9 33

9_32

9_34

Visit 9 33's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 33's page at Knotilus! Visit 9 33's page at the original Knot Atlas! 9 33 Quick Notes

9 33 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X12,8,13,7 X8394 X2,9,3,10 X18,13,1,14 X14,5,15,6 X6,17,7,18 X16,12,17,11 X10,16,11,15
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]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

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]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n55, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

Vassiliev invariants

V2 and V3:

(1, -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 $4$ $-8$ $8$ ${\frac {62}{3}}$ ${\frac {10}{3}}$ $-32$ $-{\frac {176}{3}}$ ${\frac {64}{3}}$ $-40$ ${\frac {32}{3}}$ $32$ ${\frac {248}{3}}$ ${\frac {40}{3}}$ ${\frac {5071}{30}}$ $-{\frac {422}{15}}$ ${\frac {3062}{45}}$ ${\frac {593}{18}}$ ${\frac {271}{30}}$

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 1 7                 3   -3 5               4 1   3 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                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{12}-4q^{11}+3q^{10}+11q^{9}-25q^{8}+6q^{7}+43q^{6}-59q^{5}-3q^{4}+86q^{3}-83q^{2}-22q+115-83q^{-1}-39q^{-2}+113q^{-3}-60q^{-4}-46q^{-5}+83q^{-6}-27q^{-7}-37q^{-8}+40q^{-9}-4q^{-10}-17q^{-11}+10q^{-12}+q^{-13}-3q^{-14}+q^{-15}$
3	$q^{24}-4q^{23}+3q^{22}+7q^{21}-5q^{20}-20q^{19}+7q^{18}+53q^{17}-14q^{16}-96q^{15}-q^{14}+166q^{13}+34q^{12}-247q^{11}-94q^{10}+326q^{9}+179q^{8}-392q^{7}-276q^{6}+430q^{5}+382q^{4}-452q^{3}-460q^{2}+431q+536-407q^{-1}-567q^{-2}+342q^{-3}+595q^{-4}-282q^{-5}-577q^{-6}+193q^{-7}+550q^{-8}-111q^{-9}-486q^{-10}+23q^{-11}+411q^{-12}+38q^{-13}-312q^{-14}-82q^{-15}+214q^{-16}+97q^{-17}-131q^{-18}-83q^{-19}+64q^{-20}+61q^{-21}-25q^{-22}-36q^{-23}+7q^{-24}+17q^{-25}-2q^{-26}-5q^{-27}-q^{-28}+3q^{-29}-q^{-30}$
4	$q^{40}-4q^{39}+3q^{38}+7q^{37}-9q^{36}-19q^{34}+27q^{33}+46q^{32}-47q^{31}-34q^{30}-99q^{29}+113q^{28}+237q^{27}-73q^{26}-189q^{25}-438q^{24}+202q^{23}+752q^{22}+180q^{21}-384q^{20}-1285q^{19}-56q^{18}+1494q^{17}+1012q^{16}-227q^{15}-2483q^{14}-948q^{13}+1963q^{12}+2229q^{11}+557q^{10}-3454q^{9}-2208q^{8}+1822q^{7}+3236q^{6}+1682q^{5}-3797q^{4}-3254q^{3}+1225q^{2}+3666q+2665-3560q^{-1}-3782q^{-2}+478q^{-3}+3546q^{-4}+3288q^{-5}-2904q^{-6}-3814q^{-7}-316q^{-8}+2982q^{-9}+3566q^{-10}-1903q^{-11}-3396q^{-12}-1092q^{-13}+2012q^{-14}+3422q^{-15}-701q^{-16}-2498q^{-17}-1596q^{-18}+806q^{-19}+2717q^{-20}+288q^{-21}-1290q^{-22}-1506q^{-23}-172q^{-24}+1590q^{-25}+641q^{-26}-268q^{-27}-904q^{-28}-513q^{-29}+584q^{-30}+423q^{-31}+161q^{-32}-298q^{-33}-342q^{-34}+97q^{-35}+119q^{-36}+137q^{-37}-33q^{-38}-111q^{-39}+2q^{-40}+4q^{-41}+38q^{-42}+5q^{-43}-20q^{-44}+2q^{-45}-3q^{-46}+5q^{-47}+q^{-48}-3q^{-49}+q^{-50}$
5	$q^{60}-4q^{59}+3q^{58}+7q^{57}-9q^{56}-4q^{55}+q^{54}+q^{53}+20q^{52}+23q^{51}-37q^{50}-72q^{49}-21q^{48}+71q^{47}+165q^{46}+111q^{45}-130q^{44}-403q^{43}-335q^{42}+213q^{41}+781q^{40}+791q^{39}-80q^{38}-1353q^{37}-1752q^{36}-338q^{35}+2021q^{34}+3150q^{33}+1461q^{32}-2440q^{31}-5175q^{30}-3440q^{29}+2350q^{28}+7426q^{27}+6404q^{26}-1275q^{25}-9570q^{24}-10233q^{23}-936q^{22}+11121q^{21}+14476q^{20}+4271q^{19}-11655q^{18}-18631q^{17}-8472q^{16}+11117q^{15}+22129q^{14}+12986q^{13}-9472q^{12}-24715q^{11}-17328q^{10}+7175q^{9}+26127q^{8}+21050q^{7}-4385q^{6}-26648q^{5}-23971q^{4}+1744q^{3}+26187q^{2}+25992q+943-25297q^{-1}-27295q^{-2}-3159q^{-3}+23779q^{-4}+27888q^{-5}+5437q^{-6}-22014q^{-7}-28057q^{-8}-7391q^{-9}+19688q^{-10}+27652q^{-11}+9544q^{-12}-16995q^{-13}-26787q^{-14}-11484q^{-15}+13655q^{-16}+25228q^{-17}+13403q^{-18}-9926q^{-19}-22943q^{-20}-14763q^{-21}+5780q^{-22}+19810q^{-23}+15547q^{-24}-1769q^{-25}-15968q^{-26}-15251q^{-27}-1851q^{-28}+11622q^{-29}+13979q^{-30}+4553q^{-31}-7333q^{-32}-11671q^{-33}-6070q^{-34}+3483q^{-35}+8777q^{-36}+6375q^{-37}-573q^{-38}-5803q^{-39}-5601q^{-40}-1207q^{-41}+3155q^{-42}+4243q^{-43}+1950q^{-44}-1262q^{-45}-2742q^{-46}-1861q^{-47}+121q^{-48}+1473q^{-49}+1388q^{-50}+352q^{-51}-621q^{-52}-844q^{-53}-406q^{-54}+176q^{-55}+411q^{-56}+280q^{-57}+19q^{-58}-170q^{-59}-161q^{-60}-31q^{-61}+56q^{-62}+52q^{-63}+33q^{-64}-7q^{-65}-33q^{-66}-7q^{-67}+8q^{-68}+q^{-69}+3q^{-70}+3q^{-71}-5q^{-72}-q^{-73}+3q^{-74}-q^{-75}$
6	$q^{84}-4q^{83}+3q^{82}+7q^{81}-9q^{80}-4q^{79}-3q^{78}+21q^{77}-6q^{76}-3q^{75}+33q^{74}-65q^{73}-46q^{72}-3q^{71}+130q^{70}+86q^{69}+26q^{68}+45q^{67}-372q^{66}-385q^{65}-122q^{64}+585q^{63}+755q^{62}+634q^{61}+286q^{60}-1502q^{59}-2193q^{58}-1601q^{57}+1143q^{56}+3197q^{55}+4151q^{54}+3071q^{53}-2969q^{52}-7713q^{51}-8728q^{50}-2120q^{49}+6490q^{48}+14254q^{47}+15272q^{46}+1604q^{45}-15319q^{44}-26770q^{43}-19157q^{42}+1159q^{41}+28074q^{40}+43332q^{39}+25450q^{38}-12018q^{37}-50779q^{36}-56897q^{35}-28710q^{34}+29035q^{33}+79230q^{32}+74624q^{31}+19753q^{30}-60527q^{29}-103623q^{28}-86828q^{27}-817q^{26}+99561q^{25}+133075q^{24}+80365q^{23}-38506q^{22}-133685q^{21}-152968q^{20}-58309q^{19}+88047q^{18}+173726q^{17}+146686q^{16}+9390q^{15}-132607q^{14}-200071q^{13}-119348q^{12}+52288q^{11}+184020q^{10}+193635q^{9}+60262q^{8}-108722q^{7}-218005q^{6}-162501q^{5}+12510q^{4}+172178q^{3}+214259q^{2}+97330q-78957-214818q^{-1}-184164q^{-2}-19189q^{-3}+151725q^{-4}+216694q^{-5}+120052q^{-6}-51346q^{-7}-201224q^{-8}-192445q^{-9}-44618q^{-10}+127353q^{-11}+209410q^{-12}+136239q^{-13}-22434q^{-14}-179367q^{-15}-193704q^{-16}-70883q^{-17}+94486q^{-18}+192292q^{-19}+149869q^{-20}+13884q^{-21}-143716q^{-22}-185196q^{-23}-99105q^{-24}+48156q^{-25}+158360q^{-26}+154703q^{-27}+55581q^{-28}-90163q^{-29}-157590q^{-30}-118833q^{-31}-6329q^{-32}+103658q^{-33}+138130q^{-34}+87871q^{-35}-26967q^{-36}-106483q^{-37}-114093q^{-38}-50107q^{-39}+38832q^{-40}+95210q^{-41}+92289q^{-42}+23687q^{-43}-44665q^{-44}-80241q^{-45}-63304q^{-46}-11723q^{-47}+40900q^{-48}+65843q^{-49}+41954q^{-50}+2172q^{-51}-34698q^{-52}-45436q^{-53}-29796q^{-54}+1254q^{-55}+28461q^{-56}+30486q^{-57}+18440q^{-58}-2915q^{-59}-17972q^{-60}-21421q^{-61}-11473q^{-62}+3842q^{-63}+11205q^{-64}+12832q^{-65}+6416q^{-66}-1467q^{-67}-7641q^{-68}-7524q^{-69}-2882q^{-70}+764q^{-71}+4009q^{-72}+3883q^{-73}+2079q^{-74}-915q^{-75}-2121q^{-76}-1606q^{-77}-1015q^{-78}+333q^{-79}+916q^{-80}+1000q^{-81}+220q^{-82}-212q^{-83}-266q^{-84}-391q^{-85}-127q^{-86}+60q^{-87}+215q^{-88}+65q^{-89}+7q^{-90}+12q^{-91}-63q^{-92}-34q^{-93}-12q^{-94}+36q^{-95}+2q^{-96}-6q^{-97}+11q^{-98}-6q^{-99}-3q^{-100}-3q^{-101}+5q^{-102}+q^{-103}-3q^{-104}+q^{-105}$
7	$q^{112}-4q^{111}+3q^{110}+7q^{109}-9q^{108}-4q^{107}-3q^{106}+17q^{105}+14q^{104}-29q^{103}+7q^{102}+5q^{101}-39q^{100}-18q^{99}+4q^{98}+111q^{97}+132q^{96}-62q^{95}-86q^{94}-180q^{93}-291q^{92}-86q^{91}+121q^{90}+684q^{89}+935q^{88}+297q^{87}-399q^{86}-1462q^{85}-2136q^{84}-1371q^{83}+160q^{82}+2973q^{81}+5199q^{80}+4209q^{79}+933q^{78}-5092q^{77}-10410q^{76}-10542q^{75}-5481q^{74}+6189q^{73}+18797q^{72}+23424q^{71}+16832q^{70}-3684q^{69}-29195q^{68}-44109q^{67}-40107q^{66}-9504q^{65}+36910q^{64}+73834q^{63}+81192q^{62}+41025q^{61}-34268q^{60}-107299q^{59}-141435q^{58}-101281q^{57}+7287q^{56}+134706q^{55}+219360q^{54}+196159q^{53}+56536q^{52}-139663q^{51}-302390q^{50}-324942q^{49}-169637q^{48}+103593q^{47}+373033q^{46}+477633q^{45}+333623q^{44}-11592q^{43}-408448q^{42}-633728q^{41}-539777q^{40}-143541q^{39}+388993q^{38}+768599q^{37}+768535q^{36}+354914q^{35}-304353q^{34}-858018q^{33}-992330q^{32}-604378q^{31}+155533q^{30}+886693q^{29}+1185063q^{28}+865421q^{27}+41939q^{26}-850809q^{25}-1325913q^{24}-1110482q^{23}-265534q^{22}+759123q^{21}+1406486q^{20}+1316820q^{19}+489369q^{18}-628897q^{17}-1428674q^{16}-1472214q^{15}-692247q^{14}+481739q^{13}+1404365q^{12}+1573547q^{11}+859516q^{10}-335270q^{9}-1348375q^{8}-1628437q^{7}-987518q^{6}+204023q^{5}+1277395q^{4}+1646488q^{3}+1077731q^{2}-92091q-1201173-1641850q^{-1}-1140218q^{-2}+335q^{-3}+1128107q^{-4}+1623094q^{-5}+1182743q^{-6}+78632q^{-7}-1056484q^{-8}-1598273q^{-9}-1217127q^{-10}-151677q^{-11}+984896q^{-12}+1568187q^{-13}+1247728q^{-14}+228507q^{-15}-904233q^{-16}-1531416q^{-17}-1279794q^{-18}-315323q^{-19}+808575q^{-20}+1481362q^{-21}+1310013q^{-22}+416318q^{-23}-688591q^{-24}-1410389q^{-25}-1334094q^{-26}-530059q^{-27}+541117q^{-28}+1308914q^{-29}+1340844q^{-30}+650820q^{-31}-364860q^{-32}-1170292q^{-33}-1319360q^{-34}-765645q^{-35}+167085q^{-36}+990462q^{-37}+1256627q^{-38}+859423q^{-39}+40157q^{-40}-774451q^{-41}-1145254q^{-42}-913617q^{-43}-236456q^{-44}+533062q^{-45}+982955q^{-46}+914623q^{-47}+400916q^{-48}-286916q^{-49}-779577q^{-50}-854456q^{-51}-512171q^{-52}+60416q^{-53}+551937q^{-54}+736826q^{-55}+558450q^{-56}+122500q^{-57}-326102q^{-58}-577412q^{-59}-537227q^{-60}-243628q^{-61}+127750q^{-62}+399023q^{-63}+460243q^{-64}+297765q^{-65}+22031q^{-66}-229324q^{-67}-348951q^{-68}-290825q^{-69}-112897q^{-70}+90343q^{-71}+228197q^{-72}+241102q^{-73}+148133q^{-74}+4595q^{-75}-121531q^{-76}-171176q^{-77}-140049q^{-78}-54247q^{-79}+42625q^{-80}+101582q^{-81}+107960q^{-82}+67702q^{-83}+4057q^{-84}-46924q^{-85}-69150q^{-86}-58261q^{-87}-23549q^{-88}+12089q^{-89}+35831q^{-90}+39979q^{-91}+25351q^{-92}+4737q^{-93}-13638q^{-94}-22439q^{-95}-18756q^{-96}-9240q^{-97}+1991q^{-98}+9813q^{-99}+10926q^{-100}+7989q^{-101}+2220q^{-102}-3152q^{-103}-5102q^{-104}-4780q^{-105}-2525q^{-106}+206q^{-107}+1678q^{-108}+2427q^{-109}+1790q^{-110}+395q^{-111}-420q^{-112}-917q^{-113}-791q^{-114}-376q^{-115}-118q^{-116}+288q^{-117}+409q^{-118}+188q^{-119}+54q^{-120}-88q^{-121}-88q^{-122}-43q^{-123}-85q^{-124}+55q^{-126}+29q^{-127}+13q^{-128}-17q^{-129}-5q^{-130}+11q^{-131}-13q^{-132}-6q^{-133}+6q^{-134}+3q^{-135}+3q^{-136}-5q^{-137}-q^{-138}+3q^{-139}-q^{-140}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 33]]
Out[2]=	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[3]:=	GaussCode[Knot[9, 33]]
Out[3]=	GaussCode[1, -4, 3, -1, 6, -7, 2, -3, 4, -9, 8, -2, 5, -6, 9, -8, 7, -5]
In[4]:=	DTCode[Knot[9, 33]]
Out[4]=	DTCode[4, 8, 14, 12, 2, 16, 18, 10, 6]
In[5]:=	br = BR[Knot[9, 33]]
Out[5]=	BR[4, {-1, 2, -1, 2, 2, -1, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 33]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 33]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 33]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 33]][t]
Out[10]=	-3 6 14 2 3 19 - t + -- - -- - 14 t + 6 t - t 2 t t
In[11]:=	Conway[Knot[9, 33]][z]
Out[11]=	2 6 1 + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 33], Knot[11, NonAlternating, 55]}
In[13]:=	{KnotDet[Knot[9, 33]], KnotSignature[Knot[9, 33]]}
Out[13]=	{61, 0}
In[14]:=	Jones[Knot[9, 33]][q]
Out[14]=	-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[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 33]}
In[16]:=	A2Invariant[Knot[9, 33]][q]
Out[16]=	-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[17]:=	HOMFLYPT[Knot[9, 33]][a, z]
Out[17]=	2 4 2 4 2 z 2 2 4 2 4 z 2 4 6 2 a - a - 3 z + -- + 4 a z - a z - 3 z + -- + 2 a z - z 2 2 a a
In[18]:=	Kauffman[Knot[9, 33]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[9, 33]], Vassiliev[3][Knot[9, 33]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[9, 33]][q, t]
Out[20]=	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
In[21]:=	ColouredJones[Knot[9, 33], 2][q]
Out[21]=	-15 3 -13 10 17 4 40 37 27 83 46 115 + q - --- + q + --- - --- - --- + -- - -- - -- + -- - -- - 14 12 11 10 9 8 7 6 5 q q q q q q q q q 60 113 39 83 2 3 4 5 6 -- + --- - -- - -- - 22 q - 83 q + 86 q - 3 q - 59 q + 43 q + 4 3 2 q q q q 7 8 9 10 11 12 6 q - 25 q + 11 q + 3 q - 4 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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