9 44

9_43

9_45

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 44's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 44 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 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	[22,21,2-]

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

[edit Notes on presentations of 9 44]

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 44"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,10,6,11 X3948 X9,3,10,2 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]=

[22,21,2-]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-3] Hyperbolic Volume 7.40677 A-Polynomial See Data:9 44/A-polynomial

[edit Notes for 9 44's three dimensional invariants] 9_44 has girth 4. See arXiv:math.GT/0508590 and a forthcoming paper by the same authors.

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 9 44's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}-4t+7-4t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q^{2}-2q+3-3q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}-a^{4}+z^{4}a^{2}+3z^{2}a^{2}+3a^{2}-2z^{2}-2+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{7}+az^{7}+2a^{4}z^{6}+3a^{2}z^{6}+z^{6}+a^{5}z^{5}-2a^{3}z^{5}-3az^{5}-7a^{4}z^{4}-10a^{2}z^{4}-3z^{4}-3a^{5}z^{3}-a^{3}z^{3}+4az^{3}+2z^{3}a^{-1}+5a^{4}z^{2}+10a^{2}z^{2}+z^{2}a^{-2}+6z^{2}+a^{5}z+a^{3}z-az-za^{-1}-a^{4}-3a^{2}-a^{-2}-2$
The A2 invariant	$-q^{16}+2q^{8}+q^{6}+q^{4}-1-q^{-4}+q^{-6}+q^{-8}$
The G2 invariant	$q^{80}-q^{78}+2q^{76}-3q^{74}+q^{72}-4q^{68}+6q^{66}-5q^{64}+3q^{62}-q^{60}-4q^{58}+5q^{56}-4q^{54}+2q^{50}-4q^{48}+4q^{46}-4q^{42}+7q^{40}-5q^{38}+3q^{36}-2q^{32}+6q^{30}-4q^{28}+6q^{26}-2q^{24}+2q^{22}+4q^{20}-4q^{18}+4q^{16}-2q^{14}+q^{12}+3q^{10}-4q^{8}+2q^{6}-4q^{2}+5-6q^{-2}+2q^{-6}-6q^{-8}+5q^{-10}-3q^{-12}+q^{-14}+q^{-16}-3q^{-18}+2q^{-20}+q^{-24}+q^{-26}+q^{-32}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 9_44.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{9}+q^{5}+q^{-1}-q^{-3}+q^{-5}$
2	$q^{32}-q^{30}-2q^{28}+2q^{26}+q^{24}-2q^{22}+2q^{18}-q^{16}-q^{14}+2q^{12}-q^{8}+q^{6}+q^{4}-q^{2}-1+3q^{-2}-2q^{-6}+2q^{-8}+q^{-10}-q^{-12}$
3	$-q^{63}+q^{61}+2q^{59}-3q^{55}-3q^{53}+3q^{51}+4q^{49}-4q^{45}-3q^{43}+3q^{41}+5q^{39}-q^{37}-6q^{35}-3q^{33}+6q^{31}+4q^{29}-4q^{27}-4q^{25}+4q^{23}+5q^{21}-3q^{19}-4q^{17}+2q^{15}+3q^{13}-2q^{11}-3q^{9}+3q^{5}+3q^{3}-2q-4q^{-1}+2q^{-3}+7q^{-5}+3q^{-7}-7q^{-9}-4q^{-11}+5q^{-13}+4q^{-15}-2q^{-17}-5q^{-19}+3q^{-23}+q^{-25}-q^{-29}$
4	$q^{104}-q^{102}-2q^{100}+q^{96}+5q^{94}+q^{92}-3q^{90}-5q^{88}-6q^{86}+5q^{84}+7q^{82}+5q^{80}-10q^{76}-7q^{74}-q^{72}+9q^{70}+14q^{68}+q^{66}-11q^{64}-16q^{62}-5q^{60}+15q^{58}+17q^{56}+2q^{54}-18q^{52}-18q^{50}+5q^{48}+21q^{46}+14q^{44}-9q^{42}-20q^{40}-5q^{38}+14q^{36}+13q^{34}-4q^{32}-15q^{30}-5q^{28}+9q^{26}+8q^{24}-2q^{22}-9q^{20}-3q^{18}+7q^{16}+6q^{14}+2q^{12}-4q^{10}-7q^{8}-q^{6}+6q^{4}+13q^{2}+3-15q^{-2}-15q^{-4}-q^{-6}+22q^{-8}+20q^{-10}-9q^{-12}-25q^{-14}-15q^{-16}+15q^{-18}+26q^{-20}+5q^{-22}-14q^{-24}-18q^{-26}-2q^{-28}+13q^{-30}+9q^{-32}-7q^{-36}-5q^{-38}+q^{-40}+2q^{-42}+2q^{-44}-q^{-48}$
5	$-q^{155}+q^{153}+2q^{151}-q^{147}-3q^{145}-3q^{143}-q^{141}+5q^{139}+7q^{137}+4q^{135}-q^{133}-7q^{131}-11q^{129}-8q^{127}+5q^{125}+11q^{123}+13q^{121}+9q^{119}-4q^{117}-16q^{115}-20q^{113}-10q^{111}+6q^{109}+24q^{107}+29q^{105}+13q^{103}-16q^{101}-37q^{99}-34q^{97}-8q^{95}+32q^{93}+50q^{91}+32q^{89}-13q^{87}-54q^{85}-54q^{83}-12q^{81}+43q^{79}+67q^{77}+38q^{75}-25q^{73}-67q^{71}-53q^{69}+3q^{67}+59q^{65}+62q^{63}+13q^{61}-46q^{59}-62q^{57}-22q^{55}+32q^{53}+54q^{51}+25q^{49}-23q^{47}-45q^{45}-23q^{43}+18q^{41}+34q^{39}+17q^{37}-13q^{35}-26q^{33}-10q^{31}+12q^{29}+19q^{27}+8q^{25}-9q^{23}-16q^{21}-10q^{19}+4q^{17}+17q^{15}+17q^{13}+6q^{11}-14q^{9}-29q^{7}-21q^{5}+9q^{3}+41q+40q^{-1}+5q^{-3}-44q^{-5}-63q^{-7}-27q^{-9}+42q^{-11}+81q^{-13}+49q^{-15}-23q^{-17}-82q^{-19}-70q^{-21}+q^{-23}+69q^{-25}+78q^{-27}+19q^{-29}-44q^{-31}-66q^{-33}-36q^{-35}+18q^{-37}+47q^{-39}+36q^{-41}+q^{-43}-23q^{-45}-26q^{-47}-11q^{-49}+6q^{-51}+14q^{-53}+9q^{-55}-3q^{-59}-4q^{-61}-2q^{-63}+q^{-65}+q^{-67}$

A2 Invariants.

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+q^{32}-2q^{30}+2q^{28}-2q^{26}+q^{24}-q^{22}+q^{20}+q^{18}-q^{16}+4q^{14}-2q^{12}+4q^{10}-3q^{8}+3q^{6}-2q^{4}+q^{2}-1-q^{-2}+q^{-4}-2q^{-6}+2q^{-8}-2q^{-10}+2q^{-12}+q^{-16}$
1,0	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle q^{56}-q^{52}-q^{50}+q^{48}+q^{46}-2q^{44}-2q^{42}+q^{40}+2q^{38}-2q^{34}-q^{32}+2q^{30}+q^{28}-q^{24}+q^{22}+q^{20}+q^{18}-q^{16}+2q^{12}+q^{10}-q^{8}-q^{6}+q^{4}+2q^{2}-2q^{-2}+2q^{-6}-2q^{-10}-q^{-12}+q^{-14}+2q^{-16}-q^{-20}+q^{-26}}

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-q^{44}+q^{42}-2q^{40}+2q^{38}-2q^{36}-2q^{32}-q^{28}-q^{26}+q^{24}-q^{22}+3q^{20}+5q^{16}+5q^{12}-q^{10}+3q^{8}-q^{6}+q^{4}-2q^{2}-1-q^{-2}-2q^{-4}+q^{-6}-2q^{-8}+q^{-10}-q^{-12}+3q^{-14}+q^{-18}+q^{-22}$

G2 Invariants.

Weight	Invariant
1,0	$q^{80}-q^{78}+2q^{76}-3q^{74}+q^{72}-4q^{68}+6q^{66}-5q^{64}+3q^{62}-q^{60}-4q^{58}+5q^{56}-4q^{54}+2q^{50}-4q^{48}+4q^{46}-4q^{42}+7q^{40}-5q^{38}+3q^{36}-2q^{32}+6q^{30}-4q^{28}+6q^{26}-2q^{24}+2q^{22}+4q^{20}-4q^{18}+4q^{16}-2q^{14}+q^{12}+3q^{10}-4q^{8}+2q^{6}-4q^{2}+5-6q^{-2}+2q^{-6}-6q^{-8}+5q^{-10}-3q^{-12}+q^{-14}+q^{-16}-3q^{-18}+2q^{-20}+q^{-24}+q^{-26}+q^{-32}+q^{-38}$

.

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 44"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{2}-4t+7-4t^{-1}+t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}+1$

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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{7}+az^{7}+2a^{4}z^{6}+3a^{2}z^{6}+z^{6}+a^{5}z^{5}-2a^{3}z^{5}-3az^{5}-7a^{4}z^{4}-10a^{2}z^{4}-3z^{4}-3a^{5}z^{3}-a^{3}z^{3}+4az^{3}+2z^{3}a^{-1}+5a^{4}z^{2}+10a^{2}z^{2}+z^{2}a^{-2}+6z^{2}+a^{5}z+a^{3}z-az-za^{-1}-a^{4}-3a^{2}-a^{-2}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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 44"];

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]=

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

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]=

{}

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:

(0, -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 $0$ $-8$ $0$ $0$ $-8$ $0$ 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 \frac{16}{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{64}{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 -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 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 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 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 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 48} 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{56}{3}} ${\frac {104}{3}}$ 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{16}{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 0}

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 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} ). 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 44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 χ 5               1 1 3             1   -1 1           2 1   1 -1         2 2     0 -3       1 1       0 -5     1 2         1 -7   1 1           0 -9   1             1 -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}} $i=-1$ $i=1$ $r=-5$ 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}\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}\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=-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}\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=-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}\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=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}^{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}^{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}\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=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 {\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^5+2 q^4+q^3-5 q^2+4 q+4-9 q^{-1} +4 q^{-2} +6 q^{-3} -9 q^{-4} +2 q^{-5} +7 q^{-6} -7 q^{-7} - q^{-8} +7 q^{-9} -4 q^{-10} -3 q^{-11} +5 q^{-12} - q^{-13} -2 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^{13}+q^{12}+q^{11}+2 q^{10}-4 q^9-4 q^8+4 q^7+8 q^6-3 q^5-13 q^4+q^3+18 q^2+q-18-5 q^{-1} +20 q^{-2} +6 q^{-3} -18 q^{-4} -8 q^{-5} +17 q^{-6} +7 q^{-7} -13 q^{-8} -9 q^{-9} +11 q^{-10} +8 q^{-11} -5 q^{-12} -10 q^{-13} +3 q^{-14} +8 q^{-15} +3 q^{-16} -8 q^{-17} -6 q^{-18} +5 q^{-19} +8 q^{-20} -2 q^{-21} -8 q^{-22} - q^{-23} +7 q^{-24} +2 q^{-25} -4 q^{-26} -2 q^{-27} + q^{-28} +2 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^{22}+q^{21}+2 q^{20}-q^{18}-7 q^{17}-q^{16}+9 q^{15}+9 q^{14}+3 q^{13}-22 q^{12}-17 q^{11}+13 q^{10}+28 q^9+24 q^8-33 q^7-47 q^6+3 q^5+44 q^4+53 q^3-31 q^2-70 q-11+44 q^{-1} +71 q^{-2} -21 q^{-3} -77 q^{-4} -18 q^{-5} +38 q^{-6} +74 q^{-7} -15 q^{-8} -73 q^{-9} -17 q^{-10} +28 q^{-11} +68 q^{-12} -8 q^{-13} -63 q^{-14} -16 q^{-15} +14 q^{-16} +58 q^{-17} +3 q^{-18} -46 q^{-19} -15 q^{-20} -5 q^{-21} +43 q^{-22} +14 q^{-23} -23 q^{-24} -8 q^{-25} -21 q^{-26} +20 q^{-27} +14 q^{-28} -3 q^{-29} +7 q^{-30} -23 q^{-31} +3 q^{-33} +2 q^{-34} +19 q^{-35} -10 q^{-36} -5 q^{-37} -7 q^{-38} -4 q^{-39} +16 q^{-40} -5 q^{-43} -6 q^{-44} +5 q^{-45} + q^{-46} +2 q^{-47} - q^{-48} -2 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^{31}-3 q^{29}-2 q^{28}+q^{27}+3 q^{26}+10 q^{25}+5 q^{24}-11 q^{23}-19 q^{22}-14 q^{21}+6 q^{20}+34 q^{19}+40 q^{18}-48 q^{16}-68 q^{15}-24 q^{14}+56 q^{13}+103 q^{12}+59 q^{11}-57 q^{10}-136 q^9-95 q^8+44 q^7+162 q^6+131 q^5-25 q^4-175 q^3-164 q^2+8 q+181+180 q^{-1} +10 q^{-2} -174 q^{-3} -196 q^{-4} -22 q^{-5} +173 q^{-6} +195 q^{-7} +30 q^{-8} -163 q^{-9} -196 q^{-10} -35 q^{-11} +159 q^{-12} +189 q^{-13} +37 q^{-14} -146 q^{-15} -185 q^{-16} -42 q^{-17} +137 q^{-18} +173 q^{-19} +50 q^{-20} -116 q^{-21} -168 q^{-22} -58 q^{-23} +96 q^{-24} +151 q^{-25} +72 q^{-26} -68 q^{-27} -139 q^{-28} -80 q^{-29} +42 q^{-30} +111 q^{-31} +88 q^{-32} -9 q^{-33} -90 q^{-34} -83 q^{-35} -14 q^{-36} +55 q^{-37} +74 q^{-38} +33 q^{-39} -27 q^{-40} -54 q^{-41} -38 q^{-42} +32 q^{-44} +33 q^{-45} +14 q^{-46} -9 q^{-47} -20 q^{-48} -18 q^{-49} -8 q^{-50} +7 q^{-51} +11 q^{-52} +12 q^{-53} +9 q^{-54} -2 q^{-55} -13 q^{-56} -11 q^{-57} -5 q^{-58} +2 q^{-59} +13 q^{-60} +10 q^{-61} -7 q^{-63} -7 q^{-64} -4 q^{-65} +7 q^{-67} +4 q^{-68} - q^{-69} -2 q^{-70} - q^{-71} -2 q^{-72} + q^{-73} +2 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^{47}-q^{46}-q^{45}-q^{42}-q^{41}+8 q^{40}+3 q^{39}-2 q^{37}-8 q^{36}-17 q^{35}-16 q^{34}+17 q^{33}+26 q^{32}+31 q^{31}+25 q^{30}-6 q^{29}-65 q^{28}-88 q^{27}-25 q^{26}+31 q^{25}+100 q^{24}+134 q^{23}+82 q^{22}-83 q^{21}-213 q^{20}-173 q^{19}-66 q^{18}+128 q^{17}+297 q^{16}+285 q^{15}+9 q^{14}-288 q^{13}-365 q^{12}-266 q^{11}+47 q^{10}+399 q^9+506 q^8+182 q^7-259 q^6-478 q^5-452 q^4-93 q^3+397 q^2+625 q+328-179 q^{-1} -490 q^{-2} -539 q^{-3} -199 q^{-4} +349 q^{-5} +645 q^{-6} +390 q^{-7} -120 q^{-8} -460 q^{-9} -549 q^{-10} -240 q^{-11} +310 q^{-12} +624 q^{-13} +399 q^{-14} -96 q^{-15} -428 q^{-16} -530 q^{-17} -248 q^{-18} +279 q^{-19} +587 q^{-20} +395 q^{-21} -71 q^{-22} -384 q^{-23} -500 q^{-24} -261 q^{-25} +221 q^{-26} +524 q^{-27} +395 q^{-28} -12 q^{-29} -301 q^{-30} -454 q^{-31} -296 q^{-32} +115 q^{-33} +420 q^{-34} +391 q^{-35} +84 q^{-36} -168 q^{-37} -372 q^{-38} -330 q^{-39} -28 q^{-40} +264 q^{-41} +347 q^{-42} +175 q^{-43} - q^{-44} -234 q^{-45} -311 q^{-46} -151 q^{-47} +75 q^{-48} +231 q^{-49} +193 q^{-50} +135 q^{-51} -59 q^{-52} -203 q^{-53} -182 q^{-54} -74 q^{-55} +73 q^{-56} +109 q^{-57} +163 q^{-58} +67 q^{-59} -51 q^{-60} -105 q^{-61} -105 q^{-62} -29 q^{-63} -11 q^{-64} +85 q^{-65} +76 q^{-66} +36 q^{-67} -7 q^{-68} -43 q^{-69} -22 q^{-70} -58 q^{-71} +2 q^{-72} +16 q^{-73} +25 q^{-74} +18 q^{-75} +8 q^{-76} +25 q^{-77} -28 q^{-78} -10 q^{-79} -16 q^{-80} -6 q^{-81} -7 q^{-82} +2 q^{-83} +33 q^{-84} +7 q^{-86} -5 q^{-87} -6 q^{-88} -15 q^{-89} -10 q^{-90} +12 q^{-91} + q^{-92} +8 q^{-93} +3 q^{-94} +3 q^{-95} -7 q^{-96} -6 q^{-97} +3 q^{-98} -2 q^{-99} +2 q^{-100} + q^{-101} +2 q^{-102} - q^{-103} -2 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^{63}-q^{62}-q^{61}-q^{60}+2 q^{58}+q^{57}+2 q^{56}+5 q^{55}+q^{54}-5 q^{53}-11 q^{52}-14 q^{51}-3 q^{50}+q^{49}+14 q^{48}+34 q^{47}+36 q^{46}+16 q^{45}-23 q^{44}-66 q^{43}-74 q^{42}-62 q^{41}-14 q^{40}+83 q^{39}+152 q^{38}+166 q^{37}+84 q^{36}-76 q^{35}-217 q^{34}-294 q^{33}-243 q^{32}-15 q^{31}+257 q^{30}+461 q^{29}+466 q^{28}+182 q^{27}-225 q^{26}-602 q^{25}-725 q^{24}-446 q^{23}+93 q^{22}+683 q^{21}+1000 q^{20}+770 q^{19}+109 q^{18}-684 q^{17}-1216 q^{16}-1089 q^{15}-386 q^{14}+595 q^{13}+1362 q^{12}+1380 q^{11}+672 q^{10}-460 q^9-1421 q^8-1593 q^7-911 q^6+284 q^5+1403 q^4+1723 q^3+1113 q^2-125 q-1358-1784 q^{-1} -1228 q^{-2} +3 q^{-3} +1281 q^{-4} +1788 q^{-5} +1300 q^{-6} +89 q^{-7} -1221 q^{-8} -1778 q^{-9} -1318 q^{-10} -135 q^{-11} +1171 q^{-12} +1745 q^{-13} +1319 q^{-14} +163 q^{-15} -1133 q^{-16} -1720 q^{-17} -1309 q^{-18} -173 q^{-19} +1103 q^{-20} +1689 q^{-21} +1291 q^{-22} +187 q^{-23} -1064 q^{-24} -1655 q^{-25} -1278 q^{-26} -208 q^{-27} +1010 q^{-28} +1608 q^{-29} +1266 q^{-30} +250 q^{-31} -934 q^{-32} -1544 q^{-33} -1252 q^{-34} -313 q^{-35} +818 q^{-36} +1456 q^{-37} +1250 q^{-38} +398 q^{-39} -679 q^{-40} -1339 q^{-41} -1222 q^{-42} -503 q^{-43} +484 q^{-44} +1189 q^{-45} +1197 q^{-46} +617 q^{-47} -285 q^{-48} -1000 q^{-49} -1117 q^{-50} -717 q^{-51} +45 q^{-52} +762 q^{-53} +1020 q^{-54} +791 q^{-55} +171 q^{-56} -512 q^{-57} -842 q^{-58} -798 q^{-59} -377 q^{-60} +236 q^{-61} +635 q^{-62} +751 q^{-63} +500 q^{-64} +10 q^{-65} -380 q^{-66} -617 q^{-67} -557 q^{-68} -209 q^{-69} +139 q^{-70} +428 q^{-71} +514 q^{-72} +321 q^{-73} +77 q^{-74} -220 q^{-75} -402 q^{-76} -337 q^{-77} -211 q^{-78} +28 q^{-79} +238 q^{-80} +277 q^{-81} +260 q^{-82} +108 q^{-83} -88 q^{-84} -164 q^{-85} -227 q^{-86} -163 q^{-87} -23 q^{-88} +42 q^{-89} +146 q^{-90} +154 q^{-91} +79 q^{-92} +34 q^{-93} -63 q^{-94} -94 q^{-95} -63 q^{-96} -76 q^{-97} -12 q^{-98} +41 q^{-99} +37 q^{-100} +64 q^{-101} +22 q^{-102} +4 q^{-103} +15 q^{-104} -33 q^{-105} -32 q^{-106} -18 q^{-107} -23 q^{-108} +8 q^{-109} +2 q^{-110} +4 q^{-111} +37 q^{-112} +12 q^{-113} +6 q^{-114} -20 q^{-116} -7 q^{-117} -13 q^{-118} -15 q^{-119} +7 q^{-120} +9 q^{-121} +12 q^{-122} +12 q^{-123} -5 q^{-124} + q^{-125} -3 q^{-126} -10 q^{-127} -3 q^{-128} -2 q^{-129} +4 q^{-130} +6 q^{-131} - q^{-132} +2 q^{-134} -2 q^{-135} - q^{-136} -2 q^{-137} + q^{-138} +2 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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