9 27

9_26

9_28

Visit 9 27's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 27's page at Knotilus! Visit 9 27's page at the original Knot Atlas!

9 27 Quick Notes

9 27 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,18 X5,13,6,12 X13,17,14,16 X7,14,8,15 X15,6,16,7 X17,9,18,8 X9,2,10,3
Gauss code	-1, 9, -2, 1, -4, 7, -6, 8, -9, 2, -3, 4, -5, 6, -7, 5, -8, 3
Dowker-Thistlethwaite code	4 10 12 14 2 18 16 6 8
Conway Notation	[212112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 11. A-Polynomial See Data:9 27/A-polynomial

[edit Notes for 9 27's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 27's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 49, 0 }
Jones polynomial	$q^{4}-3q^{3}+5q^{2}-7q+9-8q^{-1}+7q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-4z^{4}-a^{4}z^{2}+5a^{2}z^{2}+2z^{2}a^{-2}-6z^{2}-a^{4}+3a^{2}+a^{-2}-2$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+6a^{2}z^{6}+4z^{6}a^{-2}+7z^{6}+a^{5}z^{5}-4a^{3}z^{5}-8az^{5}+3z^{5}a^{-3}-7a^{4}z^{4}-17a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-16z^{4}-2a^{5}z^{3}-2a^{3}z^{3}-4z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+12a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+12z^{2}+a^{5}z+2a^{3}z+2az+2za^{-1}+za^{-3}-a^{4}-3a^{2}-a^{-2}-2$
The A2 invariant	$-q^{16}+q^{12}-q^{10}+2q^{8}+2q^{2}-1+2q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+7q^{72}-4q^{70}-6q^{68}+19q^{66}-29q^{64}+33q^{62}-29q^{60}+6q^{58}+22q^{56}-50q^{54}+65q^{52}-56q^{50}+32q^{48}+6q^{46}-42q^{44}+61q^{42}-58q^{40}+33q^{38}+3q^{36}-32q^{34}+41q^{32}-25q^{30}-q^{28}+36q^{26}-53q^{24}+50q^{22}-24q^{20}-20q^{18}+64q^{16}-89q^{14}+88q^{12}-53q^{10}+8q^{8}+45q^{6}-81q^{4}+87q^{2}-67+26q^{-2}+16q^{-4}-47q^{-6}+48q^{-8}-25q^{-10}-5q^{-12}+31q^{-14}-41q^{-16}+24q^{-18}+q^{-20}-35q^{-22}+58q^{-24}-59q^{-26}+43q^{-28}-9q^{-30}-22q^{-32}+45q^{-34}-51q^{-36}+45q^{-38}-28q^{-40}+7q^{-42}+11q^{-44}-23q^{-46}+24q^{-48}-18q^{-50}+13q^{-52}-4q^{-54}-2q^{-56}+4q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_27.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-2q^{7}+2q^{5}-q^{3}+q+2q^{-1}-2q^{-3}+2q^{-5}-2q^{-7}+q^{-9}$
2	$q^{32}-2q^{30}-2q^{28}+7q^{26}-2q^{24}-9q^{22}+11q^{20}+2q^{18}-15q^{16}+9q^{14}+7q^{12}-13q^{10}+3q^{8}+8q^{6}-4q^{4}-4q^{2}+5+9q^{-2}-11q^{-4}-3q^{-6}+15q^{-8}-10q^{-10}-7q^{-12}+13q^{-14}-4q^{-16}-5q^{-18}+6q^{-20}-q^{-22}-2q^{-24}+q^{-26}$
3	$-q^{63}+2q^{61}+2q^{59}-3q^{57}-7q^{55}+2q^{53}+16q^{51}+q^{49}-23q^{47}-11q^{45}+29q^{43}+26q^{41}-31q^{39}-42q^{37}+26q^{35}+55q^{33}-13q^{31}-65q^{29}-q^{27}+67q^{25}+14q^{23}-63q^{21}-25q^{19}+52q^{17}+34q^{15}-39q^{13}-36q^{11}+22q^{9}+39q^{7}-4q^{5}-36q^{3}-18q+34q^{-1}+42q^{-3}-27q^{-5}-55q^{-7}+13q^{-9}+68q^{-11}-2q^{-13}-69q^{-15}-12q^{-17}+62q^{-19}+20q^{-21}-48q^{-23}-23q^{-25}+34q^{-27}+21q^{-29}-21q^{-31}-16q^{-33}+11q^{-35}+11q^{-37}-7q^{-39}-5q^{-41}+3q^{-43}+3q^{-45}-q^{-47}-2q^{-49}+q^{-51}$
4	$q^{104}-2q^{102}-2q^{100}+3q^{98}+3q^{96}+7q^{94}-9q^{92}-16q^{90}-q^{88}+11q^{86}+41q^{84}-2q^{82}-47q^{80}-41q^{78}-7q^{76}+100q^{74}+62q^{72}-42q^{70}-117q^{68}-108q^{66}+117q^{64}+173q^{62}+62q^{60}-141q^{58}-259q^{56}+16q^{54}+223q^{52}+230q^{50}-46q^{48}-346q^{46}-148q^{44}+156q^{42}+333q^{40}+100q^{38}-305q^{36}-255q^{34}+30q^{32}+318q^{30}+195q^{28}-190q^{26}-261q^{24}-69q^{22}+225q^{20}+214q^{18}-58q^{16}-208q^{14}-135q^{12}+100q^{10}+195q^{8}+88q^{6}-124q^{4}-197q^{2}-61+154q^{-2}+254q^{-4}-2q^{-6}-232q^{-8}-236q^{-10}+52q^{-12}+355q^{-14}+151q^{-16}-171q^{-18}-343q^{-20}-94q^{-22}+321q^{-24}+243q^{-26}-34q^{-28}-299q^{-30}-185q^{-32}+176q^{-34}+204q^{-36}+72q^{-38}-160q^{-40}-163q^{-42}+51q^{-44}+96q^{-46}+80q^{-48}-49q^{-50}-85q^{-52}+7q^{-54}+23q^{-56}+41q^{-58}-9q^{-60}-29q^{-62}+4q^{-64}+12q^{-68}-2q^{-70}-8q^{-72}+3q^{-74}+3q^{-78}-q^{-80}-2q^{-82}+q^{-84}$
5	$-q^{155}+2q^{153}+2q^{151}-3q^{149}-3q^{147}-3q^{145}+9q^{141}+16q^{139}+q^{137}-20q^{135}-29q^{133}-19q^{131}+20q^{129}+61q^{127}+62q^{125}-11q^{123}-97q^{121}-121q^{119}-47q^{117}+108q^{115}+220q^{113}+161q^{111}-78q^{109}-309q^{107}-326q^{105}-59q^{103}+348q^{101}+541q^{99}+291q^{97}-284q^{95}-722q^{93}-609q^{91}+62q^{89}+811q^{87}+963q^{85}+295q^{83}-745q^{81}-1264q^{79}-732q^{77}+497q^{75}+1431q^{73}+1192q^{71}-123q^{69}-1433q^{67}-1558q^{65}-315q^{63}+1258q^{61}+1785q^{59}+742q^{57}-978q^{55}-1843q^{53}-1070q^{51}+644q^{49}+1750q^{47}+1273q^{45}-324q^{43}-1556q^{41}-1345q^{39}+50q^{37}+1315q^{35}+1314q^{33}+149q^{31}-1043q^{29}-1228q^{27}-316q^{25}+799q^{23}+1121q^{21}+450q^{19}-541q^{17}-1013q^{15}-623q^{13}+280q^{11}+929q^{9}+805q^{7}+27q^{5}-818q^{3}-1032q-389q^{-1}+672q^{-3}+1265q^{-5}+782q^{-7}-438q^{-9}-1410q^{-11}-1218q^{-13}+110q^{-15}+1477q^{-17}+1587q^{-19}+274q^{-21}-1350q^{-23}-1852q^{-25}-701q^{-27}+1098q^{-29}+1925q^{-31}+1056q^{-33}-717q^{-35}-1801q^{-37}-1283q^{-39}+301q^{-41}+1502q^{-43}+1336q^{-45}+71q^{-47}-1114q^{-49}-1217q^{-51}-315q^{-53}+702q^{-55}+984q^{-57}+430q^{-59}-366q^{-61}-710q^{-63}-417q^{-65}+137q^{-67}+449q^{-69}+335q^{-71}-3q^{-73}-256q^{-75}-239q^{-77}-36q^{-79}+127q^{-81}+138q^{-83}+47q^{-85}-51q^{-87}-81q^{-89}-32q^{-91}+23q^{-93}+34q^{-95}+17q^{-97}-5q^{-99}-13q^{-101}-10q^{-103}+3q^{-105}+8q^{-107}-q^{-109}-3q^{-111}+3q^{-119}-q^{-121}-2q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{12}-q^{10}+2q^{8}+2q^{2}-1+2q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+52q^{36}-86q^{34}+130q^{32}-180q^{30}+222q^{28}-248q^{26}+258q^{24}-236q^{22}+176q^{20}-88q^{18}-24q^{16}+144q^{14}-267q^{12}+376q^{10}-450q^{8}+492q^{6}-483q^{4}+440q^{2}-352+244q^{-2}-120q^{-4}-4q^{-6}+102q^{-8}-176q^{-10}+222q^{-12}-238q^{-14}+228q^{-16}-196q^{-18}+162q^{-20}-126q^{-22}+88q^{-24}-58q^{-26}+36q^{-28}-20q^{-30}+10q^{-32}-4q^{-34}+q^{-36}$
2,0	$q^{42}-2q^{38}-2q^{36}+2q^{34}+4q^{32}-3q^{30}-3q^{28}+4q^{26}+4q^{24}-5q^{22}-5q^{20}+5q^{18}+2q^{16}-6q^{14}+6q^{10}-2q^{8}-q^{6}+6q^{4}+q^{2}-2+3q^{-2}+5q^{-4}-7q^{-6}-5q^{-8}+7q^{-10}+2q^{-12}-7q^{-14}+6q^{-18}-3q^{-22}+2q^{-26}-q^{-28}-q^{-30}+q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+3q^{28}-7q^{26}+3q^{24}+4q^{22}-12q^{20}+5q^{18}+8q^{16}-10q^{14}+5q^{12}+10q^{10}-5q^{8}-q^{6}+4q^{4}+q^{2}-4-4q^{-2}+9q^{-4}-5q^{-6}-8q^{-8}+13q^{-10}-2q^{-12}-8q^{-14}+10q^{-16}-6q^{-20}+4q^{-22}-2q^{-26}+q^{-28}$
1,0,0	$-q^{21}-q^{17}+q^{15}-q^{13}+3q^{11}+2q^{7}+q^{3}-q^{-1}+q^{-3}-2q^{-5}+q^{-7}-q^{-9}+2q^{-11}-q^{-13}+q^{-15}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-5q^{30}+7q^{28}-9q^{26}+11q^{24}-12q^{22}+12q^{20}-9q^{18}+6q^{16}-5q^{12}+12q^{10}-17q^{8}+21q^{6}-22q^{4}+23q^{2}-20+16q^{-2}-9q^{-4}+3q^{-6}+2q^{-8}-7q^{-10}+10q^{-12}-12q^{-14}+12q^{-16}-10q^{-18}+8q^{-20}-6q^{-22}+4q^{-24}-2q^{-26}+q^{-28}$
1,0	$q^{56}-2q^{52}-2q^{50}+3q^{48}+5q^{46}-2q^{44}-8q^{42}-3q^{40}+9q^{38}+8q^{36}-7q^{34}-13q^{32}+13q^{28}+7q^{26}-9q^{24}-9q^{22}+5q^{20}+11q^{18}-8q^{14}-q^{12}+8q^{10}+3q^{8}-7q^{6}-5q^{4}+7q^{2}+7-5q^{-2}-9q^{-4}+4q^{-6}+10q^{-8}-q^{-10}-12q^{-12}-4q^{-14}+11q^{-16}+9q^{-18}-6q^{-20}-12q^{-22}+11q^{-26}+6q^{-28}-5q^{-30}-7q^{-32}+5q^{-36}+2q^{-38}-2q^{-40}-2q^{-42}+q^{-46}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-2q^{44}+3q^{42}-4q^{40}+6q^{38}-8q^{36}+7q^{34}-10q^{32}+9q^{30}-11q^{28}+6q^{26}-6q^{24}+6q^{22}-q^{18}+9q^{16}-4q^{14}+15q^{12}-14q^{10}+16q^{8}-16q^{6}+18q^{4}-19q^{2}+12-15q^{-2}+10q^{-4}-6q^{-6}+q^{-8}-q^{-10}-2q^{-12}+9q^{-14}-6q^{-16}+8q^{-18}-9q^{-20}+11q^{-22}-7q^{-24}+6q^{-26}-7q^{-28}+5q^{-30}-3q^{-32}+2q^{-34}-2q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+7q^{72}-4q^{70}-6q^{68}+19q^{66}-29q^{64}+33q^{62}-29q^{60}+6q^{58}+22q^{56}-50q^{54}+65q^{52}-56q^{50}+32q^{48}+6q^{46}-42q^{44}+61q^{42}-58q^{40}+33q^{38}+3q^{36}-32q^{34}+41q^{32}-25q^{30}-q^{28}+36q^{26}-53q^{24}+50q^{22}-24q^{20}-20q^{18}+64q^{16}-89q^{14}+88q^{12}-53q^{10}+8q^{8}+45q^{6}-81q^{4}+87q^{2}-67+26q^{-2}+16q^{-4}-47q^{-6}+48q^{-8}-25q^{-10}-5q^{-12}+31q^{-14}-41q^{-16}+24q^{-18}+q^{-20}-35q^{-22}+58q^{-24}-59q^{-26}+43q^{-28}-9q^{-30}-22q^{-32}+45q^{-34}-51q^{-36}+45q^{-38}-28q^{-40}+7q^{-42}+11q^{-44}-23q^{-46}+24q^{-48}-18q^{-50}+13q^{-52}-4q^{-54}-2q^{-56}+4q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

.

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 27"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+6a^{2}z^{6}+4z^{6}a^{-2}+7z^{6}+a^{5}z^{5}-4a^{3}z^{5}-8az^{5}+3z^{5}a^{-3}-7a^{4}z^{4}-17a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-16z^{4}-2a^{5}z^{3}-2a^{3}z^{3}-4z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+12a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+12z^{2}+a^{5}z+2a^{3}z+2az+2za^{-1}+za^{-3}-a^{4}-3a^{2}-a^{-2}-2$

Vassiliev invariants

V2 and V3:

(0, -1)

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 $0$ $-8$ $0$ $16$ $8$ $0$ ${\frac {16}{3}}$ ${\frac {64}{3}}$ $-8$ $0$ $32$ $0$ $0$ $56$ $56$ $-{\frac {152}{3}}$ ${\frac {40}{3}}$ $-24$

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 27. 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						5	3				2
-1					4	5					1
-3				3	4						-1
-5			2	4							2
-7		1	3								-2
-9		2									2
-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, 27]]
Out[2]=	9
In[3]:=	PD[Knot[9, 27]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 18], X[5, 13, 6, 12], X[13, 17, 14, 16], X[7, 14, 8, 15], X[15, 6, 16, 7], X[17, 9, 18, 8], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[9, 27]]
Out[4]=	GaussCode[-1, 9, -2, 1, -4, 7, -6, 8, -9, 2, -3, 4, -5, 6, -7, 5, -8, 3]
In[5]:=	BR[Knot[9, 27]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, 2, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 27]][t]
Out[6]=	-3 5 11 2 3 15 - t + -- - -- - 11 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[9, 27]][z]
Out[7]=	4 6 1 - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 27], Knot[11, NonAlternating, 4], Knot[11, NonAlternating, 21], Knot[11, NonAlternating, 172]}
In[9]:=	{KnotDet[Knot[9, 27]], KnotSignature[Knot[9, 27]]}
Out[9]=	{49, 0}
In[10]:=	J=Jones[Knot[9, 27]][q]
Out[10]=	-5 3 5 7 8 2 3 4 9 - 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, 27], Knot[11, NonAlternating, 83]}
In[12]:=	A2Invariant[Knot[9, 27]][q]
Out[12]=	-16 -12 -10 2 2 2 4 8 10 12 -1 - q + q - q + -- + -- + 2 q - 2 q + q - q + q 8 2 q q
In[13]:=	Kauffman[Knot[9, 27]][a, z]
Out[13]=	2 -2 2 4 z 2 z 3 5 2 z -2 - a - 3 a - a + -- + --- + 2 a z + 2 a z + a z + 12 z - -- + 3 a 4 a a 2 3 3 3 z 2 2 4 2 4 z 4 z 3 3 5 3 4 ---- + 12 a z + 4 a z - ---- - ---- - 2 a z - 2 a z - 16 z + 2 3 a a a 4 4 5 z 5 z 2 4 4 4 3 z 5 3 5 5 5 -- - ---- - 17 a z - 7 a z + ---- - 8 a z - 4 a z + a z + 4 2 3 a a a 6 7 6 4 z 2 6 4 6 3 z 7 3 7 8 2 8 7 z + ---- + 6 a z + 3 a z + ---- + 6 a z + 3 a z + z + a z 2 a a
In[14]:=	{Vassiliev[2][Knot[9, 27]], Vassiliev[3][Knot[9, 27]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[9, 27]][q, t]
Out[15]=	5 1 2 1 3 2 4 3 - + 5 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 4 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