9 30

9_29

9_31

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 30's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 30 at Knotilus!

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	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{2, 12}, {1, 6}, {11, 4}, {12, 10}, {8, 11}, {7, 9}, {6, 8}, {3, 5}, {4, 7}, {5, 2}, {9, 3}, {10, 1}]

[edit Notes on presentations of 9 30]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 30"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, 2, -9, 5, -3, 4, -2, 7, -8, 9, -5, 6, -7, 8, -6

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 2 16 6 18 12

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[211,21,2]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(4,\{-1,-1,2,2,-1,2,-3,2,-3\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 4, 9, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{2, 12}, {1, 6}, {11, 4}, {12, 10}, {8, 11}, {7, 9}, {6, 8}, {3, 5}, {4, 7}, {5, 2}, {9, 3}, {10, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

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	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 30"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ $-t^{3}+5t^{2}-12t+17-12t^{-1}+5t^{-2}-t^{-3}$, $q^{4}-3q^{3}+6q^{2}-8q+9-9q^{-1}+8q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n130,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{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 $sl(2)$)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	$J_{n}$
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-3 q^{11}+2 q^{10}+8 q^9-18 q^8+4 q^7+31 q^6-43 q^5-q^4+63 q^3-63 q^2-13 q+85-66 q^{-1} -25 q^{-2} +85 q^{-3} -50 q^{-4} -31 q^{-5} +64 q^{-6} -25 q^{-7} -26 q^{-8} +33 q^{-9} -6 q^{-10} -13 q^{-11} +10 q^{-12} -3 q^{-14} + q^{-15} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{24}-3 q^{23}+2 q^{22}+4 q^{21}-2 q^{20}-14 q^{19}+5 q^{18}+32 q^{17}-6 q^{16}-61 q^{15}+q^{14}+99 q^{13}+21 q^{12}-153 q^{11}-49 q^{10}+201 q^9+97 q^8-248 q^7-151 q^6+282 q^5+209 q^4-303 q^3-260 q^2+306 q+302-293 q^{-1} -330 q^{-2} +264 q^{-3} +347 q^{-4} -226 q^{-5} -344 q^{-6} +173 q^{-7} +332 q^{-8} -121 q^{-9} -297 q^{-10} +60 q^{-11} +262 q^{-12} -21 q^{-13} -203 q^{-14} -19 q^{-15} +152 q^{-16} +34 q^{-17} -99 q^{-18} -39 q^{-19} +59 q^{-20} +32 q^{-21} -29 q^{-22} -23 q^{-23} +13 q^{-24} +13 q^{-25} -5 q^{-26} -5 q^{-27} +3 q^{-29} - q^{-30} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{40}-3 q^{39}+2 q^{38}+4 q^{37}-6 q^{36}+2 q^{35}-13 q^{34}+16 q^{33}+27 q^{32}-28 q^{31}-13 q^{30}-61 q^{29}+61 q^{28}+129 q^{27}-46 q^{26}-82 q^{25}-238 q^{24}+112 q^{23}+387 q^{22}+55 q^{21}-172 q^{20}-661 q^{19}+24 q^{18}+779 q^{17}+411 q^{16}-133 q^{15}-1284 q^{14}-332 q^{13}+1105 q^{12}+970 q^{11}+164 q^{10}-1870 q^9-886 q^8+1191 q^7+1502 q^6+637 q^5-2198 q^4-1404 q^3+1031 q^2+1806 q+1104-2213 q^{-1} -1721 q^{-2} +718 q^{-3} +1834 q^{-4} +1448 q^{-5} -1949 q^{-6} -1806 q^{-7} +308 q^{-8} +1616 q^{-9} +1647 q^{-10} -1454 q^{-11} -1671 q^{-12} -138 q^{-13} +1180 q^{-14} +1652 q^{-15} -810 q^{-16} -1308 q^{-17} -496 q^{-18} +611 q^{-19} +1401 q^{-20} -220 q^{-21} -783 q^{-22} -599 q^{-23} +106 q^{-24} +928 q^{-25} +105 q^{-26} -288 q^{-27} -439 q^{-28} -147 q^{-29} +445 q^{-30} +143 q^{-31} -14 q^{-32} -200 q^{-33} -153 q^{-34} +145 q^{-35} +65 q^{-36} +45 q^{-37} -53 q^{-38} -73 q^{-39} +32 q^{-40} +12 q^{-41} +23 q^{-42} -6 q^{-43} -20 q^{-44} +5 q^{-45} +5 q^{-47} -3 q^{-49} + q^{-50} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-3 q^{59}+2 q^{58}+4 q^{57}-6 q^{56}-2 q^{55}+3 q^{54}-2 q^{53}+11 q^{52}+15 q^{51}-22 q^{50}-37 q^{49}-6 q^{48}+26 q^{47}+78 q^{46}+63 q^{45}-61 q^{44}-184 q^{43}-145 q^{42}+82 q^{41}+334 q^{40}+352 q^{39}-46 q^{38}-575 q^{37}-711 q^{36}-120 q^{35}+856 q^{34}+1284 q^{33}+500 q^{32}-1082 q^{31}-2062 q^{30}-1232 q^{29}+1154 q^{28}+3039 q^{27}+2281 q^{26}-936 q^{25}-3985 q^{24}-3742 q^{23}+325 q^{22}+4883 q^{21}+5394 q^{20}+663 q^{19}-5450 q^{18}-7149 q^{17}-2033 q^{16}+5724 q^{15}+8776 q^{14}+3590 q^{13}-5597 q^{12}-10153 q^{11}-5200 q^{10}+5155 q^9+11178 q^8+6701 q^7-4480 q^6-11822 q^5-7986 q^4+3677 q^3+12089 q^2+9009 q-2802-12056 q^{-1} -9757 q^{-2} +1923 q^{-3} +11735 q^{-4} +10252 q^{-5} -1006 q^{-6} -11181 q^{-7} -10548 q^{-8} +93 q^{-9} +10376 q^{-10} +10624 q^{-11} +901 q^{-12} -9355 q^{-13} -10500 q^{-14} -1882 q^{-15} +8046 q^{-16} +10132 q^{-17} +2900 q^{-18} -6571 q^{-19} -9486 q^{-20} -3729 q^{-21} +4840 q^{-22} +8528 q^{-23} +4453 q^{-24} -3165 q^{-25} -7283 q^{-26} -4739 q^{-27} +1488 q^{-28} +5784 q^{-29} +4767 q^{-30} -146 q^{-31} -4248 q^{-32} -4284 q^{-33} -884 q^{-34} +2735 q^{-35} +3600 q^{-36} +1429 q^{-37} -1485 q^{-38} -2692 q^{-39} -1593 q^{-40} +543 q^{-41} +1830 q^{-42} +1422 q^{-43} +36 q^{-44} -1072 q^{-45} -1113 q^{-46} -301 q^{-47} +540 q^{-48} +746 q^{-49} +347 q^{-50} -193 q^{-51} -446 q^{-52} -294 q^{-53} +34 q^{-54} +236 q^{-55} +190 q^{-56} +26 q^{-57} -95 q^{-58} -116 q^{-59} -42 q^{-60} +45 q^{-61} +58 q^{-62} +18 q^{-63} -6 q^{-64} -21 q^{-65} -23 q^{-66} +6 q^{-67} +13 q^{-68} +2 q^{-69} -5 q^{-72} +3 q^{-74} - q^{-75} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}-3 q^{83}+2 q^{82}+4 q^{81}-6 q^{80}-2 q^{79}-q^{78}+14 q^{77}-7 q^{76}-q^{75}+21 q^{74}-36 q^{73}-24 q^{72}-3 q^{71}+68 q^{70}+26 q^{69}+10 q^{68}+47 q^{67}-165 q^{66}-166 q^{65}-60 q^{64}+255 q^{63}+255 q^{62}+222 q^{61}+183 q^{60}-583 q^{59}-805 q^{58}-583 q^{57}+505 q^{56}+1060 q^{55}+1356 q^{54}+1143 q^{53}-1175 q^{52}-2662 q^{51}-2873 q^{50}-298 q^{49}+2254 q^{48}+4613 q^{47}+4970 q^{46}-202 q^{45}-5504 q^{44}-8702 q^{43}-5038 q^{42}+1365 q^{41}+9631 q^{40}+13916 q^{39}+6142 q^{38}-6085 q^{37}-17419 q^{36}-16308 q^{35}-6154 q^{34}+12392 q^{33}+26764 q^{32}+20456 q^{31}+655 q^{30}-24126 q^{29}-32162 q^{28}-22522 q^{27}+7565 q^{26}+37724 q^{25}+39809 q^{24}+16353 q^{23}-23280 q^{22}-46107 q^{21}-43764 q^{20}-5863 q^{19}+41144 q^{18}+57063 q^{17}+36348 q^{16}-14213 q^{15}-52623 q^{14}-62465 q^{13}-23006 q^{12}+36529 q^{11}+66871 q^{10}+53604 q^9-1437 q^8-51360 q^7-73801 q^6-37744 q^5+27760 q^4+68954 q^3+64272 q^2+10197 q-45553-77764 q^{-1} -47408 q^{-2} +18458 q^{-3} +65948 q^{-4} +68839 q^{-5} +19152 q^{-6} -37814 q^{-7} -76599 q^{-8} -52952 q^{-9} +9292 q^{-10} +59637 q^{-11} +69304 q^{-12} +26675 q^{-13} -28185 q^{-14} -71459 q^{-15} -56063 q^{-16} -1056 q^{-17} +49597 q^{-18} +66234 q^{-19} +33915 q^{-20} -15434 q^{-21} -61480 q^{-22} -56500 q^{-23} -13004 q^{-24} +34702 q^{-25} +58150 q^{-26} +39502 q^{-27} +65 q^{-28} -45519 q^{-29} -51820 q^{-30} -23870 q^{-31} +16060 q^{-32} +43584 q^{-33} +39740 q^{-34} +14429 q^{-35} -25239 q^{-36} -39997 q^{-37} -28734 q^{-38} -1411 q^{-39} +24455 q^{-40} +32015 q^{-41} +21942 q^{-42} -6334 q^{-43} -23166 q^{-44} -24707 q^{-45} -11495 q^{-46} +6871 q^{-47} +18726 q^{-48} +19971 q^{-49} +4875 q^{-50} -7707 q^{-51} -14685 q^{-52} -12040 q^{-53} -3182 q^{-54} +6384 q^{-55} +12097 q^{-56} +6920 q^{-57} +903 q^{-58} -5206 q^{-59} -7072 q^{-60} -5060 q^{-61} -208 q^{-62} +4665 q^{-63} +4072 q^{-64} +2696 q^{-65} -342 q^{-66} -2336 q^{-67} -3022 q^{-68} -1544 q^{-69} +917 q^{-70} +1252 q^{-71} +1561 q^{-72} +669 q^{-73} -200 q^{-74} -1062 q^{-75} -880 q^{-76} -27 q^{-77} +95 q^{-78} +487 q^{-79} +368 q^{-80} +189 q^{-81} -234 q^{-82} -281 q^{-83} -50 q^{-84} -78 q^{-85} +83 q^{-86} +97 q^{-87} +106 q^{-88} -37 q^{-89} -61 q^{-90} -3 q^{-91} -35 q^{-92} +6 q^{-93} +12 q^{-94} +32 q^{-95} -6 q^{-96} -13 q^{-97} +5 q^{-98} -7 q^{-99} +5 q^{-102} -3 q^{-104} + q^{-105} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{112}-3 q^{111}+2 q^{110}+4 q^{109}-6 q^{108}-2 q^{107}-q^{106}+10 q^{105}+9 q^{104}-19 q^{103}+5 q^{102}+7 q^{101}-23 q^{100}-11 q^{99}-3 q^{98}+54 q^{97}+66 q^{96}-43 q^{95}-26 q^{94}-48 q^{93}-120 q^{92}-41 q^{91}+10 q^{90}+256 q^{89}+360 q^{88}+57 q^{87}-118 q^{86}-439 q^{85}-697 q^{84}-416 q^{83}-11 q^{82}+938 q^{81}+1661 q^{80}+1136 q^{79}+234 q^{78}-1533 q^{77}-3073 q^{76}-2855 q^{75}-1462 q^{74}+2017 q^{73}+5545 q^{72}+6200 q^{71}+4188 q^{70}-1697 q^{69}-8598 q^{68}-11668 q^{67}-9984 q^{66}-939 q^{65}+11553 q^{64}+19699 q^{63}+20162 q^{62}+7772 q^{61}-12632 q^{60}-29561 q^{59}-35566 q^{58}-21280 q^{57}+8931 q^{56}+39285 q^{55}+56377 q^{54}+43541 q^{53}+2667 q^{52}-45848 q^{51}-80761 q^{50}-74920 q^{49}-25322 q^{48}+44770 q^{47}+105238 q^{46}+114826 q^{45}+60818 q^{44}-32714 q^{43}-125498 q^{42}-159320 q^{41}-108154 q^{40}+6363 q^{39}+136276 q^{38}+204052 q^{37}+165116 q^{36}+34010 q^{35}-134696 q^{34}-243111 q^{33}-225975 q^{32}-86496 q^{31}+118414 q^{30}+272168 q^{29}+285618 q^{28}+146556 q^{27}-89189 q^{26}-288323 q^{25}-338081 q^{24}-208618 q^{23}+49860 q^{22}+290962 q^{21}+379836 q^{20}+267301 q^{19}-5088 q^{18}-281941 q^{17}-409021 q^{16}-318196 q^{15}-40378 q^{14}+264251 q^{13}+425772 q^{12}+358976 q^{11}+82746 q^{10}-241485 q^9-432054 q^8-389088 q^7-119422 q^6+216898 q^5+430356 q^4+409333 q^3+149583 q^2-192532 q-423183-421788 q^{-1} -173788 q^{-2} +169564 q^{-3} +412620 q^{-4} +428330 q^{-5} +193300 q^{-6} -147605 q^{-7} -399497 q^{-8} -430936 q^{-9} -210137 q^{-10} +125684 q^{-11} +384105 q^{-12} +430664 q^{-13} +225747 q^{-14} -102197 q^{-15} -365409 q^{-16} -427749 q^{-17} -241544 q^{-18} +75535 q^{-19} +342270 q^{-20} +421529 q^{-21} +257631 q^{-22} -44562 q^{-23} -312837 q^{-24} -410688 q^{-25} -273437 q^{-26} +9110 q^{-27} +276036 q^{-28} +393037 q^{-29} +286912 q^{-30} +30306 q^{-31} -231134 q^{-32} -367093 q^{-33} -295693 q^{-34} -70743 q^{-35} +179153 q^{-36} +330748 q^{-37} +296342 q^{-38} +109569 q^{-39} -121869 q^{-40} -284915 q^{-41} -286634 q^{-42} -141581 q^{-43} +63812 q^{-44} +230046 q^{-45} +264303 q^{-46} +163756 q^{-47} -9038 q^{-48} -170584 q^{-49} -230685 q^{-50} -172069 q^{-51} -36434 q^{-52} +110653 q^{-53} +187209 q^{-54} +166245 q^{-55} +69551 q^{-56} -56314 q^{-57} -139368 q^{-58} -147372 q^{-59} -87038 q^{-60} +12099 q^{-61} +91670 q^{-62} +119293 q^{-63} +90197 q^{-64} +18811 q^{-65} -50124 q^{-66} -87023 q^{-67} -81246 q^{-68} -35776 q^{-69} +17916 q^{-70} +55884 q^{-71} +64964 q^{-72} +40523 q^{-73} +3154 q^{-74} -29803 q^{-75} -46006 q^{-76} -36794 q^{-77} -13917 q^{-78} +11200 q^{-79} +28515 q^{-80} +28303 q^{-81} +16730 q^{-82} +106 q^{-83} -14834 q^{-84} -18971 q^{-85} -14824 q^{-86} -5135 q^{-87} +5906 q^{-88} +10887 q^{-89} +10739 q^{-90} +6184 q^{-91} -907 q^{-92} -5177 q^{-93} -6764 q^{-94} -5221 q^{-95} -1033 q^{-96} +1882 q^{-97} +3592 q^{-98} +3494 q^{-99} +1452 q^{-100} -168 q^{-101} -1623 q^{-102} -2149 q^{-103} -1162 q^{-104} -277 q^{-105} +598 q^{-106} +1047 q^{-107} +671 q^{-108} +429 q^{-109} -79 q^{-110} -538 q^{-111} -402 q^{-112} -265 q^{-113} +9 q^{-114} +202 q^{-115} +121 q^{-116} +170 q^{-117} +90 q^{-118} -86 q^{-119} -87 q^{-120} -87 q^{-121} -16 q^{-122} +42 q^{-123} -5 q^{-124} +31 q^{-125} +35 q^{-126} -6 q^{-127} -12 q^{-128} -23 q^{-129} -3 q^{-130} +13 q^{-131} -5 q^{-132} +7 q^{-134} -5 q^{-137} +3 q^{-139} - q^{-140} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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