9 37

9_36

9_38

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 37's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 37 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code	-1, 4, -3, 1, -5, 6, -2, 9, -8, 3, -4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code	4 10 14 12 16 2 6 18 8
Conway Notation	[3,21,21]

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

[edit Notes on presentations of 9 37]

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 37"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 12 16 2 6 18 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,21,21]

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]=

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 \textrm{BR}(5,\{-1,-1,2,-1,-3,2,1,4,-3,2,-3,4\})}

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]=

{ 5, 12, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index 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 \{4,7\}} Nakanishi index 2 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.9894 A-Polynomial See Data:9 37/A-polynomial

[edit Notes for 9 37's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 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 1} Topological 4 genus 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 1} Concordance genus 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 1} Rasmussen s-Invariant 0

[edit Notes for 9 37's four dimensional invariants]

Polynomial invariants

Alexander polynomial	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 2 t^2-11 t+19-11 t^{-1} +2 t^{-2} }
Conway polynomial	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 2 z^4-3 z^2+1}
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 45, 0 }
Jones polynomial	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^4-2 q^3+5 q^2-7 q+7-8 q^{-1} +7 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5} }
HOMFLY-PT polynomial (db, data sources)	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 -z^2 a^4+z^4 a^2+z^2 a^2+2 a^2+z^4-z^2-2-2 z^2 a^{-2} + a^{-4} }
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+5a^{2}z^{6}+3z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-6a^{3}z^{5}-13az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-17a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-13z^{4}-2a^{5}z^{3}+3a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}-2z^{3}a^{-3}+5a^{4}z^{2}+14a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}-2a^{3}z-7az-5za^{-1}-2a^{2}+a^{-4}-2$
The A2 invariant	$-q^{16}+q^{14}+q^{12}-q^{10}+3q^{8}+q^{6}-3-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-4q^{70}-4q^{68}+16q^{66}-23q^{64}+29q^{62}-22q^{60}+6q^{58}+16q^{56}-38q^{54}+52q^{52}-49q^{50}+24q^{48}+8q^{46}-33q^{44}+50q^{42}-44q^{40}+24q^{38}+7q^{36}-27q^{34}+33q^{32}-28q^{30}-6q^{28}+40q^{26}-46q^{24}+41q^{22}-18q^{20}-17q^{18}+57q^{16}-71q^{14}+65q^{12}-48q^{10}+4q^{8}+43q^{6}-65q^{4}+65q^{2}-47+16q^{-2}+18q^{-4}-39q^{-6}+32q^{-8}-22q^{-10}-7q^{-12}+33q^{-14}-38q^{-16}+22q^{-18}+6q^{-20}-33q^{-22}+50q^{-24}-48q^{-26}+29q^{-28}-8q^{-30}-20q^{-32}+38q^{-34}-40q^{-36}+34q^{-38}-14q^{-40}+2q^{-42}+9q^{-44}-15q^{-46}+15q^{-48}-12q^{-50}+8q^{-52}-2q^{-54}-q^{-56}+3q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_37.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-q^{7}+3q^{5}-q^{3}-q-2q^{-3}+3q^{-5}-q^{-7}+q^{-9}$
2	$q^{32}-2q^{30}-2q^{28}+6q^{26}-2q^{24}-7q^{22}+10q^{20}+3q^{18}-12q^{16}+8q^{14}+6q^{12}-13q^{10}+6q^{6}-3q^{4}-4q^{2}+5+9q^{-2}-8q^{-4}-2q^{-6}+13q^{-8}-9q^{-10}-7q^{-12}+11q^{-14}-3q^{-16}-5q^{-18}+4q^{-20}-q^{-24}+q^{-26}$
3	$-q^{63}+2q^{61}+2q^{59}-3q^{57}-6q^{55}+2q^{53}+13q^{51}-q^{49}-19q^{47}-7q^{45}+26q^{43}+19q^{41}-25q^{39}-34q^{37}+22q^{35}+46q^{33}-8q^{31}-56q^{29}-5q^{27}+54q^{25}+15q^{23}-51q^{21}-30q^{19}+45q^{17}+32q^{15}-26q^{13}-33q^{11}+18q^{9}+34q^{7}+3q^{5}-32q^{3}-18q+26q^{-1}+32q^{-3}-20q^{-5}-51q^{-7}+10q^{-9}+56q^{-11}+4q^{-13}-58q^{-15}-11q^{-17}+49q^{-19}+24q^{-21}-38q^{-23}-25q^{-25}+25q^{-27}+21q^{-29}-10q^{-31}-16q^{-33}+4q^{-35}+8q^{-37}-2q^{-39}-4q^{-41}+q^{-43}+q^{-45}-q^{-49}+q^{-51}$
4	$q^{104}-2q^{102}-2q^{100}+3q^{98}+3q^{96}+6q^{94}-9q^{92}-13q^{90}+2q^{88}+10q^{86}+31q^{84}-8q^{82}-41q^{80}-26q^{78}+5q^{76}+82q^{74}+38q^{72}-48q^{70}-91q^{68}-69q^{66}+104q^{64}+137q^{62}+39q^{60}-122q^{58}-208q^{56}+11q^{54}+186q^{52}+197q^{50}-30q^{48}-287q^{46}-149q^{44}+110q^{42}+288q^{40}+120q^{38}-235q^{36}-241q^{34}-11q^{32}+262q^{30}+200q^{28}-122q^{26}-225q^{24}-87q^{22}+164q^{20}+187q^{18}-17q^{16}-167q^{14}-127q^{12}+61q^{10}+153q^{8}+85q^{6}-91q^{4}-160q^{2}-60+115q^{-2}+209q^{-4}+16q^{-6}-183q^{-8}-204q^{-10}+30q^{-12}+292q^{-14}+149q^{-16}-128q^{-18}-296q^{-20}-105q^{-22}+253q^{-24}+229q^{-26}+7q^{-28}-247q^{-30}-196q^{-32}+108q^{-34}+185q^{-36}+105q^{-38}-106q^{-40}-159q^{-42}-5q^{-44}+68q^{-46}+92q^{-48}-3q^{-50}-63q^{-52}-22q^{-54}+34q^{-58}+9q^{-60}-13q^{-62}-2q^{-64}-6q^{-66}+6q^{-68}+q^{-70}-3q^{-72}+2q^{-74}-2q^{-76}+q^{-78}-q^{-82}+q^{-84}$
5	$-q^{155}+2q^{153}+2q^{151}-3q^{149}-3q^{147}-3q^{145}+q^{143}+9q^{141}+13q^{139}-2q^{137}-20q^{135}-22q^{133}-7q^{131}+24q^{129}+49q^{127}+36q^{125}-30q^{123}-87q^{121}-77q^{119}+q^{117}+113q^{115}+163q^{113}+70q^{111}-122q^{109}-251q^{107}-195q^{105}+42q^{103}+322q^{101}+387q^{99}+126q^{97}-304q^{95}-569q^{93}-409q^{91}+149q^{89}+686q^{87}+733q^{85}+168q^{83}-652q^{81}-1042q^{79}-583q^{77}+432q^{75}+1201q^{73}+1037q^{71}-51q^{69}-1202q^{67}-1392q^{65}-396q^{63}+997q^{61}+1595q^{59}+828q^{57}-676q^{55}-1618q^{53}-1126q^{51}+330q^{49}+1462q^{47}+1286q^{45}+2q^{43}-1231q^{41}-1298q^{39}-210q^{37}+938q^{35}+1177q^{33}+372q^{31}-702q^{29}-1041q^{27}-424q^{25}+476q^{23}+875q^{21}+485q^{19}-293q^{17}-756q^{15}-533q^{13}+112q^{11}+662q^{9}+652q^{7}+95q^{5}-573q^{3}-803q-357q^{-1}+462q^{-3}+977q^{-5}+682q^{-7}-284q^{-9}-1125q^{-11}-1033q^{-13}+10q^{-15}+1166q^{-17}+1377q^{-19}+345q^{-21}-1089q^{-23}-1605q^{-25}-733q^{-27}+812q^{-29}+1696q^{-31}+1087q^{-33}-457q^{-35}-1549q^{-37}-1305q^{-39}+15q^{-41}+1246q^{-43}+1358q^{-45}+342q^{-47}-825q^{-49}-1191q^{-51}-576q^{-53}+399q^{-55}+907q^{-57}+640q^{-59}-75q^{-61}-579q^{-63}-543q^{-65}-127q^{-67}+285q^{-69}+390q^{-71}+185q^{-73}-99q^{-75}-223q^{-77}-155q^{-79}-2q^{-81}+105q^{-83}+96q^{-85}+29q^{-87}-36q^{-89}-49q^{-91}-20q^{-93}+9q^{-95}+16q^{-97}+11q^{-99}+4q^{-101}-8q^{-103}-5q^{-105}+2q^{-107}-q^{-109}+3q^{-113}-q^{-115}-2q^{-117}+q^{-119}-q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+q^{12}-q^{10}+3q^{8}+q^{6}-3-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}$
1,1	$q^{44}-4q^{42}+10q^{40}-22q^{38}+42q^{36}-70q^{34}+100q^{32}-136q^{30}+169q^{28}-186q^{26}+186q^{24}-156q^{22}+117q^{20}-40q^{18}-44q^{16}+138q^{14}-229q^{12}+284q^{10}-348q^{8}+352q^{6}-343q^{4}+300q^{2}-214+146q^{-2}-42q^{-4}-30q^{-6}+104q^{-8}-154q^{-10}+166q^{-12}-170q^{-14}+152q^{-16}-132q^{-18}+99q^{-20}-72q^{-22}+50q^{-24}-32q^{-26}+21q^{-28}-10q^{-30}+6q^{-32}-2q^{-34}+q^{-36}$
2,0	$q^{42}-q^{40}-2q^{38}+3q^{34}+q^{32}-6q^{30}+8q^{26}+5q^{24}-6q^{22}-2q^{20}+8q^{18}+4q^{16}-9q^{14}-5q^{12}+2q^{10}-5q^{8}-4q^{6}+2q^{2}+2+8q^{-2}+6q^{-4}-3q^{-6}-q^{-8}+8q^{-10}-11q^{-14}+7q^{-18}-8q^{-22}-3q^{-24}+4q^{-26}+2q^{-28}-q^{-30}+q^{-34}+q^{-36}$

A3 Invariants.

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

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-4q^{70}-4q^{68}+16q^{66}-23q^{64}+29q^{62}-22q^{60}+6q^{58}+16q^{56}-38q^{54}+52q^{52}-49q^{50}+24q^{48}+8q^{46}-33q^{44}+50q^{42}-44q^{40}+24q^{38}+7q^{36}-27q^{34}+33q^{32}-28q^{30}-6q^{28}+40q^{26}-46q^{24}+41q^{22}-18q^{20}-17q^{18}+57q^{16}-71q^{14}+65q^{12}-48q^{10}+4q^{8}+43q^{6}-65q^{4}+65q^{2}-47+16q^{-2}+18q^{-4}-39q^{-6}+32q^{-8}-22q^{-10}-7q^{-12}+33q^{-14}-38q^{-16}+22q^{-18}+6q^{-20}-33q^{-22}+50q^{-24}-48q^{-26}+29q^{-28}-8q^{-30}-20q^{-32}+38q^{-34}-40q^{-36}+34q^{-38}-14q^{-40}+2q^{-42}+9q^{-44}-15q^{-46}+15q^{-48}-12q^{-50}+8q^{-52}-2q^{-54}-q^{-56}+3q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

.

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 37"];

In[4]:=

Alexander[K][t]

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

Out[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 2 t^2-11 t+19-11 t^{-1} +2 t^{-2} }

In[5]:=

Conway[K][z]

Out[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 2 z^4-3 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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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 -z^2 a^4+z^4 a^2+z^2 a^2+2 a^2+z^4-z^2-2-2 z^2 a^{-2} + a^{-4} }

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+5a^{2}z^{6}+3z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-6a^{3}z^{5}-13az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-17a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-13z^{4}-2a^{5}z^{3}+3a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}-2z^{3}a^{-3}+5a^{4}z^{2}+14a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}-2a^{3}z-7az-5za^{-1}-2a^{2}+a^{-4}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n100,}

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

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 37"];

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]=

{ $2t^{2}-11t+19-11t^{-1}+2t^{-2}$, $q^{4}-2q^{3}+5q^{2}-7q+7-8q^{-1}+7q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$ }

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]=

{K11n100,}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-3, -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 $-12$ $-8$ 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 72} 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 82} 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 22} 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 96} 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 \frac{496}{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 \frac{160}{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 24} 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 -288} $32$ 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 -984} 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 -264} 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 -\frac{8191}{10}} 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 -\frac{974}{15}} 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 -\frac{1954}{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 \frac{85}{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 -\frac{671}{10}}

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 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 t^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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 j} , alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where 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 j-2r=s+1} or 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 j-2r=s-1} , where 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 s=} 0 is the signature of 9 37. 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                 1   -1 5               4 1   3 3             3 1     -2 1           4 4       0 -1         5 4         -1 -3       2 3           -1 -5     2 5             3 -7   1 2               -1 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} 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 i=-1} 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 i=1} 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 r=-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 {\mathbb Z}} 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 r=-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 {\mathbb Z}^{2}\oplus{\mathbb Z}_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 {\mathbb Z}} 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 r=-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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-1} 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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=0} 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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=1} 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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}\oplus{\mathbb Z}_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 {\mathbb Z}} 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 r=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 {\mathbb Z}_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 {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the 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+1} -dimensional representation of 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 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}	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 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}-2 q^{11}+q^{10}+5 q^9-11 q^8+3 q^7+19 q^6-29 q^5+q^4+41 q^3-44 q^2-5 q+58-48 q^{-1} -14 q^{-2} +59 q^{-3} -39 q^{-4} -20 q^{-5} +46 q^{-6} -20 q^{-7} -18 q^{-8} +26 q^{-9} -5 q^{-10} -11 q^{-11} +9 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}-2 q^{23}+q^{22}+q^{21}+q^{20}-7 q^{19}+3 q^{18}+11 q^{17}-3 q^{16}-27 q^{15}+9 q^{14}+42 q^{13}+q^{12}-77 q^{11}-4 q^{10}+104 q^9+26 q^8-137 q^7-51 q^6+166 q^5+78 q^4-183 q^3-112 q^2+197 q+130-189 q^{-1} -156 q^{-2} +183 q^{-3} +165 q^{-4} -158 q^{-5} -172 q^{-6} +132 q^{-7} +172 q^{-8} -100 q^{-9} -159 q^{-10} +57 q^{-11} +151 q^{-12} -34 q^{-13} -120 q^{-14} -2 q^{-15} +100 q^{-16} +14 q^{-17} -66 q^{-18} -26 q^{-19} +44 q^{-20} +23 q^{-21} -22 q^{-22} -19 q^{-23} +11 q^{-24} +11 q^{-25} -4 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}-2 q^{39}+q^{38}+q^{37}-3 q^{36}+5 q^{35}-7 q^{34}+5 q^{33}+6 q^{32}-15 q^{31}+9 q^{30}-18 q^{29}+27 q^{28}+31 q^{27}-49 q^{26}-13 q^{25}-59 q^{24}+87 q^{23}+126 q^{22}-73 q^{21}-86 q^{20}-213 q^{19}+140 q^{18}+337 q^{17}+7 q^{16}-163 q^{15}-517 q^{14}+89 q^{13}+591 q^{12}+229 q^{11}-139 q^{10}-875 q^9-102 q^8+759 q^7+506 q^6+4 q^5-1137 q^4-336 q^3+780 q^2+705 q+197-1231 q^{-1} -511 q^{-2} +680 q^{-3} +774 q^{-4} +373 q^{-5} -1163 q^{-6} -603 q^{-7} +492 q^{-8} +734 q^{-9} +523 q^{-10} -959 q^{-11} -626 q^{-12} +241 q^{-13} +596 q^{-14} +626 q^{-15} -637 q^{-16} -564 q^{-17} -32 q^{-18} +366 q^{-19} +632 q^{-20} -282 q^{-21} -396 q^{-22} -210 q^{-23} +107 q^{-24} +494 q^{-25} -25 q^{-26} -169 q^{-27} -221 q^{-28} -68 q^{-29} +275 q^{-30} +61 q^{-31} -8 q^{-32} -123 q^{-33} -101 q^{-34} +102 q^{-35} +39 q^{-36} +35 q^{-37} -37 q^{-38} -57 q^{-39} +25 q^{-40} +8 q^{-41} +20 q^{-42} -4 q^{-43} -18 q^{-44} +4 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}-2 q^{59}+q^{58}+q^{57}-3 q^{56}+q^{55}+5 q^{54}-5 q^{53}+4 q^{51}-10 q^{50}-2 q^{49}+17 q^{48}+2 q^{47}+5 q^{46}-3 q^{45}-39 q^{44}-31 q^{43}+30 q^{42}+67 q^{41}+72 q^{40}+6 q^{39}-146 q^{38}-184 q^{37}-38 q^{36}+191 q^{35}+356 q^{34}+211 q^{33}-251 q^{32}-596 q^{31}-454 q^{30}+155 q^{29}+860 q^{28}+926 q^{27}+16 q^{26}-1104 q^{25}-1429 q^{24}-460 q^{23}+1226 q^{22}+2093 q^{21}+1032 q^{20}-1216 q^{19}-2660 q^{18}-1780 q^{17}+982 q^{16}+3185 q^{15}+2576 q^{14}-607 q^{13}-3544 q^{12}-3325 q^{11}+110 q^{10}+3701 q^9+4010 q^8+425 q^7-3755 q^6-4481 q^5-933 q^4+3609 q^3+4851 q^2+1391 q-3460-4996 q^{-1} -1752 q^{-2} +3163 q^{-3} +5081 q^{-4} +2059 q^{-5} -2903 q^{-6} -4986 q^{-7} -2302 q^{-8} +2518 q^{-9} +4858 q^{-10} +2522 q^{-11} -2125 q^{-12} -4596 q^{-13} -2701 q^{-14} +1618 q^{-15} +4241 q^{-16} +2861 q^{-17} -1051 q^{-18} -3791 q^{-19} -2940 q^{-20} +470 q^{-21} +3153 q^{-22} +2928 q^{-23} +182 q^{-24} -2507 q^{-25} -2764 q^{-26} -662 q^{-27} +1697 q^{-28} +2436 q^{-29} +1124 q^{-30} -1003 q^{-31} -1997 q^{-32} -1260 q^{-33} +304 q^{-34} +1440 q^{-35} +1314 q^{-36} +148 q^{-37} -909 q^{-38} -1096 q^{-39} -465 q^{-40} +425 q^{-41} +855 q^{-42} +538 q^{-43} -89 q^{-44} -531 q^{-45} -512 q^{-46} -112 q^{-47} +297 q^{-48} +378 q^{-49} +176 q^{-50} -101 q^{-51} -251 q^{-52} -177 q^{-53} +17 q^{-54} +141 q^{-55} +120 q^{-56} +28 q^{-57} -59 q^{-58} -84 q^{-59} -33 q^{-60} +29 q^{-61} +42 q^{-62} +18 q^{-63} -2 q^{-64} -18 q^{-65} -20 q^{-66} +4 q^{-67} +11 q^{-68} +3 q^{-69} -5 q^{-72} +3 q^{-74} - q^{-75} }

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.

9_36 9_38