9 34

9_33

9_35

Visit 9 34's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 34's page at Knotilus! Visit 9 34's page at the original Knot Atlas! 9 34 Quick Notes

9 34 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X18,11,1,12 X12,17,13,18
Gauss code	1, -4, 3, -7, 6, -1, 2, -3, 5, -6, 8, -9, 7, -5, 4, -2, 9, -8
Dowker-Thistlethwaite code	6 8 10 16 14 18 4 2 12
Conway Notation	[8*20]

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

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible 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 14.3446 A-Polynomial See Data:9 34/A-polynomial

[edit Notes for 9 34'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 34's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-16t+23-16t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 69, 0 }
Jones polynomial	$q^{4}-4q^{3}+8q^{2}-10q+12-12q^{-1}+10q^{-2}-7q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+3a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+a^{2}+a^{-2}-1$
Kauffman polynomial (db, data sources)	$3a^{2}z^{8}+3z^{8}+6a^{3}z^{7}+14az^{7}+8z^{7}a^{-1}+4a^{4}z^{6}+5a^{2}z^{6}+8z^{6}a^{-2}+9z^{6}+a^{5}z^{5}-11a^{3}z^{5}-26az^{5}-10z^{5}a^{-1}+4z^{5}a^{-3}-7a^{4}z^{4}-19a^{2}z^{4}-10z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}-a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}+3a^{4}z^{2}+10a^{2}z^{2}+4z^{2}a^{-2}+11z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$
The A2 invariant	$-q^{16}+q^{14}+2q^{12}-2q^{10}+2q^{8}-q^{6}-q^{4}+2q^{2}-2+3q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+14q^{72}-12q^{70}-q^{68}+27q^{66}-55q^{64}+83q^{62}-84q^{60}+44q^{58}+24q^{56}-112q^{54}+181q^{52}-188q^{50}+127q^{48}-11q^{46}-114q^{44}+205q^{42}-213q^{40}+135q^{38}-7q^{36}-116q^{34}+167q^{32}-131q^{30}+24q^{28}+103q^{26}-178q^{24}+183q^{22}-102q^{20}-37q^{18}+174q^{16}-269q^{14}+272q^{12}-182q^{10}+32q^{8}+130q^{6}-244q^{4}+280q^{2}-217+85q^{-2}+58q^{-4}-170q^{-6}+191q^{-8}-117q^{-10}-6q^{-12}+123q^{-14}-165q^{-16}+125q^{-18}-16q^{-20}-113q^{-22}+195q^{-24}-206q^{-26}+141q^{-28}-26q^{-30}-88q^{-32}+159q^{-34}-165q^{-36}+126q^{-38}-55q^{-40}-10q^{-42}+48q^{-44}-68q^{-46}+60q^{-48}-38q^{-50}+18q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_34.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+3q^{9}-3q^{7}+3q^{5}-2q^{3}+2q^{-1}-2q^{-3}+4q^{-5}-3q^{-7}+q^{-9}$
2	$q^{32}-3q^{30}-q^{28}+12q^{26}-8q^{24}-16q^{22}+25q^{20}+q^{18}-32q^{16}+22q^{14}+16q^{12}-28q^{10}+6q^{8}+19q^{6}-10q^{4}-13q^{2}+11+14q^{-2}-25q^{-4}-2q^{-6}+33q^{-8}-21q^{-10}-15q^{-12}+30q^{-14}-7q^{-16}-15q^{-18}+10q^{-20}+q^{-22}-3q^{-24}+q^{-26}$
3	$-q^{63}+3q^{61}+q^{59}-8q^{57}-7q^{55}+16q^{53}+30q^{51}-24q^{49}-64q^{47}+6q^{45}+107q^{43}+40q^{41}-138q^{39}-106q^{37}+135q^{35}+178q^{33}-96q^{31}-232q^{29}+37q^{27}+250q^{25}+28q^{23}-234q^{21}-86q^{19}+198q^{17}+119q^{15}-144q^{13}-137q^{11}+85q^{9}+150q^{7}-25q^{5}-151q^{3}-42q+150q^{-1}+110q^{-3}-132q^{-5}-180q^{-7}+98q^{-9}+234q^{-11}-44q^{-13}-255q^{-15}-22q^{-17}+239q^{-19}+81q^{-21}-187q^{-23}-113q^{-25}+121q^{-27}+105q^{-29}-53q^{-31}-82q^{-33}+15q^{-35}+48q^{-37}-2q^{-39}-18q^{-41}-3q^{-43}+7q^{-45}+q^{-47}-3q^{-49}+q^{-51}$
4	$q^{104}-3q^{102}-q^{100}+8q^{98}+3q^{96}-q^{94}-30q^{92}-20q^{90}+45q^{88}+67q^{86}+55q^{84}-123q^{82}-211q^{80}-29q^{78}+231q^{76}+437q^{74}+26q^{72}-539q^{70}-606q^{68}-39q^{66}+965q^{64}+866q^{62}-237q^{60}-1287q^{58}-1138q^{56}+669q^{54}+1720q^{52}+979q^{50}-1012q^{48}-2141q^{46}-546q^{44}+1536q^{42}+2020q^{40}+111q^{38}-2061q^{36}-1539q^{34}+565q^{32}+2042q^{30}+996q^{28}-1229q^{26}-1668q^{24}-264q^{22}+1418q^{20}+1232q^{18}-399q^{16}-1348q^{14}-721q^{12}+758q^{10}+1234q^{8}+313q^{6}-1013q^{4}-1169q^{2}+46+1278q^{-2}+1203q^{-4}-513q^{-6}-1679q^{-8}-981q^{-10}+1003q^{-12}+2097q^{-14}+482q^{-16}-1631q^{-18}-2013q^{-20}+11q^{-22}+2191q^{-24}+1535q^{-26}-628q^{-28}-2103q^{-30}-1083q^{-32}+1160q^{-34}+1627q^{-36}+495q^{-38}-1100q^{-40}-1210q^{-42}+49q^{-44}+779q^{-46}+690q^{-48}-151q^{-50}-563q^{-52}-227q^{-54}+109q^{-56}+291q^{-58}+71q^{-60}-111q^{-62}-74q^{-64}-24q^{-66}+53q^{-68}+22q^{-70}-13q^{-72}-6q^{-74}-6q^{-76}+7q^{-78}+q^{-80}-3q^{-82}+q^{-84}$
5	$-q^{155}+3q^{153}+q^{151}-8q^{149}-3q^{147}+5q^{145}+15q^{143}+20q^{141}-q^{139}-59q^{137}-86q^{135}-15q^{133}+123q^{131}+240q^{129}+177q^{127}-129q^{125}-545q^{123}-631q^{121}-100q^{119}+802q^{117}+1419q^{115}+982q^{113}-623q^{111}-2384q^{109}-2611q^{107}-542q^{105}+2761q^{103}+4754q^{101}+3135q^{99}-1778q^{97}-6538q^{95}-6794q^{93}-1213q^{91}+6706q^{89}+10559q^{87}+6079q^{85}-4446q^{83}-12971q^{81}-11708q^{79}-320q^{77}+12798q^{75}+16578q^{73}+6682q^{71}-9776q^{69}-19250q^{67}-13013q^{65}+4536q^{63}+19015q^{61}+17848q^{59}+1556q^{57}-16212q^{55}-20243q^{53}-7017q^{51}+11839q^{49}+20042q^{47}+10881q^{45}-7083q^{43}-17954q^{41}-12766q^{39}+2989q^{37}+14851q^{35}+12931q^{33}+56q^{31}-11650q^{29}-12114q^{27}-1995q^{25}+8933q^{23}+11006q^{21}+3244q^{19}-6807q^{17}-10249q^{15}-4442q^{13}+5184q^{11}+10107q^{9}+6046q^{7}-3522q^{5}-10451q^{3}-8507q+1299q^{-1}+10908q^{-3}+11682q^{-5}+1980q^{-7}-10674q^{-9}-15162q^{-11}-6449q^{-13}+9048q^{-15}+18039q^{-17}+11723q^{-19}-5574q^{-21}-19230q^{-23}-16847q^{-25}+368q^{-27}+17919q^{-29}+20551q^{-31}+5661q^{-33}-13978q^{-35}-21629q^{-37}-11107q^{-39}+8117q^{-41}+19635q^{-43}+14545q^{-45}-1807q^{-47}-15140q^{-49}-15083q^{-51}-3294q^{-53}+9385q^{-55}+12979q^{-57}+6171q^{-59}-4122q^{-61}-9312q^{-63}-6523q^{-65}+394q^{-67}+5411q^{-69}+5270q^{-71}+1385q^{-73}-2442q^{-75}-3377q^{-77}-1635q^{-79}+677q^{-81}+1733q^{-83}+1211q^{-85}+37q^{-87}-739q^{-89}-650q^{-91}-151q^{-93}+224q^{-95}+283q^{-97}+113q^{-99}-60q^{-101}-110q^{-103}-41q^{-105}+20q^{-107}+27q^{-109}+11q^{-111}-q^{-113}-9q^{-115}-6q^{-117}+7q^{-119}+q^{-121}-3q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+2q^{12}-2q^{10}+2q^{8}-q^{6}-q^{4}+2q^{2}-2+3q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}$
1,1	$q^{44}-6q^{42}+20q^{40}-50q^{38}+107q^{36}-204q^{34}+344q^{32}-514q^{30}+701q^{28}-864q^{26}+950q^{24}-938q^{22}+791q^{20}-502q^{18}+98q^{16}+374q^{14}-840q^{12}+1276q^{10}-1606q^{8}+1788q^{6}-1806q^{4}+1650q^{2}-1352+934q^{-2}-456q^{-4}-16q^{-6}+432q^{-8}-726q^{-10}+888q^{-12}-918q^{-14}+850q^{-16}-712q^{-18}+538q^{-20}-376q^{-22}+248q^{-24}-148q^{-26}+77q^{-28}-38q^{-30}+18q^{-32}-6q^{-34}+q^{-36}$
2,0	$q^{42}-q^{40}-3q^{38}+7q^{34}+5q^{32}-10q^{30}-7q^{28}+9q^{26}+7q^{24}-13q^{22}-7q^{20}+13q^{18}+8q^{16}-11q^{14}-q^{12}+13q^{10}-5q^{8}-3q^{6}+4q^{4}-q^{2}-7+5q^{-2}+7q^{-4}-13q^{-6}-4q^{-8}+16q^{-10}+6q^{-12}-17q^{-14}+q^{-16}+14q^{-18}-9q^{-22}-3q^{-24}+6q^{-26}-2q^{-30}+q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-3q^{32}+q^{30}+6q^{28}-12q^{26}+6q^{24}+13q^{22}-22q^{20}+10q^{18}+15q^{16}-24q^{14}+7q^{12}+14q^{10}-13q^{8}-2q^{6}+7q^{4}+3q^{2}-6-6q^{-2}+18q^{-4}-6q^{-6}-17q^{-8}+25q^{-10}-5q^{-12}-18q^{-14}+20q^{-16}-3q^{-18}-10q^{-20}+9q^{-22}-3q^{-26}+q^{-28}$
1,0,0	$-q^{21}+q^{19}+2q^{15}-2q^{13}+3q^{11}-2q^{9}+q^{7}-q^{5}+q^{3}-q^{-1}+2q^{-3}-2q^{-5}+2q^{-7}-q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}-q^{42}-3q^{40}+2q^{38}+4q^{36}-4q^{34}-5q^{32}+9q^{30}+7q^{28}-14q^{26}-4q^{24}+20q^{22}-q^{20}-20q^{18}+9q^{16}+17q^{14}-14q^{12}-12q^{10}+16q^{8}+2q^{6}-19q^{4}+10q^{2}+18-16q^{-2}-5q^{-4}+24q^{-6}-5q^{-8}-21q^{-10}+8q^{-12}+15q^{-14}-12q^{-16}-10q^{-18}+14q^{-20}+7q^{-22}-10q^{-24}-q^{-26}+7q^{-28}-2q^{-30}-2q^{-32}+q^{-34}$
1,0,0,0	$-q^{26}+q^{24}+2q^{18}-2q^{16}+3q^{14}-q^{12}+q^{8}-q^{6}+q^{4}-q^{2}+1-2q^{-2}+2q^{-4}-2q^{-6}+2q^{-8}+3q^{-14}-2q^{-16}+q^{-18}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+3q^{32}-7q^{30}+12q^{28}-18q^{26}+24q^{24}-25q^{22}+26q^{20}-22q^{18}+15q^{16}-4q^{14}-9q^{12}+22q^{10}-35q^{8}+44q^{6}-49q^{4}+49q^{2}-44+34q^{-2}-20q^{-4}+8q^{-6}+5q^{-8}-15q^{-10}+23q^{-12}-26q^{-14}+26q^{-16}-23q^{-18}+18q^{-20}-11q^{-22}+6q^{-24}-3q^{-26}+q^{-28}$
1,0	$q^{56}-3q^{52}-3q^{50}+4q^{48}+9q^{46}-q^{44}-15q^{42}-8q^{40}+18q^{38}+19q^{36}-10q^{34}-27q^{32}-3q^{30}+28q^{28}+15q^{26}-21q^{24}-22q^{22}+10q^{20}+23q^{18}-q^{16}-20q^{14}-3q^{12}+18q^{10}+7q^{8}-16q^{6}-11q^{4}+13q^{2}+16-10q^{-2}-20q^{-4}+7q^{-6}+23q^{-8}+q^{-10}-25q^{-12}-10q^{-14}+23q^{-16}+22q^{-18}-14q^{-20}-27q^{-22}+q^{-24}+23q^{-26}+10q^{-28}-13q^{-30}-14q^{-32}+3q^{-34}+10q^{-36}+3q^{-38}-3q^{-40}-3q^{-42}+q^{-46}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-3q^{44}+4q^{42}-6q^{40}+10q^{38}-15q^{36}+17q^{34}-18q^{32}+22q^{30}-21q^{28}+18q^{26}-15q^{24}+14q^{22}-5q^{20}-5q^{18}+10q^{16}-14q^{14}+25q^{12}-33q^{10}+33q^{8}-35q^{6}+40q^{4}-37q^{2}+31-29q^{-2}+25q^{-4}-12q^{-6}+4q^{-8}-2q^{-10}-5q^{-12}+16q^{-14}-17q^{-16}+18q^{-18}-22q^{-20}+23q^{-22}-16q^{-24}+14q^{-26}-14q^{-28}+10q^{-30}-4q^{-32}+3q^{-34}-3q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-3q^{78}+7q^{76}-13q^{74}+14q^{72}-12q^{70}-q^{68}+27q^{66}-55q^{64}+83q^{62}-84q^{60}+44q^{58}+24q^{56}-112q^{54}+181q^{52}-188q^{50}+127q^{48}-11q^{46}-114q^{44}+205q^{42}-213q^{40}+135q^{38}-7q^{36}-116q^{34}+167q^{32}-131q^{30}+24q^{28}+103q^{26}-178q^{24}+183q^{22}-102q^{20}-37q^{18}+174q^{16}-269q^{14}+272q^{12}-182q^{10}+32q^{8}+130q^{6}-244q^{4}+280q^{2}-217+85q^{-2}+58q^{-4}-170q^{-6}+191q^{-8}-117q^{-10}-6q^{-12}+123q^{-14}-165q^{-16}+125q^{-18}-16q^{-20}-113q^{-22}+195q^{-24}-206q^{-26}+141q^{-28}-26q^{-30}-88q^{-32}+159q^{-34}-165q^{-36}+126q^{-38}-55q^{-40}-10q^{-42}+48q^{-44}-68q^{-46}+60q^{-48}-38q^{-50}+18q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

.

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 34"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+6t^{2}-16t+23-16t^{-1}+6t^{-2}-t^{-3}$

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]=

{ 69, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{2}z^{8}+3z^{8}+6a^{3}z^{7}+14az^{7}+8z^{7}a^{-1}+4a^{4}z^{6}+5a^{2}z^{6}+8z^{6}a^{-2}+9z^{6}+a^{5}z^{5}-11a^{3}z^{5}-26az^{5}-10z^{5}a^{-1}+4z^{5}a^{-3}-7a^{4}z^{4}-19a^{2}z^{4}-10z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}-a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}+3a^{4}z^{2}+10a^{2}z^{2}+4z^{2}a^{-2}+11z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n32, K11n119, ...}

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

Vassiliev invariants

V2 and V3:

(-1, 0)

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 $-4$ $0$ $8$ ${\frac {34}{3}}$ ${\frac {14}{3}}$ $0$ $0$ $32$ $-32$ $-{\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ $-{\frac {56}{3}}$ $-{\frac {1231}{30}}$ $-{\frac {338}{15}}$ $-{\frac {1742}{45}}$ ${\frac {655}{18}}$ $-{\frac {271}{30}}$

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 34. 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               5 1   4 3             5 3     -2 1           7 5       2 -1         6 6         0 -3       4 6           -2 -5     3 6             3 -7   1 4               -3 -9   3                 3 -11 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{12}-4q^{11}+4q^{10}+10q^{9}-29q^{8}+12q^{7}+47q^{6}-74q^{5}+6q^{4}+101q^{3}-109q^{2}-17q+140-112q^{-1}-41q^{-2}+143q^{-3}-83q^{-4}-54q^{-5}+109q^{-6}-39q^{-7}-48q^{-8}+55q^{-9}-6q^{-10}-24q^{-11}+14q^{-12}+2q^{-13}-4q^{-14}+q^{-15}$
3	$q^{24}-4q^{23}+4q^{22}+6q^{21}-9q^{20}-19q^{19}+20q^{18}+56q^{17}-42q^{16}-116q^{15}+49q^{14}+214q^{13}-26q^{12}-350q^{11}-25q^{10}+482q^{9}+132q^{8}-611q^{7}-258q^{6}+693q^{5}+410q^{4}-747q^{3}-536q^{2}+741q+652-707q^{-1}-728q^{-2}+632q^{-3}+778q^{-4}-532q^{-5}-793q^{-6}+410q^{-7}+771q^{-8}-269q^{-9}-714q^{-10}+126q^{-11}+623q^{-12}-7q^{-13}-492q^{-14}-87q^{-15}+354q^{-16}+129q^{-17}-218q^{-18}-130q^{-19}+113q^{-20}+97q^{-21}-40q^{-22}-63q^{-23}+12q^{-24}+27q^{-25}-9q^{-27}-2q^{-28}+4q^{-29}-q^{-30}$
4	$q^{40}-4q^{39}+4q^{38}+6q^{37}-13q^{36}+q^{35}-11q^{34}+39q^{33}+37q^{32}-90q^{31}-49q^{30}-48q^{29}+221q^{28}+257q^{27}-272q^{26}-385q^{25}-384q^{24}+633q^{23}+1098q^{22}-183q^{21}-1115q^{20}-1643q^{19}+743q^{18}+2693q^{17}+949q^{16}-1582q^{15}-3886q^{14}-277q^{13}+4168q^{12}+3112q^{11}-926q^{10}-6066q^{9}-2301q^{8}+4550q^{7}+5225q^{6}+689q^{5}-7160q^{4}-4285q^{3}+3852q^{2}+6391q+2405-7085q^{-1}-5517q^{-2}+2637q^{-3}+6547q^{-4}+3731q^{-5}-6164q^{-6}-5993q^{-7}+1158q^{-8}+5920q^{-9}+4680q^{-10}-4533q^{-11}-5807q^{-12}-524q^{-13}+4516q^{-14}+5119q^{-15}-2308q^{-16}-4761q^{-17}-2001q^{-18}+2412q^{-19}+4597q^{-20}-136q^{-21}-2852q^{-22}-2485q^{-23}+330q^{-24}+3002q^{-25}+993q^{-26}-861q^{-27}-1744q^{-28}-721q^{-29}+1195q^{-30}+844q^{-31}+189q^{-32}-641q^{-33}-622q^{-34}+191q^{-35}+277q^{-36}+256q^{-37}-76q^{-38}-211q^{-39}-15q^{-40}+17q^{-41}+74q^{-42}+12q^{-43}-33q^{-44}-3q^{-45}-5q^{-46}+9q^{-47}+2q^{-48}-4q^{-49}+q^{-50}$
5	$q^{60}-4q^{59}+4q^{58}+6q^{57}-13q^{56}-3q^{55}+9q^{54}+8q^{53}+20q^{52}-q^{51}-74q^{50}-72q^{49}+59q^{48}+181q^{47}+190q^{46}-60q^{45}-449q^{44}-571q^{43}-30q^{42}+957q^{41}+1364q^{40}+462q^{39}-1505q^{38}-2883q^{37}-1772q^{36}+1892q^{35}+5191q^{34}+4347q^{33}-1364q^{32}-7900q^{31}-8689q^{30}-897q^{29}+10381q^{28}+14640q^{27}+5444q^{26}-11494q^{25}-21368q^{24}-12686q^{23}+10324q^{22}+27973q^{21}+21796q^{20}-6404q^{19}-32886q^{18}-31910q^{17}-198q^{16}+35624q^{15}+41435q^{14}+8486q^{13}-35518q^{12}-49461q^{11}-17413q^{10}+33241q^{9}+55091q^{8}+25783q^{7}-29202q^{6}-58452q^{5}-32910q^{4}+24528q^{3}+59579q^{2}+38437q-19500-59226q^{-1}-42519q^{-2}+14722q^{-3}+57635q^{-4}+45366q^{-5}-9932q^{-6}-55165q^{-7}-47442q^{-8}+5096q^{-9}+51828q^{-10}+48808q^{-11}+119q^{-12}-47403q^{-13}-49515q^{-14}-5832q^{-15}+41709q^{-16}+49272q^{-17}+11825q^{-18}-34528q^{-19}-47595q^{-20}-17694q^{-21}+25954q^{-22}+44084q^{-23}+22696q^{-24}-16564q^{-25}-38434q^{-26}-25897q^{-27}+7098q^{-28}+30858q^{-29}+26727q^{-30}+1204q^{-31}-22142q^{-32}-24730q^{-33}-7381q^{-34}+13309q^{-35}+20490q^{-36}+10678q^{-37}-5684q^{-38}-14834q^{-39}-11161q^{-40}+191q^{-41}+9102q^{-42}+9415q^{-43}+2841q^{-44}-4309q^{-45}-6681q^{-46}-3662q^{-47}+1183q^{-48}+3834q^{-49}+3097q^{-50}+451q^{-51}-1768q^{-52}-2043q^{-53}-810q^{-54}+531q^{-55}+1028q^{-56}+678q^{-57}-7q^{-58}-438q^{-59}-373q^{-60}-86q^{-61}+126q^{-62}+147q^{-63}+79q^{-64}-22q^{-65}-67q^{-66}-23q^{-67}+9q^{-68}+9q^{-69}+8q^{-70}+5q^{-71}-9q^{-72}-2q^{-73}+4q^{-74}-q^{-75}$
6	$q^{84}-4q^{83}+4q^{82}+6q^{81}-13q^{80}-3q^{79}+5q^{78}+28q^{77}-11q^{76}-18q^{75}+15q^{74}-87q^{73}-16q^{72}+93q^{71}+214q^{70}+60q^{69}-172q^{68}-226q^{67}-596q^{66}-201q^{65}+616q^{64}+1477q^{63}+1095q^{62}-260q^{61}-1676q^{60}-3767q^{59}-2705q^{58}+1083q^{57}+6329q^{56}+7879q^{55}+4218q^{54}-3277q^{53}-14376q^{52}-16750q^{51}-7389q^{50}+12518q^{49}+28184q^{48}+29057q^{47}+10529q^{46}-27268q^{45}-53389q^{44}-48973q^{43}-4566q^{42}+51710q^{41}+87565q^{40}+72590q^{39}-6610q^{38}-94578q^{37}-136646q^{36}-85215q^{35}+30912q^{34}+151409q^{33}+191466q^{32}+92051q^{31}-82084q^{30}-229420q^{29}-229647q^{28}-78266q^{27}+154170q^{26}+312752q^{25}+257079q^{24}+23502q^{23}-255646q^{22}-371665q^{21}-251033q^{20}+63988q^{19}+363413q^{18}+413536q^{17}+186123q^{16}-191750q^{15}-441171q^{14}-408894q^{13}-75882q^{12}+328123q^{11}+497760q^{10}+330085q^{9}-83203q^{8}-430541q^{7}-498059q^{6}-197997q^{5}+250860q^{4}+508808q^{3}+414062q^{2}+14014q-380135-523707q^{-1}-273997q^{-2}+175604q^{-3}+482484q^{-4}+448797q^{-5}+83170q^{-6}-323134q^{-7}-517342q^{-8}-318813q^{-9}+109774q^{-10}+442844q^{-11}+462028q^{-12}+141818q^{-13}-259898q^{-14}-495577q^{-15}-356319q^{-16}+34442q^{-17}+384479q^{-18}+463670q^{-19}+209578q^{-20}-170599q^{-21}-447370q^{-22}-389077q^{-23}-65372q^{-24}+286036q^{-25}+435588q^{-26}+280278q^{-27}-44606q^{-28}-348258q^{-29}-389585q^{-30}-173593q^{-31}+140530q^{-32}+348024q^{-33}+315023q^{-34}+92020q^{-35}-193995q^{-36}-321541q^{-37}-239573q^{-38}-16250q^{-39}+199049q^{-40}+271427q^{-41}+178557q^{-42}-29574q^{-43}-186479q^{-44}-217687q^{-45}-115811q^{-46}+42172q^{-47}+156660q^{-48}+170542q^{-49}+71966q^{-50}-45861q^{-51}-123612q^{-52}-118831q^{-53}-48992q^{-54}+38852q^{-55}+94087q^{-56}+80016q^{-57}+30049q^{-58}-29789q^{-59}-61282q^{-60}-55241q^{-61}-18946q^{-62}+22114q^{-63}+38216q^{-64}+33982q^{-65}+11198q^{-66}-11458q^{-67}-24328q^{-68}-20110q^{-69}-5535q^{-70}+5890q^{-71}+13020q^{-72}+10756q^{-73}+4175q^{-74}-3742q^{-75}-6640q^{-76}-4888q^{-77}-2203q^{-78}+1492q^{-79}+2913q^{-80}+2695q^{-81}+623q^{-82}-680q^{-83}-1006q^{-84}-1063q^{-85}-320q^{-86}+212q^{-87}+552q^{-88}+242q^{-89}+59q^{-90}-27q^{-91}-163q^{-92}-92q^{-93}-26q^{-94}+72q^{-95}+16q^{-96}+2q^{-97}+15q^{-98}-14q^{-99}-8q^{-100}-5q^{-101}+9q^{-102}+2q^{-103}-4q^{-104}+q^{-105}$
7	$q^{112}-4q^{111}+4q^{110}+6q^{109}-13q^{108}-3q^{107}+5q^{106}+24q^{105}+9q^{104}-49q^{103}-2q^{102}+2q^{101}-31q^{100}+28q^{99}+75q^{98}+158q^{97}+48q^{96}-291q^{95}-318q^{94}-260q^{93}-66q^{92}+534q^{91}+917q^{90}+1208q^{89}+482q^{88}-1471q^{87}-2844q^{86}-3329q^{85}-1711q^{84}+2371q^{83}+6470q^{82}+9138q^{81}+6836q^{80}-2254q^{79}-13194q^{78}-21631q^{77}-19779q^{76}-3481q^{75}+20489q^{74}+43605q^{73}+49176q^{72}+25233q^{71}-21814q^{70}-75414q^{69}-103193q^{68}-77368q^{67}+217q^{66}+106234q^{65}+185323q^{64}+179387q^{63}+71464q^{62}-113375q^{61}-286642q^{60}-341928q^{59}-223202q^{58}+56051q^{57}+375169q^{56}+559280q^{55}+480840q^{54}+112560q^{53}-400811q^{52}-797484q^{51}-843686q^{50}-430400q^{49}+298839q^{48}+993820q^{47}+1278907q^{46}+910274q^{45}-15765q^{44}-1073367q^{43}-1718343q^{42}-1520174q^{41}-470669q^{40}+964613q^{39}+2073346q^{38}+2192067q^{37}+1139954q^{36}-633602q^{35}-2263880q^{34}-2829249q^{33}-1922465q^{32}+89307q^{31}+2235800q^{30}+3341119q^{29}+2725638q^{28}+609765q^{27}-1986923q^{26}-3661296q^{25}-3449927q^{24}-1377286q^{23}+1555703q^{22}+3768099q^{21}+4023424q^{20}+2119469q^{19}-1015128q^{18}-3681754q^{17}-4408420q^{16}-2762212q^{15}+444233q^{14}+3454987q^{13}+4609291q^{12}+3261430q^{11}+87493q^{10}-3149968q^{9}-4658362q^{8}-3610169q^{7}-537899q^{6}+2826220q^{5}+4604709q^{4}+3825894q^{3}+890834q^{2}-2522244q-4494135-3945409q^{-1}-1157376q^{-2}+2257688q^{-3}+4364984q^{-4}+4006145q^{-5}+1360966q^{-6}-2029292q^{-7}-4235883q^{-8}-4042274q^{-9}-1535296q^{-10}+1820136q^{-11}+4112104q^{-12}+4076227q^{-13}+1709872q^{-14}-1604177q^{-15}-3981713q^{-16}-4116003q^{-17}-1909743q^{-18}+1351280q^{-19}+3823412q^{-20}+4156866q^{-21}+2147420q^{-22}-1036621q^{-23}-3607912q^{-24}-4177817q^{-25}-2420560q^{-26}+641548q^{-27}+3304202q^{-28}+4148097q^{-29}+2710187q^{-30}-164342q^{-31}-2887912q^{-32}-4027779q^{-33}-2978499q^{-34}-377124q^{-35}+2346613q^{-36}+3778094q^{-37}+3175544q^{-38}+942327q^{-39}-1691930q^{-40}-3373205q^{-41}-3244729q^{-42}-1469069q^{-43}+962632q^{-44}+2808633q^{-45}+3139347q^{-46}+1886766q^{-47}-224497q^{-48}-2115194q^{-49}-2838187q^{-50}-2127516q^{-51}-438709q^{-52}+1355500q^{-53}+2354958q^{-54}+2150543q^{-55}+946552q^{-56}-617661q^{-57}-1746377q^{-58}-1954553q^{-59}-1238823q^{-60}-6866q^{-61}+1097392q^{-62}+1583164q^{-63}+1299853q^{-64}+447271q^{-65}-504567q^{-66}-1116612q^{-67}-1160225q^{-68}-672613q^{-69}+46196q^{-70}+648950q^{-71}+889664q^{-72}+700205q^{-73}+237065q^{-74}-260227q^{-75}-574945q^{-76}-587886q^{-77}-349646q^{-78}-1773q^{-79}+292569q^{-80}+407982q^{-81}+333928q^{-82}+133949q^{-83}-89221q^{-84}-229710q^{-85}-249493q^{-86}-162516q^{-87}-23369q^{-88}+93644q^{-89}+149147q^{-90}+132781q^{-91}+63605q^{-92}-14057q^{-93}-69180q^{-94}-83861q^{-95}-59274q^{-96}-18707q^{-97}+19848q^{-98}+41277q^{-99}+39464q^{-100}+23395q^{-101}+1684q^{-102}-15056q^{-103}-19946q^{-104}-16215q^{-105}-7040q^{-106}+2517q^{-107}+7576q^{-108}+8655q^{-109}+5620q^{-110}+985q^{-111}-1979q^{-112}-3381q^{-113}-2854q^{-114}-1311q^{-115}-33q^{-116}+1072q^{-117}+1289q^{-118}+675q^{-119}+178q^{-120}-253q^{-121}-344q^{-122}-232q^{-123}-197q^{-124}+20q^{-125}+141q^{-126}+92q^{-127}+39q^{-128}-24q^{-129}-21q^{-130}+5q^{-131}-26q^{-132}-10q^{-133}+14q^{-134}+8q^{-135}+5q^{-136}-9q^{-137}-2q^{-138}+4q^{-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, 34]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[8, 3, 9, 4], X[2, 15, 3, 16], X[14, 9, 15, 10], X[10, 6, 11, 5], X[4, 14, 5, 13], X[18, 11, 1, 12], X[12, 17, 13, 18]]
In[3]:=	GaussCode[Knot[9, 34]]
Out[3]=	GaussCode[1, -4, 3, -7, 6, -1, 2, -3, 5, -6, 8, -9, 7, -5, 4, -2, 9, -8]
In[4]:=	DTCode[Knot[9, 34]]
Out[4]=	DTCode[6, 8, 10, 16, 14, 18, 4, 2, 12]
In[5]:=	br = BR[Knot[9, 34]]
Out[5]=	BR[4, {-1, 2, -1, 2, -3, 2, -1, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 34]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 34]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 34]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 34]][t]
Out[10]=	-3 6 16 2 3 23 - t + -- - -- - 16 t + 6 t - t 2 t t
In[11]:=	Conway[Knot[9, 34]][z]
Out[11]=	2 6 1 - z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 34], Knot[11, NonAlternating, 32], Knot[11, NonAlternating, 119]}
In[13]:=	{KnotDet[Knot[9, 34]], KnotSignature[Knot[9, 34]]}
Out[13]=	{69, 0}
In[14]:=	Jones[Knot[9, 34]][q]
Out[14]=	-5 4 7 10 12 2 3 4 12 - q + -- - -- + -- - -- - 10 q + 8 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, 34]}
In[16]:=	A2Invariant[Knot[9, 34]][q]
Out[16]=	-16 -14 2 2 2 -6 -4 2 2 4 6 -2 - q + q + --- - --- + -- - q - q + -- + 3 q - 2 q + q + 12 10 8 2 q q q q 8 10 12 2 q - 2 q + q
In[17]:=	HOMFLYPT[Knot[9, 34]][a, z]
Out[17]=	2 4 -2 2 2 z 2 2 4 2 4 z 2 4 6 -1 + a + a - 4 z + -- + 3 a z - a z - 3 z + -- + 2 a z - z 2 2 a a
In[18]:=	Kauffman[Knot[9, 34]][a, z]
Out[18]=	2 3 -2 2 z 2 4 z 2 2 4 2 2 z -1 - a - a - - - a z + 11 z + ---- + 10 a z + 3 a z - ---- + a 2 3 a a 3 4 4 4 z 3 3 3 5 3 4 z 10 z 2 4 ---- + 12 a z + 5 a z - a z - 23 z + -- - ----- - 19 a z - a 4 2 a a 5 5 6 4 4 4 z 10 z 5 3 5 5 5 6 8 z 7 a z + ---- - ----- - 26 a z - 11 a z + a z + 9 z + ---- + 3 a 2 a a 7 2 6 4 6 8 z 7 3 7 8 2 8 5 a z + 4 a z + ---- + 14 a z + 6 a z + 3 z + 3 a z a
In[19]:=	{Vassiliev[2][Knot[9, 34]], Vassiliev[3][Knot[9, 34]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[9, 34]][q, t]
Out[20]=	6 1 3 1 4 3 6 4 - + 7 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 6 6 3 3 2 5 2 5 3 7 3 ---- + --- + 5 q t + 5 q t + 3 q t + 5 q t + q t + 3 q t + 3 q t q t 9 4 q t
In[21]:=	ColouredJones[Knot[9, 34], 2][q]
Out[21]=	-15 4 2 14 24 6 55 48 39 109 54 140 + q - --- + --- + --- - --- - --- + -- - -- - -- + --- - -- - 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q 83 143 41 112 2 3 4 5 6 -- + --- - -- - --- - 17 q - 109 q + 101 q + 6 q - 74 q + 47 q + 4 3 2 q q q q 7 8 9 10 11 12 12 q - 29 q + 10 q + 4 q - 4 q + q

See/edit the Rolfsen_Splice_Template.