9 33

9_32

9_34

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9 33 Quick Notes

9 33 Further Notes and Views

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_12,8,13,7 X₈₃₉₄ X_2,9,3,10 X_18,13,1,14 X_14,5,15,6 X_6,17,7,18 X_16,12,17,11 X_10,16,11,15
Gauss code	1, -4, 3, -1, 6, -7, 2, -3, 4, -9, 8, -2, 5, -6, 9, -8, 7, -5
Dowker-Thistlethwaite code	4 8 14 12 2 16 18 10 6
Conway Notation	[.21.2]

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	$\{4,6\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-5]
Hyperbolic Volume	13.2805
A-Polynomial	See Data:9 33/A-polynomial

[edit Notes for 9 33's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$1$
Topological 4 genus	$1$
Concordance genus	$3$
Rasmussen s-Invariant	0

[edit Notes for 9 33's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-14t+19-14t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 61, 0 }
Jones polynomial	$q^{4}-4q^{3}+7q^{2}-9q+11-10q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+4a^{2}z^{2}+z^{2}a^{-2}-3z^{2}-a^{4}+2a^{2}$
Kauffman polynomial (db, data sources)	$2a^{2}z^{8}+2z^{8}+4a^{3}z^{7}+10az^{7}+6z^{7}a^{-1}+3a^{4}z^{6}+5a^{2}z^{6}+7z^{6}a^{-2}+9z^{6}+a^{5}z^{5}-6a^{3}z^{5}-16az^{5}-5z^{5}a^{-1}+4z^{5}a^{-3}-6a^{4}z^{4}-16a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}-20z^{4}-2a^{5}z^{3}+a^{3}z^{3}+5az^{3}-z^{3}a^{-1}-3z^{3}a^{-3}+4a^{4}z^{2}+10a^{2}z^{2}+3z^{2}a^{-2}+9z^{2}+a^{5}z+a^{3}z-a^{4}-2a^{2}$
The A2 invariant	$-q^{16}+q^{12}-2q^{10}+2q^{8}+3q^{2}-1+3q^{-2}-2q^{-4}+q^{-8}-2q^{-10}+q^{-12}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-7q^{70}-2q^{68}+17q^{66}-35q^{64}+50q^{62}-52q^{60}+28q^{58}+13q^{56}-67q^{54}+113q^{52}-123q^{50}+92q^{48}-23q^{46}-64q^{44}+129q^{42}-148q^{40}+111q^{38}-31q^{36}-54q^{34}+104q^{32}-98q^{30}+43q^{28}+39q^{26}-101q^{24}+121q^{22}-81q^{20}-5q^{18}+102q^{16}-171q^{14}+188q^{12}-138q^{10}+43q^{8}+71q^{6}-159q^{4}+198q^{2}-170+88q^{-2}+17q^{-4}-101q^{-6}+132q^{-8}-101q^{-10}+29q^{-12}+54q^{-14}-99q^{-16}+89q^{-18}-33q^{-20}-51q^{-22}+119q^{-24}-142q^{-26}+110q^{-28}-38q^{-30}-44q^{-32}+103q^{-34}-124q^{-36}+105q^{-38}-58q^{-40}+6q^{-42}+31q^{-44}-54q^{-46}+53q^{-48}-36q^{-50}+20q^{-52}-2q^{-54}-6q^{-56}+9q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_33.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-3q^{7}+3q^{5}-q^{3}+q+2q^{-1}-2q^{-3}+3q^{-5}-3q^{-7}+q^{-9}$
2	$q^{32}-2q^{30}-q^{28}+8q^{26}-6q^{24}-11q^{22}+19q^{20}-q^{18}-24q^{16}+19q^{14}+10q^{12}-23q^{10}+7q^{8}+14q^{6}-9q^{4}-7q^{2}+10+10q^{-2}-19q^{-4}+24q^{-8}-19q^{-10}-10q^{-12}+24q^{-14}-8q^{-16}-11q^{-18}+10q^{-20}-3q^{-24}+q^{-26}$
3	$-q^{63}+2q^{61}+q^{59}-4q^{57}-5q^{55}+9q^{53}+17q^{51}-14q^{49}-37q^{47}+7q^{45}+64q^{43}+17q^{41}-89q^{39}-53q^{37}+97q^{35}+98q^{33}-83q^{31}-142q^{29}+55q^{27}+160q^{25}-14q^{23}-163q^{21}-24q^{19}+146q^{17}+55q^{15}-116q^{13}-71q^{11}+78q^{9}+88q^{7}-37q^{5}-96q^{3}-7q+100q^{-1}+55q^{-3}-99q^{-5}-100q^{-7}+84q^{-9}+144q^{-11}-59q^{-13}-163q^{-15}+19q^{-17}+164q^{-19}+19q^{-21}-141q^{-23}-48q^{-25}+103q^{-27}+55q^{-29}-58q^{-31}-50q^{-33}+26q^{-35}+35q^{-37}-11q^{-39}-15q^{-41}+q^{-43}+7q^{-45}-3q^{-49}+q^{-51}$
4	$q^{104}-2q^{102}-q^{100}+4q^{98}+q^{96}+2q^{94}-15q^{92}-11q^{90}+22q^{88}+29q^{86}+29q^{84}-62q^{82}-100q^{80}-q^{78}+114q^{76}+209q^{74}-22q^{72}-287q^{70}-263q^{68}+41q^{66}+528q^{64}+357q^{62}-249q^{60}-678q^{58}-460q^{56}+546q^{54}+887q^{52}+285q^{50}-737q^{48}-1090q^{46}+37q^{44}+1015q^{42}+925q^{40}-283q^{38}-1272q^{36}-567q^{34}+639q^{32}+1143q^{30}+245q^{28}-957q^{26}-813q^{24}+157q^{22}+933q^{20}+515q^{18}-486q^{16}-764q^{14}-200q^{12}+594q^{10}+626q^{8}-30q^{6}-653q^{4}-533q^{2}+214+742q^{-2}+505q^{-4}-478q^{-6}-908q^{-8}-311q^{-10}+735q^{-12}+1078q^{-14}-47q^{-16}-1054q^{-18}-913q^{-20}+347q^{-22}+1318q^{-24}+534q^{-26}-683q^{-28}-1152q^{-30}-260q^{-32}+938q^{-34}+781q^{-36}-51q^{-38}-793q^{-40}-535q^{-42}+312q^{-44}+507q^{-46}+254q^{-48}-261q^{-50}-350q^{-52}-11q^{-54}+144q^{-56}+170q^{-58}-21q^{-60}-107q^{-62}-27q^{-64}+7q^{-66}+45q^{-68}+6q^{-70}-18q^{-72}-3q^{-74}-2q^{-76}+7q^{-78}-3q^{-82}+q^{-84}$
5	$-q^{155}+2q^{153}+q^{151}-4q^{149}-q^{147}+2q^{145}+4q^{143}+9q^{141}+3q^{139}-25q^{137}-35q^{135}-5q^{133}+46q^{131}+94q^{129}+70q^{127}-58q^{125}-221q^{123}-235q^{121}-7q^{119}+348q^{117}+555q^{115}+310q^{113}-364q^{111}-1004q^{109}-932q^{107}+45q^{105}+1342q^{103}+1869q^{101}+852q^{99}-1269q^{97}-2883q^{95}-2321q^{93}+449q^{91}+3483q^{89}+4137q^{87}+1278q^{85}-3263q^{83}-5786q^{81}-3654q^{79}+1968q^{77}+6658q^{75}+6189q^{73}+321q^{71}-6439q^{69}-8261q^{67}-3059q^{65}+5080q^{63}+9299q^{61}+5719q^{59}-2916q^{57}-9238q^{55}-7670q^{53}+518q^{51}+8149q^{49}+8637q^{47}+1662q^{45}-6495q^{43}-8639q^{41}-3221q^{39}+4654q^{37}+7933q^{35}+4089q^{33}-2980q^{31}-6839q^{29}-4415q^{27}+1618q^{25}+5711q^{23}+4441q^{21}-578q^{19}-4685q^{17}-4449q^{15}-358q^{13}+3874q^{11}+4636q^{9}+1353q^{7}-3141q^{5}-5037q^{3}-2629q+2274q^{-1}+5598q^{-3}+4247q^{-5}-1081q^{-7}-6023q^{-9}-6083q^{-11}-652q^{-13}+5991q^{-15}+7924q^{-17}+2837q^{-19}-5227q^{-21}-9225q^{-23}-5283q^{-25}+3573q^{-27}+9657q^{-29}+7474q^{-31}-1241q^{-33}-8894q^{-35}-8890q^{-37}-1354q^{-39}+7044q^{-41}+9129q^{-43}+3583q^{-45}-4489q^{-47}-8184q^{-49}-4898q^{-51}+1895q^{-53}+6290q^{-55}+5125q^{-57}+182q^{-59}-4094q^{-61}-4423q^{-63}-1321q^{-65}+2102q^{-67}+3189q^{-69}+1648q^{-71}-711q^{-73}-1951q^{-75}-1400q^{-77}+17q^{-79}+967q^{-81}+917q^{-83}+237q^{-85}-379q^{-87}-521q^{-89}-207q^{-91}+124q^{-93}+217q^{-95}+129q^{-97}-16q^{-99}-86q^{-101}-64q^{-103}+4q^{-105}+32q^{-107}+16q^{-109}-q^{-111}-6q^{-113}-6q^{-115}-2q^{-117}+7q^{-119}-3q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{12}-2q^{10}+2q^{8}+3q^{2}-1+3q^{-2}-2q^{-4}+q^{-8}-2q^{-10}+q^{-12}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+60q^{36}-114q^{34}+194q^{32}-296q^{30}+411q^{28}-526q^{26}+600q^{24}-622q^{22}+565q^{20}-416q^{18}+186q^{16}+102q^{14}-411q^{12}+716q^{10}-966q^{8}+1136q^{6}-1197q^{4}+1156q^{2}-1002+762q^{-2}-462q^{-4}+146q^{-6}+148q^{-8}-384q^{-10}+538q^{-12}-606q^{-14}+596q^{-16}-528q^{-18}+423q^{-20}-312q^{-22}+216q^{-24}-134q^{-26}+73q^{-28}-38q^{-30}+18q^{-32}-6q^{-34}+q^{-36}$
2,0	$q^{42}-2q^{38}-q^{36}+4q^{34}+4q^{32}-6q^{30}-6q^{28}+6q^{26}+4q^{24}-10q^{22}-6q^{20}+9q^{18}+6q^{16}-9q^{14}+q^{12}+11q^{10}-3q^{8}-2q^{6}+6q^{4}+q^{2}-5+4q^{-2}+6q^{-4}-10q^{-6}-5q^{-8}+11q^{-10}+4q^{-12}-13q^{-14}+q^{-16}+10q^{-18}-q^{-20}-6q^{-22}-q^{-24}+5q^{-26}-q^{-28}-2q^{-30}+q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+4q^{28}-9q^{26}+3q^{24}+9q^{22}-18q^{20}+6q^{18}+13q^{16}-19q^{14}+6q^{12}+13q^{10}-10q^{8}-q^{6}+8q^{4}+3q^{2}-3-3q^{-2}+14q^{-4}-5q^{-6}-15q^{-8}+18q^{-10}-4q^{-12}-16q^{-14}+16q^{-16}-q^{-18}-9q^{-20}+8q^{-22}-3q^{-26}+q^{-28}$
1,0,0	$-q^{21}-q^{17}+q^{15}-2q^{13}+3q^{11}-q^{9}+2q^{7}+2q^{3}+q+2q^{-3}-2q^{-5}+q^{-7}-2q^{-9}+2q^{-11}-2q^{-13}+q^{-15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}-q^{40}+q^{38}+2q^{36}-3q^{34}-5q^{32}+3q^{30}+4q^{28}-11q^{26}-5q^{24}+14q^{22}+q^{20}-14q^{18}+6q^{16}+15q^{14}-8q^{12}-9q^{10}+13q^{8}+7q^{6}-13q^{4}+8q^{2}+17-10q^{-2}-6q^{-4}+17q^{-6}-4q^{-8}-18q^{-10}+4q^{-12}+11q^{-14}-9q^{-16}-9q^{-18}+11q^{-20}+6q^{-22}-8q^{-24}-q^{-26}+6q^{-28}-2q^{-30}-2q^{-32}+q^{-34}$
1,0,0,0	$-q^{26}-q^{22}-q^{20}+q^{18}-2q^{16}+3q^{14}+q^{10}+2q^{8}+2q^{4}+2-q^{-2}+2q^{-4}-2q^{-6}+q^{-8}-q^{-10}-q^{-12}+2q^{-14}-2q^{-16}+q^{-18}$

B2 Invariants.

Weight

Invariant

0,1

-q^{34}+2q^{32}-5q^{30}+8q^{28}-13q^{26}+17q^{24}-19q^{22}+20q^{20}-18q^{18}+13q^{16}-5q^{14}-4q^{12}+15q^{10}-24q^{8}+33q^{6}-36q^{4}+39q^{2}-35+29q^{-2}-18q^{-4}+9q^{-6}+q^{-8}-10q^{-10}+16q^{-12}-20q^{-14}+20q^{-16}-19q^{-18}+15q^{-20}-10q^{-22}+6q^{-24}-3q^{-26}+q^{-28}

1,0

q^{56}-2q^{52}-2q^{50}+3q^{48}+6q^{46}-q^{44}-11q^{42}-6q^{40}+12q^{38}+14q^{36}-7q^{34}-21q^{32}-4q^{30}+21q^{28}+13q^{26}-15q^{24}-18q^{22}+6q^{20}+19q^{18}+q^{16}-15q^{14}-4q^{12}+14q^{10}+7q^{8}-10q^{6}-9q^{4}+10q^{2}+14-5q^{-2}-15q^{-4}+4q^{-6}+17q^{-8}+2q^{-10}-19q^{-12}-10q^{-14}+16q^{-16}+17q^{-18}-10q^{-20}-22q^{-22}-q^{-24}+18q^{-26}+10q^{-28}-10q^{-30}-12q^{-32}+2q^{-34}+9q^{-36}+3q^{-38}-3q^{-40}-3q^{-42}+q^{-46}

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-7q^{70}-2q^{68}+17q^{66}-35q^{64}+50q^{62}-52q^{60}+28q^{58}+13q^{56}-67q^{54}+113q^{52}-123q^{50}+92q^{48}-23q^{46}-64q^{44}+129q^{42}-148q^{40}+111q^{38}-31q^{36}-54q^{34}+104q^{32}-98q^{30}+43q^{28}+39q^{26}-101q^{24}+121q^{22}-81q^{20}-5q^{18}+102q^{16}-171q^{14}+188q^{12}-138q^{10}+43q^{8}+71q^{6}-159q^{4}+198q^{2}-170+88q^{-2}+17q^{-4}-101q^{-6}+132q^{-8}-101q^{-10}+29q^{-12}+54q^{-14}-99q^{-16}+89q^{-18}-33q^{-20}-51q^{-22}+119q^{-24}-142q^{-26}+110q^{-28}-38q^{-30}-44q^{-32}+103q^{-34}-124q^{-36}+105q^{-38}-58q^{-40}+6q^{-42}+31q^{-44}-54q^{-46}+53q^{-48}-36q^{-50}+20q^{-52}-2q^{-54}-6q^{-56}+9q^{-58}-10q^{-60}+6q^{-62}-3q^{-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.

In[3]:= K = Knot["9 33"];

In[4]:= Alexander[K][t]

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

Out[4]= $-t^{3}+6t^{2}-14t+19-14t^{-1}+6t^{-2}-t^{-3}$

In[5]:= Conway[K][z]

Out[5]= $-z^{6}+z^{2}+1$

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]= { 61, 0 }

In[8]:= Jones[K][q]

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

Out[8]= $q^{4}-4q^{3}+7q^{2}-9q+11-10q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+4a^{2}z^{2}+z^{2}a^{-2}-3z^{2}-a^{4}+2a^{2}$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $2a^{2}z^{8}+2z^{8}+4a^{3}z^{7}+10az^{7}+6z^{7}a^{-1}+3a^{4}z^{6}+5a^{2}z^{6}+7z^{6}a^{-2}+9z^{6}+a^{5}z^{5}-6a^{3}z^{5}-16az^{5}-5z^{5}a^{-1}+4z^{5}a^{-3}-6a^{4}z^{4}-16a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}-20z^{4}-2a^{5}z^{3}+a^{3}z^{3}+5az^{3}-z^{3}a^{-1}-3z^{3}a^{-3}+4a^{4}z^{2}+10a^{2}z^{2}+3z^{2}a^{-2}+9z^{2}+a^{5}z+a^{3}z-a^{4}-2a^{2}$

Vassiliev invariants

V₂ and V₃:

(1, -1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

4

-8

8

{\frac {62}{3}}

{\frac {10}{3}}

-32

-{\frac {176}{3}}

{\frac {64}{3}}

-40

{\frac {32}{3}}

32

{\frac {248}{3}}

{\frac {40}{3}}

{\frac {5071}{30}}

-{\frac {422}{15}}

{\frac {3062}{45}}

{\frac {593}{18}}

{\frac {271}{30}}

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 33. 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

7

3

-3

5

4

1

3

5

3

-2

1

6

4

2

-1

5

6

1

-3

4

5

-1

-5

2

5

3

-7

1

4

-3

-9

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, 33]]
Out[2]=	9
In[3]:=	PD[Knot[9, 33]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 8, 13, 7], X[8, 3, 9, 4], X[2, 9, 3, 10], X[18, 13, 1, 14], X[14, 5, 15, 6], X[6, 17, 7, 18], X[16, 12, 17, 11], X[10, 16, 11, 15]]
In[4]:=	GaussCode[Knot[9, 33]]
Out[4]=	GaussCode[1, -4, 3, -1, 6, -7, 2, -3, 4, -9, 8, -2, 5, -6, 9, -8, 7, -5]
In[5]:=	BR[Knot[9, 33]]
Out[5]=	BR[4, {-1, 2, -1, 2, 2, -1, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 33]][t]
Out[6]=	-3 6 14 2 3 19 - t + -- - -- - 14 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[9, 33]][z]
Out[7]=	2 6 1 + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 33], Knot[11, NonAlternating, 55]}
In[9]:=	{KnotDet[Knot[9, 33]], KnotSignature[Knot[9, 33]]}
Out[9]=	{61, 0}
In[10]:=	J=Jones[Knot[9, 33]][q]
Out[10]=	-5 3 6 9 10 2 3 4 11 - q + -- - -- + -- - -- - 9 q + 7 q - 4 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, 33]}
In[12]:=	A2Invariant[Knot[9, 33]][q]
Out[12]=	-16 -12 2 2 3 2 4 8 10 12 -1 - q + q - --- + -- + -- + 3 q - 2 q + q - 2 q + q 10 8 2 q q q
In[13]:=	Kauffman[Knot[9, 33]][a, z]
Out[13]=	2 3 2 4 3 5 2 3 z 2 2 4 2 3 z -2 a - a + a z + a z + 9 z + ---- + 10 a z + 4 a z - ---- - 2 3 a a 3 4 4 z 3 3 3 5 3 4 z 9 z 2 4 -- + 5 a z + a z - 2 a z - 20 z + -- - ---- - 16 a z - a 4 2 a a 5 5 6 4 4 4 z 5 z 5 3 5 5 5 6 7 z 6 a z + ---- - ---- - 16 a z - 6 a z + a z + 9 z + ---- + 3 a 2 a a 7 2 6 4 6 6 z 7 3 7 8 2 8 5 a z + 3 a z + ---- + 10 a z + 4 a z + 2 z + 2 a z a
In[14]:=	{Vassiliev[2][Knot[9, 33]], Vassiliev[3][Knot[9, 33]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[9, 33]][q, t]
Out[15]=	6 1 2 1 4 2 5 4 - + 6 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 5 5 3 3 2 5 2 5 3 7 3 ---- + --- + 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + 3 q t q t 9 4 q t

9 33

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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