9 31

9_30

9_32

Visit 9 31's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 31's page at Knotilus! Visit 9 31's page at the original Knot Atlas!

9 31 Quick Notes

9 31 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,18 X5,13,6,12 X17,7,18,6 X7,14,8,15 X13,16,14,17 X15,8,16,9 X9,2,10,3
Gauss code	-1, 9, -2, 1, -4, 5, -6, 8, -9, 2, -3, 4, -7, 6, -8, 7, -5, 3
Dowker-Thistlethwaite code	4 10 12 14 2 18 16 8 6
Conway Notation	[2111112]

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 [-9][-2] Hyperbolic Volume 11.6863 A-Polynomial See Data:9 31/A-polynomial

[edit Notes for 9 31'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 31's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+13t-17+13t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 55, -2 }
Jones polynomial	$-q^{2}+3q-5+8q^{-1}-9q^{-2}+10q^{-3}-8q^{-4}+6q^{-5}-4q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-2a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+4a^{2}-z^{4}-2z^{2}-1$
Kauffman polynomial (db, data sources)	$z^{4}a^{8}+4z^{5}a^{7}-4z^{3}a^{7}+6z^{6}a^{6}-8z^{4}a^{6}+3z^{2}a^{6}+4z^{7}a^{5}+z^{5}a^{5}-8z^{3}a^{5}+3za^{5}+z^{8}a^{4}+11z^{6}a^{4}-23z^{4}a^{4}+13z^{2}a^{4}-2a^{4}+7z^{7}a^{3}-7z^{5}a^{3}-5z^{3}a^{3}+5za^{3}+z^{8}a^{2}+8z^{6}a^{2}-21z^{4}a^{2}+15z^{2}a^{2}-4a^{2}+3z^{7}a-3z^{5}a-3z^{3}a+3za+3z^{6}-7z^{4}+5z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$
The A2 invariant	$q^{22}-q^{20}-2q^{18}+q^{16}-2q^{14}+q^{12}+q^{10}+3q^{6}-q^{4}+3q^{2}-q^{-2}+q^{-4}-q^{-6}$
The G2 invariant	$q^{114}-3q^{112}+6q^{110}-10q^{108}+8q^{106}-4q^{104}-5q^{102}+22q^{100}-33q^{98}+45q^{96}-41q^{94}+16q^{92}+17q^{90}-54q^{88}+80q^{86}-86q^{84}+65q^{82}-20q^{80}-33q^{78}+75q^{76}-90q^{74}+70q^{72}-28q^{70}-23q^{68}+48q^{66}-52q^{64}+24q^{62}+26q^{60}-67q^{58}+82q^{56}-60q^{54}+2q^{52}+65q^{50}-121q^{48}+136q^{46}-108q^{44}+48q^{42}+32q^{40}-96q^{38}+129q^{36}-115q^{34}+68q^{32}-4q^{30}-51q^{28}+73q^{26}-55q^{24}+22q^{22}+29q^{20}-57q^{18}+59q^{16}-26q^{14}-24q^{12}+72q^{10}-94q^{8}+83q^{6}-43q^{4}-9q^{2}+55-80q^{-2}+81q^{-4}-55q^{-6}+18q^{-8}+13q^{-10}-35q^{-12}+38q^{-14}-31q^{-16}+19q^{-18}-5q^{-20}-5q^{-22}+7q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 9_31.

A1 Invariants.

Weight	Invariant
1	$q^{15}-3q^{13}+2q^{11}-2q^{9}+2q^{7}+q^{5}-q^{3}+3q-2q^{-1}+2q^{-3}-q^{-5}$
2	$q^{42}-3q^{40}-q^{38}+10q^{36}-7q^{34}-8q^{32}+18q^{30}-7q^{28}-15q^{26}+18q^{24}-q^{22}-15q^{20}+7q^{18}+6q^{16}-5q^{14}-7q^{12}+11q^{10}+8q^{8}-17q^{6}+7q^{4}+15q^{2}-18+14q^{-4}-9q^{-6}-3q^{-8}+7q^{-10}-2q^{-12}-2q^{-14}+q^{-16}$
3	$q^{81}-3q^{79}-q^{77}+7q^{75}+5q^{73}-11q^{71}-18q^{69}+20q^{67}+29q^{65}-22q^{63}-49q^{61}+21q^{59}+71q^{57}-13q^{55}-89q^{53}-q^{51}+101q^{49}+18q^{47}-96q^{45}-38q^{43}+87q^{41}+48q^{39}-63q^{37}-60q^{35}+35q^{33}+56q^{31}-4q^{29}-56q^{27}-27q^{25}+50q^{23}+54q^{21}-36q^{19}-76q^{17}+23q^{15}+94q^{13}-3q^{11}-99q^{9}-15q^{7}+96q^{5}+36q^{3}-84q-47q^{-1}+62q^{-3}+54q^{-5}-41q^{-7}-49q^{-9}+20q^{-11}+39q^{-13}-6q^{-15}-26q^{-17}-2q^{-19}+16q^{-21}+3q^{-23}-7q^{-25}-3q^{-27}+2q^{-29}+2q^{-31}-q^{-33}$
4	$q^{132}-3q^{130}-q^{128}+7q^{126}+2q^{124}+q^{122}-21q^{120}-10q^{118}+29q^{116}+23q^{114}+19q^{112}-72q^{110}-63q^{108}+58q^{106}+96q^{104}+89q^{102}-148q^{100}-197q^{98}+31q^{96}+212q^{94}+268q^{92}-164q^{90}-396q^{88}-127q^{86}+273q^{84}+507q^{82}-34q^{80}-504q^{78}-358q^{76}+169q^{74}+631q^{72}+184q^{70}-404q^{68}-484q^{66}-45q^{64}+522q^{62}+330q^{60}-156q^{58}-424q^{56}-226q^{54}+256q^{52}+343q^{50}+102q^{48}-247q^{46}-320q^{44}-43q^{42}+286q^{40}+318q^{38}-53q^{36}-359q^{34}-306q^{32}+198q^{30}+474q^{28}+150q^{26}-327q^{24}-521q^{22}+46q^{20}+524q^{18}+348q^{16}-187q^{14}-617q^{12}-162q^{10}+402q^{8}+461q^{6}+49q^{4}-515q^{2}-312+152q^{-2}+395q^{-4}+229q^{-6}-269q^{-8}-292q^{-10}-57q^{-12}+201q^{-14}+235q^{-16}-58q^{-18}-149q^{-20}-106q^{-22}+39q^{-24}+128q^{-26}+19q^{-28}-34q^{-30}-59q^{-32}-13q^{-34}+41q^{-36}+14q^{-38}+2q^{-40}-16q^{-42}-10q^{-44}+7q^{-46}+3q^{-48}+3q^{-50}-2q^{-52}-2q^{-54}+q^{-56}$
5	$q^{195}-3q^{193}-q^{191}+7q^{189}+2q^{187}-2q^{185}-9q^{183}-13q^{181}-q^{179}+29q^{177}+37q^{175}-3q^{173}-55q^{171}-74q^{169}-16q^{167}+86q^{165}+165q^{163}+74q^{161}-160q^{159}-290q^{157}-171q^{155}+177q^{153}+496q^{151}+398q^{149}-184q^{147}-756q^{145}-715q^{143}+53q^{141}+1009q^{139}+1206q^{137}+231q^{135}-1204q^{133}-1773q^{131}-710q^{129}+1230q^{127}+2331q^{125}+1378q^{123}-1030q^{121}-2774q^{119}-2105q^{117}+580q^{115}+2947q^{113}+2787q^{111}+59q^{109}-2838q^{107}-3243q^{105}-771q^{103}+2405q^{101}+3437q^{99}+1405q^{97}-1779q^{95}-3268q^{93}-1888q^{91}+1030q^{89}+2882q^{87}+2136q^{85}-324q^{83}-2268q^{81}-2189q^{79}-336q^{77}+1640q^{75}+2098q^{73}+849q^{71}-986q^{69}-1933q^{67}-1298q^{65}+391q^{63}+1769q^{61}+1674q^{59}+133q^{57}-1609q^{55}-2038q^{53}-646q^{51}+1454q^{49}+2397q^{47}+1152q^{45}-1274q^{43}-2698q^{41}-1691q^{39}+994q^{37}+2926q^{35}+2239q^{33}-586q^{31}-2993q^{29}-2738q^{27}+37q^{25}+2839q^{23}+3121q^{21}+621q^{19}-2429q^{17}-3304q^{15}-1265q^{13}+1794q^{11}+3190q^{9}+1818q^{7}-1007q^{5}-2819q^{3}-2140q+240q^{-1}+2189q^{-3}+2180q^{-5}+426q^{-7}-1472q^{-9}-1953q^{-11}-829q^{-13}+769q^{-15}+1529q^{-17}+981q^{-19}-222q^{-21}-1034q^{-23}-908q^{-25}-123q^{-27}+590q^{-29}+702q^{-31}+260q^{-33}-254q^{-35}-458q^{-37}-277q^{-39}+66q^{-41}+261q^{-43}+205q^{-45}+20q^{-47}-117q^{-49}-132q^{-51}-46q^{-53}+49q^{-55}+71q^{-57}+33q^{-59}-13q^{-61}-29q^{-63}-23q^{-65}-2q^{-67}+16q^{-69}+10q^{-71}-3q^{-75}-3q^{-77}-3q^{-79}+2q^{-81}+2q^{-83}-q^{-85}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-q^{20}-2q^{18}+q^{16}-2q^{14}+q^{12}+q^{10}+3q^{6}-q^{4}+3q^{2}-q^{-2}+q^{-4}-q^{-6}$
1,1	$q^{60}-6q^{58}+18q^{56}-38q^{54}+69q^{52}-120q^{50}+186q^{48}-254q^{46}+324q^{44}-382q^{42}+414q^{40}-396q^{38}+326q^{36}-208q^{34}+40q^{32}+156q^{30}-368q^{28}+554q^{26}-708q^{24}+792q^{22}-811q^{20}+756q^{18}-632q^{16}+466q^{14}-253q^{12}+60q^{10}+126q^{8}-270q^{6}+367q^{4}-406q^{2}+396-354q^{-2}+292q^{-4}-218q^{-6}+150q^{-8}-96q^{-10}+54q^{-12}-28q^{-14}+12q^{-16}-4q^{-18}+q^{-20}$
2,0	$q^{56}-q^{54}-3q^{52}+5q^{48}+3q^{46}-6q^{44}+8q^{40}-9q^{36}+7q^{32}-5q^{30}-10q^{28}+2q^{26}+2q^{24}-7q^{22}+3q^{20}+7q^{18}-q^{16}+q^{14}+10q^{12}+2q^{10}-8q^{8}+2q^{6}+10q^{4}-4q^{2}-8+5q^{-2}+5q^{-4}-4q^{-6}-3q^{-8}+2q^{-10}+2q^{-12}-2q^{-14}-q^{-16}+q^{-18}$

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}-q^{60}-3q^{58}+q^{56}+4q^{54}-q^{52}-4q^{50}+6q^{48}+8q^{46}-6q^{44}-8q^{42}+9q^{40}+2q^{38}-13q^{36}-2q^{34}+7q^{32}-8q^{30}-12q^{28}+4q^{26}+q^{24}-10q^{22}+4q^{20}+17q^{18}-2q^{16}+19q^{12}+9q^{10}-9q^{8}+2q^{6}+9q^{4}-4q^{2}-10+q^{-2}+3q^{-4}-5q^{-6}-2q^{-8}+3q^{-10}-q^{-14}+q^{-16}$
1,0,0,0	$q^{36}-q^{34}-2q^{28}+q^{26}-3q^{24}-q^{20}-q^{18}+q^{16}+q^{14}+3q^{12}+2q^{10}+4q^{8}+3q^{4}-q^{2}-2q^{-4}+q^{-6}-q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-3q^{46}+6q^{44}-9q^{42}+12q^{40}-15q^{38}+15q^{36}-15q^{34}+11q^{32}-8q^{30}+8q^{26}-16q^{24}+23q^{22}-26q^{20}+30q^{18}-28q^{16}+26q^{14}-18q^{12}+11q^{10}-3q^{8}-3q^{6}+10q^{4}-13q^{2}+16-14q^{-2}+13q^{-4}-10q^{-6}+7q^{-8}-5q^{-10}+2q^{-12}-q^{-14}$
1,0	$q^{78}-3q^{74}-3q^{72}+3q^{70}+8q^{68}+q^{66}-10q^{64}-7q^{62}+10q^{60}+14q^{58}-2q^{56}-17q^{54}-7q^{52}+12q^{50}+12q^{48}-8q^{46}-15q^{44}+q^{42}+12q^{40}+2q^{38}-12q^{36}-5q^{34}+9q^{32}+7q^{30}-8q^{28}-8q^{26}+6q^{24}+11q^{22}-3q^{20}-10q^{18}+3q^{16}+16q^{14}+5q^{12}-13q^{10}-10q^{8}+11q^{6}+16q^{4}-2q^{2}-16-6q^{-2}+10q^{-4}+10q^{-6}-4q^{-8}-9q^{-10}-2q^{-12}+5q^{-14}+3q^{-16}-2q^{-18}-2q^{-20}+q^{-24}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-3q^{64}+3q^{62}-4q^{60}+8q^{58}-10q^{56}+10q^{54}-10q^{52}+14q^{50}-12q^{48}+8q^{46}-9q^{44}+8q^{42}-2q^{40}-5q^{38}+4q^{36}-11q^{34}+15q^{32}-21q^{30}+16q^{28}-25q^{26}+23q^{24}-20q^{22}+20q^{20}-15q^{18}+18q^{16}-q^{14}+9q^{12}+2q^{10}-2q^{8}+10q^{6}-9q^{4}+9q^{2}-14+11q^{-2}-11q^{-4}+8q^{-6}-9q^{-8}+6q^{-10}-4q^{-12}+3q^{-14}-2q^{-16}+q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-3q^{112}+6q^{110}-10q^{108}+8q^{106}-4q^{104}-5q^{102}+22q^{100}-33q^{98}+45q^{96}-41q^{94}+16q^{92}+17q^{90}-54q^{88}+80q^{86}-86q^{84}+65q^{82}-20q^{80}-33q^{78}+75q^{76}-90q^{74}+70q^{72}-28q^{70}-23q^{68}+48q^{66}-52q^{64}+24q^{62}+26q^{60}-67q^{58}+82q^{56}-60q^{54}+2q^{52}+65q^{50}-121q^{48}+136q^{46}-108q^{44}+48q^{42}+32q^{40}-96q^{38}+129q^{36}-115q^{34}+68q^{32}-4q^{30}-51q^{28}+73q^{26}-55q^{24}+22q^{22}+29q^{20}-57q^{18}+59q^{16}-26q^{14}-24q^{12}+72q^{10}-94q^{8}+83q^{6}-43q^{4}-9q^{2}+55-80q^{-2}+81q^{-4}-55q^{-6}+18q^{-8}+13q^{-10}-35q^{-12}+38q^{-14}-31q^{-16}+19q^{-18}-5q^{-20}-5q^{-22}+7q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

.

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["9 31"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+13t-17+13t^{-1}-5t^{-2}+t^{-3}$

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]=

{ 55, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-2a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+4a^{2}-z^{4}-2z^{2}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{8}+4z^{5}a^{7}-4z^{3}a^{7}+6z^{6}a^{6}-8z^{4}a^{6}+3z^{2}a^{6}+4z^{7}a^{5}+z^{5}a^{5}-8z^{3}a^{5}+3za^{5}+z^{8}a^{4}+11z^{6}a^{4}-23z^{4}a^{4}+13z^{2}a^{4}-2a^{4}+7z^{7}a^{3}-7z^{5}a^{3}-5z^{3}a^{3}+5za^{3}+z^{8}a^{2}+8z^{6}a^{2}-21z^{4}a^{2}+15z^{2}a^{2}-4a^{2}+3z^{7}a-3z^{5}a-3z^{3}a+3za+3z^{6}-7z^{4}+5z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$

Vassiliev invariants

V2 and V3:

(2, -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 $8$ $-16$ $32$ ${\frac {172}{3}}$ ${\frac {20}{3}}$ $-128$ $-{\frac {640}{3}}$ $-{\frac {160}{3}}$ $-16$ ${\frac {256}{3}}$ $128$ ${\frac {1376}{3}}$ ${\frac {160}{3}}$ ${\frac {11911}{15}}$ ${\frac {916}{15}}$ ${\frac {12604}{45}}$ $-{\frac {55}{9}}$ ${\frac {631}{15}}$

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 $s=$-2 is the signature of 9 31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-6	-5	-4	-3	-2	-1	0	1	2	3	χ
5										1	-1
3									2		2
1								3	1		-2
-1							5	2			3
-3						5	4				-1
-5					5	4					1
-7				3	5						2
-9			3	5							-2
-11		1	3								2
-13		3									-3
-15	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, 31]]
Out[2]=	9
In[3]:=	PD[Knot[9, 31]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 18], X[5, 13, 6, 12], X[17, 7, 18, 6], X[7, 14, 8, 15], X[13, 16, 14, 17], X[15, 8, 16, 9], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[9, 31]]
Out[4]=	GaussCode[-1, 9, -2, 1, -4, 5, -6, 8, -9, 2, -3, 4, -7, 6, -8, 7, -5, 3]
In[5]:=	BR[Knot[9, 31]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[9, 31]][t]
Out[6]=	-3 5 13 2 3 -17 + t - -- + -- + 13 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[9, 31]][z]
Out[7]=	2 4 6 1 + 2 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 31], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 22], Knot[11, NonAlternating, 112], Knot[11, NonAlternating, 127]}
In[9]:=	{KnotDet[Knot[9, 31]], KnotSignature[Knot[9, 31]]}
Out[9]=	{55, -2}
In[10]:=	J=Jones[Knot[9, 31]][q]
Out[10]=	-7 4 6 8 10 9 8 2 -5 + q - -- + -- - -- + -- - -- + - + 3 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 31]}
In[12]:=	A2Invariant[Knot[9, 31]][q]
Out[12]=	-22 -20 2 -16 2 -12 -10 3 -4 3 2 q - q - --- + q - --- + q + q + -- - q + -- - q + 18 14 6 2 q q q q 4 6 q - q
In[13]:=	Kauffman[Knot[9, 31]][a, z]
Out[13]=	2 4 z 3 5 2 2 2 -1 - 4 a - 2 a + - + 3 a z + 5 a z + 3 a z + 5 z + 15 a z + a 3 4 2 6 2 2 z 3 3 3 5 3 7 3 13 a z + 3 a z - ---- - 3 a z - 5 a z - 8 a z - 4 a z - a 5 4 2 4 4 4 6 4 8 4 z 5 7 z - 21 a z - 23 a z - 8 a z + a z + -- - 3 a z - a 3 5 5 5 7 5 6 2 6 4 6 6 6 7 a z + a z + 4 a z + 3 z + 8 a z + 11 a z + 6 a z + 7 3 7 5 7 2 8 4 8 3 a z + 7 a z + 4 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 31]], Vassiliev[3][Knot[9, 31]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[9, 31]][q, t]
Out[15]=	4 5 1 3 1 3 3 5 3 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 5 5 4 5 2 t 2 3 2 5 3 ----- + ----- + ---- + ---- + --- + 3 q t + q t + 2 q t + q t 7 2 5 2 5 3 q q t q t q t q t