}+10q^{64}-15q^{62}-17q^{60}+22q^{58}+22q^{56}-3q^{54}+21q^{52}+33q^{50}-3q^{48}-18q^{46}+4q^{44}-15q^{42}-43q^{40}-15q^{38}+10q^{36}-16q^{34}-15q^{32}+25q^{30}+16q^{28}-8q^{26}+4q^{24}+18q^{22}+q^{20}-6q^{18}+6q^{16}+7q^{14}-3q^{12}-q^{10}+4q^{8}-q^{4}+q^{2}$
1,0,0,0	$q^{48}-q^{46}+3q^{44}+q^{42}+q^{40}+3q^{38}-3q^{36}-6q^{32}-4q^{30}-4q^{28}-2q^{26}+2q^{22}+4q^{20}+q^{18}+5q^{16}-q^{14}+3q^{12}+2q^{6}-q^{4}+q^{2}$

B2 Invariants.

Weight	Invariant
0,1	$q^{68}-2q^{66}+4q^{64}-8q^{62}+13q^{60}-18q^{58}+25q^{56}-29q^{54}+31q^{52}-28q^{50}+22q^{48}-12q^{46}-q^{44}+16q^{42}-33q^{40}+44q^{38}-56q^{36}+58q^{34}-59q^{32}+50q^{30}-39q^{28}+24q^{26}-8q^{24}-5q^{22}+19q^{20}-25q^{18}+32q^{16}-29q^{14}+28q^{12}-22q^{10}+16q^{8}-10q^{6}+6q^{4}-2q^{2}+1$
1,0	$q^{110}-2q^{106}-2q^{104}+2q^{102}+7q^{100}+2q^{98}-10q^{96}-11q^{94}+5q^{92}+21q^{90}+6q^{88}-23q^{86}-22q^{84}+12q^{82}+30q^{80}+q^{78}-31q^{76}-15q^{74}+24q^{72}+25q^{70}-10q^{68}-22q^{66}+8q^{64}+27q^{62}+6q^{60}-22q^{58}-8q^{56}+16q^{54}+8q^{52}-21q^{50}-19q^{48}+10q^{46}+16q^{44}-13q^{42}-29q^{40}+2q^{38}+32q^{36}+15q^{34}-25q^{32}-24q^{30}+16q^{28}+34q^{26}+2q^{24}-25q^{22}-13q^{20}+17q^{18}+17q^{16}-4q^{14}-12q^{12}-2q^{10}+7q^{8}+4q^{6}-2q^{4}-2q^{2}+q^{-2}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{94}-2q^{92}+2q^{90}-3q^{88}+7q^{86}-10q^{84}+10q^{82}-13q^{80}+21q^{78}-23q^{76}+19q^{74}-25q^{72}+24q^{70}-21q^{68}+12q^{66}-11q^{64}+6q^{62}+14q^{60}-9q^{58}+28q^{56}-26q^{54}+44q^{52}-41q^{50}+41q^{48}-55q^{46}+34q^{44}-47q^{42}+24q^{40}-34q^{38}+13q^{36}-6q^{34}+3q^{32}+11q^{30}-10q^{28}+26q^{26}-19q^{24}+26q^{22}-23q^{20}+25q^{18}-18q^{16}+17q^{14}-12q^{12}+10q^{10}-5q^{8}+4q^{6}-2q^{4}+q^{2}$

G2 Invariants.

Weight

Invariant

1,0

$q^{162}-2q^{160}+4q^{158}-6q^{156}+6q^{154}-5q^{152}+10q^{148}-21q^{146}+31q^{144}-39q^{142}+33q^{140}-18q^{138}-12q^{136}+58q^{134}-94q^{132}+117q^{130}-107q^{128}+56q^{126}+17q^{124}-112q^{122}+185q^{120}-203q^{118}+154q^{116}-41q^{114}-84q^{112}+183q^{110}-192q^{108}+134q^{106}-19q^{104}-110q^{102}+175q^{100}-143q^{98}+34q^{96}+116q^{94}-230q^{92}+256q^{90}-161q^{88}-7q^{86}+163q^{84}-297q^{82}+322q^{80}-242q^{78}+78q^{76}+92q^{74}-232q^{72}+288q^{70}-230q^{68}+90q^{66}+43q^{64}-158q^{62}+191q^{60}-129q^{58}+8q^{56}+128q^{54}-194q^{52}+181q^{50}-68q^{48}-83q^{46}+204q^{44}-244q^{42}+197q^{40}-81q^{38}-51q^{36}+157q^{34}-188q^{32}+162q^{30}-84q^{28}+3q^{26}+51q^{24}-78q^{22}+69q^{20}-44q^{18}+20q^{16}+2q^{14}-11q^{12}+13q^{10}-10q^{8}+6q^{6}-2q^{4}+q^{2}$

.

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["10 98"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+9t^{2}-18t+23-18t^{-1}+9t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-3z^{4}+1$

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]=

$\left\{t^{2}-t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$1-3q^{-1}+7q^{-2}-9q^{-3}+13q^{-4}-14q^{-5}+12q^{-6}-11q^{-7}+7q^{-8}-3q^{-9}+q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-2z^{3}a^{11}+6z^{6}a^{10}-7z^{4}a^{10}+4z^{2}a^{10}+8z^{7}a^{9}-14z^{5}a^{9}+14z^{3}a^{9}-6za^{9}+6z^{8}a^{8}-7z^{6}a^{8}+2z^{4}a^{8}+2a^{8}+2z^{9}a^{7}+8z^{7}a^{7}-26z^{5}a^{7}+25z^{3}a^{7}-12za^{7}+10z^{8}a^{6}-23z^{6}a^{6}+17z^{4}a^{6}-10z^{2}a^{6}+5a^{6}+2z^{9}a^{5}+3z^{7}a^{5}-17z^{5}a^{5}+14z^{3}a^{5}-6za^{5}+4z^{8}a^{4}-9z^{6}a^{4}+4z^{4}a^{4}-2z^{2}a^{4}+3a^{4}+3z^{7}a^{3}-8z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$

Vassiliev invariants

V2 and V3:

(0, 3)

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 $0$ $24$ $0$ $-80$ $24$ $0$ $48$ $-64$ $-136$ $0$ $288$ $0$ $0$ $904$ $-{\frac {920}{3}}$ $1160$ $184$ $104$

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

\   r    \    j   \	-8	-7	-6	-5	-4	-3	-2	-1	0	1	2	χ
1											1	1
-1										2		-2
-3									5	1		4
-5								5	3			-2
-7							8	4				4
-9						6	5					-1
-11					6	8						-2
-13				5	6							1
-15			2	6								-4
-17		1	5									4
-19		2										-2
-21	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[10, 98]]
Out[2]=	10
In[3]:=	PD[Knot[10, 98]]
Out[3]=	PD[X[1, 6, 2, 7], X[3, 10, 4, 11], X[7, 18, 8, 19], X[17, 8, 18, 9], X[9, 2, 10, 3], X[11, 16, 12, 17], X[5, 15, 6, 14], X[15, 5, 16, 4], X[13, 20, 14, 1], X[19, 12, 20, 13]]
In[4]:=	GaussCode[Knot[10, 98]]
Out[4]=	GaussCode[-1, 5, -2, 8, -7, 1, -3, 4, -5, 2, -6, 10, -9, 7, -8, 6, -4, 3, -10, 9]
In[5]:=	BR[Knot[10, 98]]
Out[5]=	BR[4, {-1, -1, -2, -2, 3, -2, 1, -2, -2, 3, -2}]
In[6]:=	alex = Alexander[Knot[10, 98]][t]
Out[6]=	2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 98]][z]
Out[7]=	4 6 1 - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}
In[9]:=	{KnotDet[Knot[10, 98]], KnotSignature[Knot[10, 98]]}
Out[9]=	{81, -4}
In[10]:=	J=Jones[Knot[10, 98]][q]
Out[10]=	-10 3 7 11 12 14 13 9 7 3 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 98]}
In[12]:=	A2Invariant[Knot[10, 98]][q]
Out[12]=	-30 -28 2 2 3 5 -16 -14 5 -8 1 + q - q + --- + --- - --- - --- - q + q + --- - q + 26 24 22 18 10 q q q q q 2 -4 -2 -- + q - q 6 q
In[13]:=	Kauffman[Knot[10, 98]][a, z]
Out[13]=	2 4 6 8 5 7 9 2 2 -a + 3 a + 5 a + 2 a - 6 a z - 12 a z - 6 a z + 3 a z - 4 2 6 2 10 2 12 2 3 3 5 3 2 a z - 10 a z + 4 a z - a z + 5 a z + 14 a z + 7 3 9 3 11 3 2 4 4 4 6 4 25 a z + 14 a z - 2 a z - 3 a z + 4 a z + 17 a z + 8 4 10 4 12 4 3 5 5 5 7 5 2 a z - 7 a z + a z - 8 a z - 17 a z - 26 a z - 9 5 11 5 2 6 4 6 6 6 8 6 14 a z + 3 a z + a z - 9 a z - 23 a z - 7 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 a z + 3 a z + 3 a z + 8 a z + 8 a z + 4 a z + 6 8 8 8 5 9 7 9 10 a z + 6 a z + 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[10, 98]], Vassiliev[3][Knot[10, 98]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[10, 98]][q, t]
Out[15]=	3 5 1 2 1 5 2 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 q q q t q t q t q t q t q t 5 6 6 8 6 5 8 4 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t 5 t 2 t 2 ---- + -- + --- + q t 5 3 q q t q