9 30

9_29

9_31

Visit 9 30's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 30's page at Knotilus! Visit 9 30's page at the original Knot Atlas! 9 30 Quick Notes

9 30 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13
Gauss code	1, -4, 3, -1, 2, -9, 5, -3, 4, -2, 7, -8, 9, -5, 6, -7, 8, -6
Dowker-Thistlethwaite code	4 8 10 14 2 16 6 18 12
Conway Notation	[211,21,2]

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 11.9545 A-Polynomial See Data:9 30/A-polynomial

[edit Notes for 9 30'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 30's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-12t+17-12t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 53, 0 }
Jones polynomial	$q^{4}-3q^{3}+6q^{2}-8q+9-9q^{-1}+8q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-4z^{4}-a^{4}z^{2}+5a^{2}z^{2}+2z^{2}a^{-2}-7z^{2}-a^{4}+4a^{2}+2a^{-2}-4$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+7az^{7}+4z^{7}a^{-1}+3a^{4}z^{6}+8a^{2}z^{6}+5z^{6}a^{-2}+10z^{6}+a^{5}z^{5}-3a^{3}z^{5}-9az^{5}-2z^{5}a^{-1}+3z^{5}a^{-3}-7a^{4}z^{4}-22a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}-2a^{5}z^{3}-3a^{3}z^{3}-2z^{3}a^{-1}-3z^{3}a^{-3}+5a^{4}z^{2}+16a^{2}z^{2}+5z^{2}a^{-2}-z^{2}a^{-4}+17z^{2}+a^{5}z+2a^{3}z+az+za^{-1}+za^{-3}-a^{4}-4a^{2}-2a^{-2}-4$
The A2 invariant	$-q^{16}+q^{12}-q^{10}+3q^{8}+q^{6}+q^{2}-3+q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+7q^{72}-5q^{70}-5q^{68}+19q^{66}-31q^{64}+39q^{62}-34q^{60}+11q^{58}+19q^{56}-55q^{54}+79q^{52}-79q^{50}+49q^{48}-2q^{46}-48q^{44}+83q^{42}-84q^{40}+60q^{38}-9q^{36}-36q^{34}+61q^{32}-52q^{30}+14q^{28}+39q^{26}-69q^{24}+75q^{22}-40q^{20}-15q^{18}+77q^{16}-118q^{14}+120q^{12}-84q^{10}+16q^{8}+55q^{6}-109q^{4}+123q^{2}-97+43q^{-2}+15q^{-4}-64q^{-6}+72q^{-8}-52q^{-10}+6q^{-12}+39q^{-14}-60q^{-16}+48q^{-18}-8q^{-20}-41q^{-22}+79q^{-24}-87q^{-26}+65q^{-28}-21q^{-30}-29q^{-32}+67q^{-34}-77q^{-36}+67q^{-38}-34q^{-40}+5q^{-42}+19q^{-44}-33q^{-46}+32q^{-48}-23q^{-50}+13q^{-52}-2q^{-54}-4q^{-56}+5q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_30.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-2q^{7}+3q^{5}-q^{3}+q^{-1}-2q^{-3}+3q^{-5}-2q^{-7}+q^{-9}$
2	$q^{32}-2q^{30}-2q^{28}+7q^{26}-3q^{24}-9q^{22}+14q^{20}+q^{18}-18q^{16}+13q^{14}+8q^{12}-17q^{10}+4q^{8}+10q^{6}-6q^{4}-6q^{2}+6+9q^{-2}-13q^{-4}-q^{-6}+19q^{-8}-13q^{-10}-8q^{-12}+17q^{-14}-6q^{-16}-8q^{-18}+7q^{-20}-2q^{-24}+q^{-26}$
3	$-q^{63}+2q^{61}+2q^{59}-3q^{57}-7q^{55}+3q^{53}+16q^{51}-2q^{49}-26q^{47}-7q^{45}+39q^{43}+23q^{41}-47q^{39}-45q^{37}+48q^{35}+68q^{33}-36q^{31}-91q^{29}+19q^{27}+98q^{25}+4q^{23}-96q^{21}-26q^{19}+87q^{17}+40q^{15}-65q^{13}-50q^{11}+41q^{9}+55q^{7}-12q^{5}-57q^{3}-15q+55q^{-1}+45q^{-3}-48q^{-5}-72q^{-7}+37q^{-9}+92q^{-11}-20q^{-13}-101q^{-15}+q^{-17}+96q^{-19}+20q^{-21}-82q^{-23}-32q^{-25}+60q^{-27}+33q^{-29}-34q^{-31}-30q^{-33}+17q^{-35}+21q^{-37}-7q^{-39}-10q^{-41}+q^{-43}+4q^{-45}-2q^{-49}+q^{-51}$
4	$q^{104}-2q^{102}-2q^{100}+3q^{98}+3q^{96}+7q^{94}-10q^{92}-16q^{90}+2q^{88}+14q^{86}+41q^{84}-12q^{82}-59q^{80}-37q^{78}+16q^{76}+129q^{74}+49q^{72}-98q^{70}-157q^{68}-79q^{66}+221q^{64}+227q^{62}-12q^{60}-286q^{58}-324q^{56}+159q^{54}+412q^{52}+252q^{50}-243q^{48}-568q^{46}-95q^{44}+410q^{42}+513q^{40}-12q^{38}-602q^{36}-351q^{34}+218q^{32}+576q^{30}+213q^{28}-431q^{26}-436q^{24}+446q^{20}+311q^{18}-184q^{16}-383q^{14}-165q^{12}+245q^{10}+330q^{8}+66q^{6}-278q^{4}-306q^{2}+7+324q^{-2}+339q^{-4}-128q^{-6}-432q^{-8}-272q^{-10}+246q^{-12}+574q^{-14}+101q^{-16}-431q^{-18}-517q^{-20}+37q^{-22}+623q^{-24}+326q^{-26}-233q^{-28}-559q^{-30}-203q^{-32}+420q^{-34}+381q^{-36}+25q^{-38}-367q^{-40}-279q^{-42}+144q^{-44}+234q^{-46}+133q^{-48}-125q^{-50}-176q^{-52}+q^{-54}+70q^{-56}+88q^{-58}-14q^{-60}-59q^{-62}-11q^{-64}+4q^{-66}+26q^{-68}+3q^{-70}-11q^{-72}-q^{-74}-2q^{-76}+4q^{-78}-2q^{-82}+q^{-84}$
5	$-q^{155}+2q^{153}+2q^{151}-3q^{149}-3q^{147}-3q^{145}+10q^{141}+16q^{139}-2q^{137}-23q^{135}-29q^{133}-13q^{131}+32q^{129}+71q^{127}+52q^{125}-43q^{123}-132q^{121}-124q^{119}+8q^{117}+199q^{115}+275q^{113}+97q^{111}-254q^{109}-473q^{107}-316q^{105}+194q^{103}+700q^{101}+693q^{99}+26q^{97}-853q^{95}-1164q^{93}-488q^{91}+802q^{89}+1646q^{87}+1166q^{85}-454q^{83}-1975q^{81}-1968q^{79}-198q^{77}+1994q^{75}+2703q^{73}+1111q^{71}-1652q^{69}-3227q^{67}-2060q^{65}+989q^{63}+3361q^{61}+2906q^{59}-129q^{57}-3148q^{55}-3462q^{53}-718q^{51}+2634q^{49}+3644q^{47}+1441q^{45}-1965q^{43}-3518q^{41}-1914q^{39}+1292q^{37}+3139q^{35}+2125q^{33}-659q^{31}-2658q^{29}-2166q^{27}+164q^{25}+2139q^{23}+2091q^{21}+265q^{19}-1642q^{17}-2014q^{15}-655q^{13}+1175q^{11}+1966q^{9}+1091q^{7}-705q^{5}-1948q^{3}-1594q+160q^{-1}+1931q^{-3}+2165q^{-5}+487q^{-7}-1821q^{-9}-2732q^{-11}-1254q^{-13}+1532q^{-15}+3201q^{-17}+2084q^{-19}-1027q^{-21}-3429q^{-23}-2860q^{-25}+307q^{-27}+3311q^{-29}+3458q^{-31}+531q^{-33}-2851q^{-35}-3692q^{-37}-1334q^{-39}+2073q^{-41}+3538q^{-43}+1939q^{-45}-1174q^{-47}-3018q^{-49}-2189q^{-51}+321q^{-53}+2244q^{-55}+2098q^{-57}+317q^{-59}-1438q^{-61}-1736q^{-63}-624q^{-65}+727q^{-67}+1234q^{-69}+688q^{-71}-244q^{-73}-766q^{-75}-564q^{-77}+2q^{-79}+393q^{-81}+378q^{-83}+89q^{-85}-167q^{-87}-223q^{-89}-84q^{-91}+63q^{-93}+102q^{-95}+54q^{-97}-13q^{-99}-41q^{-101}-32q^{-103}+3q^{-105}+19q^{-107}+8q^{-109}-q^{-111}-2q^{-113}-4q^{-115}-2q^{-117}+4q^{-119}-2q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{12}-q^{10}+3q^{8}+q^{6}+q^{2}-3+q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+54q^{36}-96q^{34}+150q^{32}-214q^{30}+280q^{28}-336q^{26}+366q^{24}-352q^{22}+299q^{20}-192q^{18}+42q^{16}+138q^{14}-321q^{12}+486q^{10}-622q^{8}+698q^{6}-714q^{4}+662q^{2}-550+398q^{-2}-213q^{-4}+36q^{-6}+132q^{-8}-256q^{-10}+334q^{-12}-360q^{-14}+344q^{-16}-304q^{-18}+239q^{-20}-174q^{-22}+118q^{-24}-74q^{-26}+42q^{-28}-20q^{-30}+10q^{-32}-4q^{-34}+q^{-36}$
2,0	$q^{42}-2q^{38}-2q^{36}+2q^{34}+3q^{32}-4q^{30}-3q^{28}+6q^{26}+5q^{24}-6q^{22}-3q^{20}+8q^{18}+5q^{16}-8q^{14}-q^{12}+5q^{10}-6q^{8}-5q^{6}+2q^{4}-3+7q^{-2}+9q^{-4}-3q^{-6}-2q^{-8}+9q^{-10}+q^{-12}-12q^{-14}-q^{-16}+6q^{-18}-q^{-20}-5q^{-22}+4q^{-26}-q^{-30}+q^{-32}$

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-2q^{44}+3q^{42}-4q^{40}+6q^{38}-9q^{36}+8q^{34}-11q^{32}+11q^{30}-13q^{28}+9q^{26}-8q^{24}+9q^{22}-q^{20}+9q^{16}-4q^{14}+17q^{12}-17q^{10}+19q^{8}-20q^{6}+21q^{4}-24q^{2}+14-20q^{-2}+12q^{-4}-9q^{-6}+2q^{-8}-2q^{-10}-q^{-12}+11q^{-14}-7q^{-16}+11q^{-18}-11q^{-20}+14q^{-22}-9q^{-24}+8q^{-26}-9q^{-28}+6q^{-30}-3q^{-32}+2q^{-34}-2q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+7q^{72}-5q^{70}-5q^{68}+19q^{66}-31q^{64}+39q^{62}-34q^{60}+11q^{58}+19q^{56}-55q^{54}+79q^{52}-79q^{50}+49q^{48}-2q^{46}-48q^{44}+83q^{42}-84q^{40}+60q^{38}-9q^{36}-36q^{34}+61q^{32}-52q^{30}+14q^{28}+39q^{26}-69q^{24}+75q^{22}-40q^{20}-15q^{18}+77q^{16}-118q^{14}+120q^{12}-84q^{10}+16q^{8}+55q^{6}-109q^{4}+123q^{2}-97+43q^{-2}+15q^{-4}-64q^{-6}+72q^{-8}-52q^{-10}+6q^{-12}+39q^{-14}-60q^{-16}+48q^{-18}-8q^{-20}-41q^{-22}+79q^{-24}-87q^{-26}+65q^{-28}-21q^{-30}-29q^{-32}+67q^{-34}-77q^{-36}+67q^{-38}-34q^{-40}+5q^{-42}+19q^{-44}-33q^{-46}+32q^{-48}-23q^{-50}+13q^{-52}-2q^{-54}-4q^{-56}+5q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

.

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 30"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-12t+17-12t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 53, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-4z^{4}-a^{4}z^{2}+5a^{2}z^{2}+2z^{2}a^{-2}-7z^{2}-a^{4}+4a^{2}+2a^{-2}-4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+7az^{7}+4z^{7}a^{-1}+3a^{4}z^{6}+8a^{2}z^{6}+5z^{6}a^{-2}+10z^{6}+a^{5}z^{5}-3a^{3}z^{5}-9az^{5}-2z^{5}a^{-1}+3z^{5}a^{-3}-7a^{4}z^{4}-22a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}-2a^{5}z^{3}-3a^{3}z^{3}-2z^{3}a^{-1}-3z^{3}a^{-3}+5a^{4}z^{2}+16a^{2}z^{2}+5z^{2}a^{-2}-z^{2}a^{-4}+17z^{2}+a^{5}z+2a^{3}z+az+za^{-1}+za^{-3}-a^{4}-4a^{2}-2a^{-2}-4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n130, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n114, ...}

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 $-4$ $-8$ $8$ ${\frac {82}{3}}$ ${\frac {38}{3}}$ $32$ ${\frac {208}{3}}$ ${\frac {64}{3}}$ $24$ $-{\frac {32}{3}}$ $32$ $-{\frac {328}{3}}$ $-{\frac {152}{3}}$ $-{\frac {2431}{30}}$ ${\frac {1222}{15}}$ $-{\frac {8342}{45}}$ ${\frac {607}{18}}$ $-{\frac {1471}{30}}$

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 30. 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               4 1   3 3             4 2     -2 1           5 4       1 -1         5 5         0 -3       3 4           -1 -5     2 5             3 -7   1 3               -2 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{12}-3q^{11}+2q^{10}+8q^{9}-18q^{8}+4q^{7}+31q^{6}-43q^{5}-q^{4}+63q^{3}-63q^{2}-13q+85-66q^{-1}-25q^{-2}+85q^{-3}-50q^{-4}-31q^{-5}+64q^{-6}-25q^{-7}-26q^{-8}+33q^{-9}-6q^{-10}-13q^{-11}+10q^{-12}-3q^{-14}+q^{-15}$
3	$q^{24}-3q^{23}+2q^{22}+4q^{21}-2q^{20}-14q^{19}+5q^{18}+32q^{17}-6q^{16}-61q^{15}+q^{14}+99q^{13}+21q^{12}-153q^{11}-49q^{10}+201q^{9}+97q^{8}-248q^{7}-151q^{6}+282q^{5}+209q^{4}-303q^{3}-260q^{2}+306q+302-293q^{-1}-330q^{-2}+264q^{-3}+347q^{-4}-226q^{-5}-344q^{-6}+173q^{-7}+332q^{-8}-121q^{-9}-297q^{-10}+60q^{-11}+262q^{-12}-21q^{-13}-203q^{-14}-19q^{-15}+152q^{-16}+34q^{-17}-99q^{-18}-39q^{-19}+59q^{-20}+32q^{-21}-29q^{-22}-23q^{-23}+13q^{-24}+13q^{-25}-5q^{-26}-5q^{-27}+3q^{-29}-q^{-30}$
4	$q^{40}-3q^{39}+2q^{38}+4q^{37}-6q^{36}+2q^{35}-13q^{34}+16q^{33}+27q^{32}-28q^{31}-13q^{30}-61q^{29}+61q^{28}+129q^{27}-46q^{26}-82q^{25}-238q^{24}+112q^{23}+387q^{22}+55q^{21}-172q^{20}-661q^{19}+24q^{18}+779q^{17}+411q^{16}-133q^{15}-1284q^{14}-332q^{13}+1105q^{12}+970q^{11}+164q^{10}-1870q^{9}-886q^{8}+1191q^{7}+1502q^{6}+637q^{5}-2198q^{4}-1404q^{3}+1031q^{2}+1806q+1104-2213q^{-1}-1721q^{-2}+718q^{-3}+1834q^{-4}+1448q^{-5}-1949q^{-6}-1806q^{-7}+308q^{-8}+1616q^{-9}+1647q^{-10}-1454q^{-11}-1671q^{-12}-138q^{-13}+1180q^{-14}+1652q^{-15}-810q^{-16}-1308q^{-17}-496q^{-18}+611q^{-19}+1401q^{-20}-220q^{-21}-783q^{-22}-599q^{-23}+106q^{-24}+928q^{-25}+105q^{-26}-288q^{-27}-439q^{-28}-147q^{-29}+445q^{-30}+143q^{-31}-14q^{-32}-200q^{-33}-153q^{-34}+145q^{-35}+65q^{-36}+45q^{-37}-53q^{-38}-73q^{-39}+32q^{-40}+12q^{-41}+23q^{-42}-6q^{-43}-20q^{-44}+5q^{-45}+5q^{-47}-3q^{-49}+q^{-50}$
5	$q^{60}-3q^{59}+2q^{58}+4q^{57}-6q^{56}-2q^{55}+3q^{54}-2q^{53}+11q^{52}+15q^{51}-22q^{50}-37q^{49}-6q^{48}+26q^{47}+78q^{46}+63q^{45}-61q^{44}-184q^{43}-145q^{42}+82q^{41}+334q^{40}+352q^{39}-46q^{38}-575q^{37}-711q^{36}-120q^{35}+856q^{34}+1284q^{33}+500q^{32}-1082q^{31}-2062q^{30}-1232q^{29}+1154q^{28}+3039q^{27}+2281q^{26}-936q^{25}-3985q^{24}-3742q^{23}+325q^{22}+4883q^{21}+5394q^{20}+663q^{19}-5450q^{18}-7149q^{17}-2033q^{16}+5724q^{15}+8776q^{14}+3590q^{13}-5597q^{12}-10153q^{11}-5200q^{10}+5155q^{9}+11178q^{8}+6701q^{7}-4480q^{6}-11822q^{5}-7986q^{4}+3677q^{3}+12089q^{2}+9009q-2802-12056q^{-1}-9757q^{-2}+1923q^{-3}+11735q^{-4}+10252q^{-5}-1006q^{-6}-11181q^{-7}-10548q^{-8}+93q^{-9}+10376q^{-10}+10624q^{-11}+901q^{-12}-9355q^{-13}-10500q^{-14}-1882q^{-15}+8046q^{-16}+10132q^{-17}+2900q^{-18}-6571q^{-19}-9486q^{-20}-3729q^{-21}+4840q^{-22}+8528q^{-23}+4453q^{-24}-3165q^{-25}-7283q^{-26}-4739q^{-27}+1488q^{-28}+5784q^{-29}+4767q^{-30}-146q^{-31}-4248q^{-32}-4284q^{-33}-884q^{-34}+2735q^{-35}+3600q^{-36}+1429q^{-37}-1485q^{-38}-2692q^{-39}-1593q^{-40}+543q^{-41}+1830q^{-42}+1422q^{-43}+36q^{-44}-1072q^{-45}-1113q^{-46}-301q^{-47}+540q^{-48}+746q^{-49}+347q^{-50}-193q^{-51}-446q^{-52}-294q^{-53}+34q^{-54}+236q^{-55}+190q^{-56}+26q^{-57}-95q^{-58}-116q^{-59}-42q^{-60}+45q^{-61}+58q^{-62}+18q^{-63}-6q^{-64}-21q^{-65}-23q^{-66}+6q^{-67}+13q^{-68}+2q^{-69}-5q^{-72}+3q^{-74}-q^{-75}$
6	$q^{84}-3q^{83}+2q^{82}+4q^{81}-6q^{80}-2q^{79}-q^{78}+14q^{77}-7q^{76}-q^{75}+21q^{74}-36q^{73}-24q^{72}-3q^{71}+68q^{70}+26q^{69}+10q^{68}+47q^{67}-165q^{66}-166q^{65}-60q^{64}+255q^{63}+255q^{62}+222q^{61}+183q^{60}-583q^{59}-805q^{58}-583q^{57}+505q^{56}+1060q^{55}+1356q^{54}+1143q^{53}-1175q^{52}-2662q^{51}-2873q^{50}-298q^{49}+2254q^{48}+4613q^{47}+4970q^{46}-202q^{45}-5504q^{44}-8702q^{43}-5038q^{42}+1365q^{41}+9631q^{40}+13916q^{39}+6142q^{38}-6085q^{37}-17419q^{36}-16308q^{35}-6154q^{34}+12392q^{33}+26764q^{32}+20456q^{31}+655q^{30}-24126q^{29}-32162q^{28}-22522q^{27}+7565q^{26}+37724q^{25}+39809q^{24}+16353q^{23}-23280q^{22}-46107q^{21}-43764q^{20}-5863q^{19}+41144q^{18}+57063q^{17}+36348q^{16}-14213q^{15}-52623q^{14}-62465q^{13}-23006q^{12}+36529q^{11}+66871q^{10}+53604q^{9}-1437q^{8}-51360q^{7}-73801q^{6}-37744q^{5}+27760q^{4}+68954q^{3}+64272q^{2}+10197q-45553-77764q^{-1}-47408q^{-2}+18458q^{-3}+65948q^{-4}+68839q^{-5}+19152q^{-6}-37814q^{-7}-76599q^{-8}-52952q^{-9}+9292q^{-10}+59637q^{-11}+69304q^{-12}+26675q^{-13}-28185q^{-14}-71459q^{-15}-56063q^{-16}-1056q^{-17}+49597q^{-18}+66234q^{-19}+33915q^{-20}-15434q^{-21}-61480q^{-22}-56500q^{-23}-13004q^{-24}+34702q^{-25}+58150q^{-26}+39502q^{-27}+65q^{-28}-45519q^{-29}-51820q^{-30}-23870q^{-31}+16060q^{-32}+43584q^{-33}+39740q^{-34}+14429q^{-35}-25239q^{-36}-39997q^{-37}-28734q^{-38}-1411q^{-39}+24455q^{-40}+32015q^{-41}+21942q^{-42}-6334q^{-43}-23166q^{-44}-24707q^{-45}-11495q^{-46}+6871q^{-47}+18726q^{-48}+19971q^{-49}+4875q^{-50}-7707q^{-51}-14685q^{-52}-12040q^{-53}-3182q^{-54}+6384q^{-55}+12097q^{-56}+6920q^{-57}+903q^{-58}-5206q^{-59}-7072q^{-60}-5060q^{-61}-208q^{-62}+4665q^{-63}+4072q^{-64}+2696q^{-65}-342q^{-66}-2336q^{-67}-3022q^{-68}-1544q^{-69}+917q^{-70}+1252q^{-71}+1561q^{-72}+669q^{-73}-200q^{-74}-1062q^{-75}-880q^{-76}-27q^{-77}+95q^{-78}+487q^{-79}+368q^{-80}+189q^{-81}-234q^{-82}-281q^{-83}-50q^{-84}-78q^{-85}+83q^{-86}+97q^{-87}+106q^{-88}-37q^{-89}-61q^{-90}-3q^{-91}-35q^{-92}+6q^{-93}+12q^{-94}+32q^{-95}-6q^{-96}-13q^{-97}+5q^{-98}-7q^{-99}+5q^{-102}-3q^{-104}+q^{-105}$
7	$q^{112}-3q^{111}+2q^{110}+4q^{109}-6q^{108}-2q^{107}-q^{106}+10q^{105}+9q^{104}-19q^{103}+5q^{102}+7q^{101}-23q^{100}-11q^{99}-3q^{98}+54q^{97}+66q^{96}-43q^{95}-26q^{94}-48q^{93}-120q^{92}-41q^{91}+10q^{90}+256q^{89}+360q^{88}+57q^{87}-118q^{86}-439q^{85}-697q^{84}-416q^{83}-11q^{82}+938q^{81}+1661q^{80}+1136q^{79}+234q^{78}-1533q^{77}-3073q^{76}-2855q^{75}-1462q^{74}+2017q^{73}+5545q^{72}+6200q^{71}+4188q^{70}-1697q^{69}-8598q^{68}-11668q^{67}-9984q^{66}-939q^{65}+11553q^{64}+19699q^{63}+20162q^{62}+7772q^{61}-12632q^{60}-29561q^{59}-35566q^{58}-21280q^{57}+8931q^{56}+39285q^{55}+56377q^{54}+43541q^{53}+2667q^{52}-45848q^{51}-80761q^{50}-74920q^{49}-25322q^{48}+44770q^{47}+105238q^{46}+114826q^{45}+60818q^{44}-32714q^{43}-125498q^{42}-159320q^{41}-108154q^{40}+6363q^{39}+136276q^{38}+204052q^{37}+165116q^{36}+34010q^{35}-134696q^{34}-243111q^{33}-225975q^{32}-86496q^{31}+118414q^{30}+272168q^{29}+285618q^{28}+146556q^{27}-89189q^{26}-288323q^{25}-338081q^{24}-208618q^{23}+49860q^{22}+290962q^{21}+379836q^{20}+267301q^{19}-5088q^{18}-281941q^{17}-409021q^{16}-318196q^{15}-40378q^{14}+264251q^{13}+425772q^{12}+358976q^{11}+82746q^{10}-241485q^{9}-432054q^{8}-389088q^{7}-119422q^{6}+216898q^{5}+430356q^{4}+409333q^{3}+149583q^{2}-192532q-423183-421788q^{-1}-173788q^{-2}+169564q^{-3}+412620q^{-4}+428330q^{-5}+193300q^{-6}-147605q^{-7}-399497q^{-8}-430936q^{-9}-210137q^{-10}+125684q^{-11}+384105q^{-12}+430664q^{-13}+225747q^{-14}-102197q^{-15}-365409q^{-16}-427749q^{-17}-241544q^{-18}+75535q^{-19}+342270q^{-20}+421529q^{-21}+257631q^{-22}-44562q^{-23}-312837q^{-24}-410688q^{-25}-273437q^{-26}+9110q^{-27}+276036q^{-28}+393037q^{-29}+286912q^{-30}+30306q^{-31}-231134q^{-32}-367093q^{-33}-295693q^{-34}-70743q^{-35}+179153q^{-36}+330748q^{-37}+296342q^{-38}+109569q^{-39}-121869q^{-40}-284915q^{-41}-286634q^{-42}-141581q^{-43}+63812q^{-44}+230046q^{-45}+264303q^{-46}+163756q^{-47}-9038q^{-48}-170584q^{-49}-230685q^{-50}-172069q^{-51}-36434q^{-52}+110653q^{-53}+187209q^{-54}+166245q^{-55}+69551q^{-56}-56314q^{-57}-139368q^{-58}-147372q^{-59}-87038q^{-60}+12099q^{-61}+91670q^{-62}+119293q^{-63}+90197q^{-64}+18811q^{-65}-50124q^{-66}-87023q^{-67}-81246q^{-68}-35776q^{-69}+17916q^{-70}+55884q^{-71}+64964q^{-72}+40523q^{-73}+3154q^{-74}-29803q^{-75}-46006q^{-76}-36794q^{-77}-13917q^{-78}+11200q^{-79}+28515q^{-80}+28303q^{-81}+16730q^{-82}+106q^{-83}-14834q^{-84}-18971q^{-85}-14824q^{-86}-5135q^{-87}+5906q^{-88}+10887q^{-89}+10739q^{-90}+6184q^{-91}-907q^{-92}-5177q^{-93}-6764q^{-94}-5221q^{-95}-1033q^{-96}+1882q^{-97}+3592q^{-98}+3494q^{-99}+1452q^{-100}-168q^{-101}-1623q^{-102}-2149q^{-103}-1162q^{-104}-277q^{-105}+598q^{-106}+1047q^{-107}+671q^{-108}+429q^{-109}-79q^{-110}-538q^{-111}-402q^{-112}-265q^{-113}+9q^{-114}+202q^{-115}+121q^{-116}+170q^{-117}+90q^{-118}-86q^{-119}-87q^{-120}-87q^{-121}-16q^{-122}+42q^{-123}-5q^{-124}+31q^{-125}+35q^{-126}-6q^{-127}-12q^{-128}-23q^{-129}-3q^{-130}+13q^{-131}-5q^{-132}+7q^{-134}-5q^{-137}+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, 30]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[14, 8, 15, 7], X[18, 15, 1, 16], X[16, 11, 17, 12], X[12, 17, 13, 18], X[6, 14, 7, 13]]
In[3]:=	GaussCode[Knot[9, 30]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, -9, 5, -3, 4, -2, 7, -8, 9, -5, 6, -7, 8, -6]
In[4]:=	DTCode[Knot[9, 30]]
Out[4]=	DTCode[4, 8, 10, 14, 2, 16, 6, 18, 12]
In[5]:=	br = BR[Knot[9, 30]]
Out[5]=	BR[4, {-1, -1, 2, 2, -1, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 30]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 30]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 30]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 30]][t]
Out[10]=	-3 5 12 2 3 17 - t + -- - -- - 12 t + 5 t - t 2 t t
In[11]:=	Conway[Knot[9, 30]][z]
Out[11]=	2 4 6 1 - z - z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 30], Knot[11, NonAlternating, 130]}
In[13]:=	{KnotDet[Knot[9, 30]], KnotSignature[Knot[9, 30]]}
Out[13]=	{53, 0}
In[14]:=	Jones[Knot[9, 30]][q]
Out[14]=	-5 3 5 8 9 2 3 4 9 - q + -- - -- + -- - - - 8 q + 6 q - 3 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, 30], Knot[11, NonAlternating, 114]}
In[16]:=	A2Invariant[Knot[9, 30]][q]
Out[16]=	-16 -12 -10 3 -6 -2 2 4 6 8 -3 - q + q - q + -- + q + q + q - 2 q + q + 2 q - 8 q 10 12 q + q
In[17]:=	HOMFLYPT[Knot[9, 30]][a, z]
Out[17]=	2 4 2 2 4 2 2 z 2 2 4 2 4 z -4 + -- + 4 a - a - 7 z + ---- + 5 a z - a z - 4 z + -- + 2 2 2 a a a 2 4 6 2 a z - z
In[18]:=	Kauffman[Knot[9, 30]][a, z]
Out[18]=	2 2 2 4 z z 3 5 2 z -4 - -- - 4 a - a + -- + - + a z + 2 a z + a z + 17 z - -- + 2 3 a 4 a a a 2 3 3 5 z 2 2 4 2 3 z 2 z 3 3 5 3 4 ---- + 16 a z + 5 a z - ---- - ---- - 3 a z - 2 a z - 23 z + 2 3 a a a 4 4 5 5 z 7 z 2 4 4 4 3 z 2 z 5 3 5 -- - ---- - 22 a z - 7 a z + ---- - ---- - 9 a z - 3 a z + 4 2 3 a a a a 6 7 5 5 6 5 z 2 6 4 6 4 z 7 3 7 a z + 10 z + ---- + 8 a z + 3 a z + ---- + 7 a z + 3 a z + 2 a a 8 2 8 z + a z
In[19]:=	{Vassiliev[2][Knot[9, 30]], Vassiliev[3][Knot[9, 30]]}
Out[19]=	{-1, -1}
In[20]:=	Kh[Knot[9, 30]][q, t]
Out[20]=	5 1 2 1 3 2 5 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 5 3 3 2 5 2 5 3 7 3 ---- + --- + 4 q t + 4 q t + 2 q t + 4 q t + q t + 2 q t + 3 q t q t 9 4 q t
In[21]:=	ColouredJones[Knot[9, 30], 2][q]
Out[21]=	-15 3 10 13 6 33 26 25 64 31 50 85 85 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- - 14 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q 25 66 2 3 4 5 6 7 8 -- - -- - 13 q - 63 q + 63 q - q - 43 q + 31 q + 4 q - 18 q + 2 q q 9 10 11 12 8 q + 2 q - 3 q + q

See/edit the Rolfsen_Splice_Template.