10 103

10_102

10_104

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10 103 Quick Notes

10 103 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X18,6,19,5 X20,13,1,14 X16,7,17,8 X10,3,11,4 X4,11,5,12 X14,9,15,10 X8,15,9,16 X12,19,13,20 X2,18,3,17
Gauss code	1, -10, 5, -6, 2, -1, 4, -8, 7, -5, 6, -9, 3, -7, 8, -4, 10, -2, 9, -3
Dowker-Thistlethwaite code	6 10 18 16 14 4 20 8 2 12
Conway Notation	[30:2:2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-10][-2] Hyperbolic Volume 13.8748 A-Polynomial See Data:10 103/A-polynomial

[edit Notes for 10 103's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant 2

[edit Notes for 10 103's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-8t^{2}+17t-21+17t^{-1}-8t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+4z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{5,t+1\}$
Determinant and Signature	{ 75, -2 }
Jones polynomial	$-q^{2}+3q-6+10q^{-1}-11q^{-2}+13q^{-3}-12q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-2z^{2}a^{6}-a^{6}+z^{6}a^{4}+3z^{4}a^{4}+3z^{2}a^{4}+z^{6}a^{2}+3z^{4}a^{2}+4z^{2}a^{2}+3a^{2}-z^{4}-2z^{2}-1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+5z^{7}a^{7}-12z^{5}a^{7}+10z^{3}a^{7}-4za^{7}+5z^{8}a^{6}-12z^{6}a^{6}+13z^{4}a^{6}-6z^{2}a^{6}+a^{6}+2z^{9}a^{5}+3z^{7}a^{5}-16z^{5}a^{5}+21z^{3}a^{5}-6za^{5}+9z^{8}a^{4}-23z^{6}a^{4}+25z^{4}a^{4}-8z^{2}a^{4}+2z^{9}a^{3}+2z^{7}a^{3}-9z^{5}a^{3}+9z^{3}a^{3}-2za^{3}+4z^{8}a^{2}-5z^{6}a^{2}+2z^{2}a^{2}-3a^{2}+4z^{7}a-5z^{5}a-2z^{3}a+za+3z^{6}-6z^{4}+3z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$
The A2 invariant	$-q^{24}+q^{22}-q^{20}-q^{18}+2q^{16}-3q^{14}+q^{12}+q^{8}+4q^{6}-q^{4}+3q^{2}-1-q^{-2}+q^{-4}-q^{-6}$
The G2 invariant	$q^{128}-2q^{126}+4q^{124}-7q^{122}+7q^{120}-6q^{118}+12q^{114}-22q^{112}+35q^{110}-42q^{108}+35q^{106}-15q^{104}-25q^{102}+72q^{100}-108q^{98}+118q^{96}-88q^{94}+18q^{92}+74q^{90}-153q^{88}+184q^{86}-149q^{84}+53q^{82}+55q^{80}-142q^{78}+160q^{76}-106q^{74}+11q^{72}+92q^{70}-143q^{68}+115q^{66}-28q^{64}-92q^{62}+180q^{60}-204q^{58}+150q^{56}-38q^{54}-94q^{52}+208q^{50}-252q^{48}+218q^{46}-117q^{44}-25q^{42}+148q^{40}-206q^{38}+189q^{36}-98q^{34}-12q^{32}+112q^{30}-142q^{28}+98q^{26}-2q^{24}-100q^{22}+159q^{20}-140q^{18}+58q^{16}+52q^{14}-136q^{12}+176q^{10}-149q^{8}+80q^{6}+3q^{4}-78q^{2}+111-107q^{-2}+78q^{-4}-34q^{-6}-3q^{-8}+28q^{-10}-42q^{-12}+39q^{-14}-29q^{-16}+15q^{-18}-3q^{-20}-6q^{-22}+8q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 10_103.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-3q^{13}+3q^{11}-3q^{9}+q^{7}+2q^{5}-q^{3}+4q-3q^{-1}+2q^{-3}-q^{-5}$
2	$q^{48}-2q^{46}+6q^{42}-8q^{40}-4q^{38}+19q^{36}-10q^{34}-19q^{32}+26q^{30}-27q^{26}+20q^{24}+11q^{22}-20q^{20}-q^{18}+13q^{16}-q^{14}-19q^{12}+13q^{10}+19q^{8}-26q^{6}+2q^{4}+27q^{2}-19-9q^{-2}+19q^{-4}-5q^{-6}-7q^{-8}+6q^{-10}-q^{-12}-2q^{-14}+q^{-16}$
3	$-q^{93}+2q^{91}-3q^{87}+7q^{83}+q^{81}-18q^{79}-5q^{77}+34q^{75}+22q^{73}-47q^{71}-58q^{69}+48q^{67}+103q^{65}-24q^{63}-143q^{61}-25q^{59}+165q^{57}+82q^{55}-158q^{53}-132q^{51}+125q^{49}+164q^{47}-83q^{45}-170q^{43}+36q^{41}+153q^{39}+16q^{37}-131q^{35}-50q^{33}+94q^{31}+91q^{29}-64q^{27}-128q^{25}+20q^{23}+159q^{21}+25q^{19}-175q^{17}-73q^{15}+168q^{13}+124q^{11}-134q^{9}-151q^{7}+82q^{5}+160q^{3}-29q-131q^{-1}-19q^{-3}+95q^{-5}+37q^{-7}-53q^{-9}-34q^{-11}+22q^{-13}+22q^{-15}-8q^{-17}-12q^{-19}+5q^{-21}+4q^{-23}-3q^{-25}-2q^{-27}+q^{-29}+2q^{-31}-q^{-33}$
4	$q^{152}-2q^{150}+3q^{146}-3q^{144}+q^{142}-5q^{140}+7q^{138}+16q^{136}-17q^{134}-17q^{132}-28q^{130}+31q^{128}+93q^{126}+6q^{124}-76q^{122}-177q^{120}-34q^{118}+246q^{116}+248q^{114}+47q^{112}-424q^{110}-459q^{108}+95q^{106}+597q^{104}+677q^{102}-224q^{100}-974q^{98}-687q^{96}+363q^{94}+1355q^{92}+666q^{90}-793q^{88}-1447q^{86}-557q^{84}+1248q^{82}+1443q^{80}+59q^{78}-1390q^{76}-1282q^{74}+494q^{72}+1410q^{70}+729q^{68}-738q^{66}-1277q^{64}-178q^{62}+874q^{60}+884q^{58}-132q^{56}-910q^{54}-543q^{52}+378q^{50}+870q^{48}+316q^{46}-600q^{44}-909q^{42}-79q^{40}+938q^{38}+866q^{36}-236q^{34}-1315q^{32}-732q^{30}+778q^{28}+1434q^{26}+479q^{24}-1294q^{22}-1398q^{20}+57q^{18}+1430q^{16}+1246q^{14}-530q^{12}-1410q^{10}-784q^{8}+623q^{6}+1304q^{4}+351q^{2}-631-903q^{-2}-218q^{-4}+619q^{-6}+523q^{-8}+85q^{-10}-394q^{-12}-359q^{-14}+53q^{-16}+187q^{-18}+189q^{-20}-26q^{-22}-128q^{-24}-41q^{-26}-7q^{-28}+58q^{-30}+17q^{-32}-16q^{-34}+2q^{-36}-13q^{-38}+6q^{-40}-q^{-42}-4q^{-44}+6q^{-46}-q^{-48}+2q^{-50}-q^{-52}-2q^{-54}+q^{-56}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{24}+q^{22}-q^{20}-q^{18}+2q^{16}-3q^{14}+q^{12}+q^{8}+4q^{6}-q^{4}+3q^{2}-1-q^{-2}+q^{-4}-q^{-6}$
1,1	$q^{68}-4q^{66}+10q^{64}-22q^{62}+46q^{60}-78q^{58}+124q^{56}-202q^{54}+301q^{52}-406q^{50}+528q^{48}-644q^{46}+711q^{44}-702q^{42}+606q^{40}-428q^{38}+145q^{36}+198q^{34}-556q^{32}+896q^{30}-1167q^{28}+1354q^{26}-1420q^{24}+1360q^{22}-1198q^{20}+912q^{18}-578q^{16}+228q^{14}+115q^{12}-388q^{10}+596q^{8}-672q^{6}+682q^{4}-632q^{2}+534-418q^{-2}+311q^{-4}-224q^{-6}+146q^{-8}-92q^{-10}+54q^{-12}-28q^{-14}+12q^{-16}-4q^{-18}+q^{-20}$
2,0	$q^{62}-q^{60}+2q^{56}-q^{54}-2q^{52}-q^{50}+7q^{48}+q^{46}-11q^{44}+9q^{40}-q^{38}-13q^{36}+2q^{34}+12q^{32}-2q^{30}-10q^{28}+5q^{26}+3q^{24}-11q^{22}+3q^{20}+5q^{18}-3q^{16}+12q^{12}+3q^{10}-10q^{8}+4q^{6}+14q^{4}-6q^{2}-11+7q^{-2}+7q^{-4}-5q^{-6}-6q^{-8}+2q^{-10}+3q^{-12}-2q^{-14}-q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-2q^{52}+4q^{48}-6q^{46}+2q^{44}+11q^{42}-15q^{40}+2q^{38}+17q^{36}-22q^{34}-q^{32}+17q^{30}-16q^{28}-4q^{26}+10q^{24}-3q^{22}-6q^{20}-q^{18}+12q^{16}-10q^{12}+22q^{10}+5q^{8}-20q^{6}+18q^{4}+2q^{2}-17+9q^{-2}+q^{-4}-8q^{-6}+4q^{-8}+q^{-10}-2q^{-12}+q^{-14}$
1,0,0	$-q^{31}+q^{29}-2q^{27}+q^{25}-2q^{23}+2q^{21}-3q^{19}+q^{17}-q^{15}+q^{13}+2q^{11}+2q^{9}+4q^{7}-q^{5}+3q^{3}-2q+q^{-1}-2q^{-3}+q^{-5}-q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{68}-q^{66}-2q^{64}+4q^{62}+q^{60}-8q^{58}+4q^{56}+11q^{54}-6q^{52}-8q^{50}+12q^{48}+9q^{46}-18q^{44}-10q^{42}+14q^{40}-6q^{38}-22q^{36}+8q^{34}+10q^{32}-18q^{30}-2q^{28}+15q^{26}-3q^{24}-9q^{22}+17q^{20}+19q^{18}-9q^{16}+2q^{14}+22q^{12}+q^{10}-16q^{8}+5q^{6}+7q^{4}-8q^{2}-8+2q^{-2}+2q^{-4}-4q^{-6}-q^{-8}+3q^{-10}-q^{-14}+q^{-16}$
1,0,0,0	$-q^{38}+q^{36}-2q^{34}-2q^{28}+2q^{26}-3q^{24}+q^{22}-q^{20}+q^{16}+2q^{14}+3q^{12}+2q^{10}+4q^{8}-q^{6}+3q^{4}-2q^{2}-2q^{-4}+q^{-6}-q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+2q^{52}-4q^{50}+8q^{48}-12q^{46}+16q^{44}-23q^{42}+25q^{40}-26q^{38}+23q^{36}-16q^{34}+9q^{32}+3q^{30}-16q^{28}+30q^{26}-42q^{24}+47q^{22}-50q^{20}+47q^{18}-42q^{16}+32q^{14}-16q^{12}+6q^{10}+9q^{8}-14q^{6}+24q^{4}-26q^{2}+25-23q^{-2}+17q^{-4}-12q^{-6}+8q^{-8}-5q^{-10}+2q^{-12}-q^{-14}$
1,0	$q^{88}-2q^{84}-2q^{82}+2q^{80}+6q^{78}-9q^{74}-6q^{72}+9q^{70}+17q^{68}-3q^{66}-23q^{64}-10q^{62}+21q^{60}+22q^{58}-12q^{56}-28q^{54}-3q^{52}+25q^{50}+12q^{48}-19q^{46}-17q^{44}+11q^{42}+17q^{40}-7q^{38}-18q^{36}+q^{34}+17q^{32}+2q^{30}-17q^{28}-6q^{26}+18q^{24}+14q^{22}-14q^{20}-16q^{18}+13q^{16}+28q^{14}-q^{12}-26q^{10}-12q^{8}+23q^{6}+21q^{4}-10q^{2}-23-4q^{-2}+16q^{-4}+9q^{-6}-7q^{-8}-10q^{-10}-q^{-12}+6q^{-14}+3q^{-16}-2q^{-18}-2q^{-20}+q^{-24}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{74}-2q^{72}+2q^{70}-4q^{68}+7q^{66}-9q^{64}+11q^{62}-13q^{60}+19q^{58}-20q^{56}+20q^{54}-19q^{52}+19q^{50}-17q^{48}+6q^{46}-7q^{44}-q^{42}+8q^{40}-21q^{38}+22q^{36}-29q^{34}+37q^{32}-38q^{30}+36q^{28}-38q^{26}+38q^{24}-27q^{22}+26q^{20}-17q^{18}+15q^{16}+4q^{14}+10q^{10}-14q^{8}+20q^{6}-20q^{4}+18q^{2}-22+17q^{-2}-15q^{-4}+10q^{-6}-10q^{-8}+7q^{-10}-4q^{-12}+3q^{-14}-2q^{-16}+q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-2q^{126}+4q^{124}-7q^{122}+7q^{120}-6q^{118}+12q^{114}-22q^{112}+35q^{110}-42q^{108}+35q^{106}-15q^{104}-25q^{102}+72q^{100}-108q^{98}+118q^{96}-88q^{94}+18q^{92}+74q^{90}-153q^{88}+184q^{86}-149q^{84}+53q^{82}+55q^{80}-142q^{78}+160q^{76}-106q^{74}+11q^{72}+92q^{70}-143q^{68}+115q^{66}-28q^{64}-92q^{62}+180q^{60}-204q^{58}+150q^{56}-38q^{54}-94q^{52}+208q^{50}-252q^{48}+218q^{46}-117q^{44}-25q^{42}+148q^{40}-206q^{38}+189q^{36}-98q^{34}-12q^{32}+112q^{30}-142q^{28}+98q^{26}-2q^{24}-100q^{22}+159q^{20}-140q^{18}+58q^{16}+52q^{14}-136q^{12}+176q^{10}-149q^{8}+80q^{6}+3q^{4}-78q^{2}+111-107q^{-2}+78q^{-4}-34q^{-6}-3q^{-8}+28q^{-10}-42q^{-12}+39q^{-14}-29q^{-16}+15q^{-18}-3q^{-20}-6q^{-22}+8q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

.

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["10 103"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-8t^{2}+17t-21+17t^{-1}-8t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{5,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 75, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-2z^{3}a^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+5z^{7}a^{7}-12z^{5}a^{7}+10z^{3}a^{7}-4za^{7}+5z^{8}a^{6}-12z^{6}a^{6}+13z^{4}a^{6}-6z^{2}a^{6}+a^{6}+2z^{9}a^{5}+3z^{7}a^{5}-16z^{5}a^{5}+21z^{3}a^{5}-6za^{5}+9z^{8}a^{4}-23z^{6}a^{4}+25z^{4}a^{4}-8z^{2}a^{4}+2z^{9}a^{3}+2z^{7}a^{3}-9z^{5}a^{3}+9z^{3}a^{3}-2za^{3}+4z^{8}a^{2}-5z^{6}a^{2}+2z^{2}a^{2}-3a^{2}+4z^{7}a-5z^{5}a-2z^{3}a+za+3z^{6}-6z^{4}+3z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$

Vassiliev invariants

V2 and V3:

(3, -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 $12$ $-32$ $72$ $126$ $2$ $-384$ $-{\frac {1760}{3}}$ $-{\frac {128}{3}}$ $-96$ $288$ $512$ $1512$ $24$ ${\frac {27311}{10}}$ ${\frac {454}{15}}$ ${\frac {4234}{5}}$ ${\frac {155}{2}}$ ${\frac {431}{10}}$

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 10 103. 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	3	χ
5											1	-1
3										2		2
1									4	1		-3
-1								6	2			4
-3							6	5				-1
-5						7	5					2
-7					5	6						1
-9				4	7							-3
-11			2	5								3
-13		1	4									-3
-15		2										2
-17	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, 103]]
Out[2]=	10
In[3]:=	PD[Knot[10, 103]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 6, 19, 5], X[20, 13, 1, 14], X[16, 7, 17, 8], X[10, 3, 11, 4], X[4, 11, 5, 12], X[14, 9, 15, 10], X[8, 15, 9, 16], X[12, 19, 13, 20], X[2, 18, 3, 17]]
In[4]:=	GaussCode[Knot[10, 103]]
Out[4]=	GaussCode[1, -10, 5, -6, 2, -1, 4, -8, 7, -5, 6, -9, 3, -7, 8, -4, 10, -2, 9, -3]
In[5]:=	BR[Knot[10, 103]]
Out[5]=	BR[4, {-1, -1, -2, 1, 3, -2, -2, 3, -2, -2, 3}]
In[6]:=	alex = Alexander[Knot[10, 103]][t]
Out[6]=	2 8 17 2 3 -21 + -- - -- + -- + 17 t - 8 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 103]][z]
Out[7]=	2 4 6 1 + 3 z + 4 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 40], Knot[10, 103]}
In[9]:=	{KnotDet[Knot[10, 103]], KnotSignature[Knot[10, 103]]}
Out[9]=	{75, -2}
In[10]:=	J=Jones[Knot[10, 103]][q]
Out[10]=	-8 3 6 9 12 13 11 10 2 -6 - q + -- - -- + -- - -- + -- - -- + -- + 3 q - q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 40], Knot[10, 103]}
In[12]:=	A2Invariant[Knot[10, 103]][q]
Out[12]=	-24 -22 -20 -18 2 3 -12 -8 4 -4 -1 - q + q - q - q + --- - --- + q + q + -- - q + 16 14 6 q q q 3 2 4 6 -- - q + q - q 2 q
In[13]:=	Kauffman[Knot[10, 103]][a, z]
Out[13]=	2 6 z 3 5 7 2 2 2 -1 - 3 a + a + - + a z - 2 a z - 6 a z - 4 a z + 3 z + 2 a z - a 3 4 2 6 2 8 2 2 z 3 3 3 5 3 8 a z - 6 a z + a z - ---- - 2 a z + 9 a z + 21 a z + a 5 7 3 9 3 4 4 4 6 4 8 4 z 10 a z - 2 a z - 6 z + 25 a z + 13 a z - 6 a z + -- - a 5 3 5 5 5 7 5 9 5 6 2 6 5 a z - 9 a z - 16 a z - 12 a z + a z + 3 z - 5 a z - 4 6 6 6 8 6 7 3 7 5 7 23 a z - 12 a z + 3 a z + 4 a z + 2 a z + 3 a z + 7 7 2 8 4 8 6 8 3 9 5 9 5 a z + 4 a z + 9 a z + 5 a z + 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[10, 103]], Vassiliev[3][Knot[10, 103]]}
Out[14]=	{0, -4}
In[15]:=	Kh[Knot[10, 103]][q, t]
Out[15]=	5 6 1 2 1 4 2 5 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 7 5 6 7 5 6 2 t 2 ----- + ----- + ----- + ----- + ---- + ---- + --- + 4 q t + q t + 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t 3 2 5 3 2 q t + q t