9 43

9_42

9_44

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 43's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 43 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X15,1,16,18 X11,17,12,16 X17,13,18,12 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	[211,3,2-]

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

[edit Notes on presentations of 9 43]

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 43"];

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 X15,1,16,18 X11,17,12,16 X17,13,18,12 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]=

[211,3,2-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 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 \{4,5\}} Nakanishi index 1 Maximal Thurston-Bennequin number [1][-10] Hyperbolic Volume 5.90409 A-Polynomial See Data:9 43/A-polynomial

[edit Notes for 9 43'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 2} Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for 9 43's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+3t^{2}-2t+1-2t^{-1}+3t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 13, 4 }
Jones polynomial	$-q^{7}+2q^{6}-2q^{5}+2q^{4}-2q^{3}+2q^{2}-q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}+z^{4}a^{-2}-5z^{4}a^{-4}+z^{4}a^{-6}+4z^{2}a^{-2}-7z^{2}a^{-4}+4z^{2}a^{-6}+3a^{-2}-4a^{-4}+3a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-3}+z^{7}a^{-5}+z^{6}a^{-2}+3z^{6}a^{-4}+2z^{6}a^{-6}-4z^{5}a^{-3}-3z^{5}a^{-5}+z^{5}a^{-7}-5z^{4}a^{-2}-13z^{4}a^{-4}-8z^{4}a^{-6}+3z^{3}a^{-3}+z^{3}a^{-5}-2z^{3}a^{-7}+7z^{2}a^{-2}+14z^{2}a^{-4}+9z^{2}a^{-6}+2z^{2}a^{-8}+za^{-7}+za^{-9}-3a^{-2}-4a^{-4}-3a^{-6}-a^{-8}$
The A2 invariant	$1+q^{-2}+q^{-4}+q^{-6}-2q^{-12}+q^{-18}+q^{-20}-q^{-26}$
The G2 invariant	$q^{-2}+2q^{-6}-q^{-8}+q^{-10}+q^{-12}-q^{-14}+4q^{-16}-q^{-18}+2q^{-20}+q^{-22}-q^{-24}+3q^{-26}-q^{-30}+2q^{-32}-q^{-34}+2q^{-38}-3q^{-40}+2q^{-42}-2q^{-44}-q^{-48}-3q^{-50}+q^{-52}-3q^{-54}+q^{-56}-2q^{-58}-q^{-64}+q^{-66}-q^{-68}+2q^{-72}+3q^{-78}-2q^{-80}+4q^{-82}+2q^{-88}-2q^{-90}+2q^{-92}-q^{-100}-q^{-102}+q^{-104}-q^{-106}-q^{-108}-q^{-112}-q^{-116}+q^{-120}$

Further Quantum Invariants

Further quantum knot invariants for 9_43.

A1 Invariants.

Weight	Invariant
1	$q+q^{-3}+q^{-13}-q^{-15}$
2	$q^{6}-q^{2}+1+q^{-2}-q^{-4}+q^{-8}+q^{-10}+q^{-14}+q^{-16}-q^{-18}+q^{-22}-q^{-24}-q^{-26}-q^{-32}+q^{-36}$
3	$q^{15}-q^{11}-q^{9}+q^{7}+2q^{5}-2q-q^{-1}+q^{-3}+2q^{-5}+q^{-7}-q^{-11}+q^{-13}+3q^{-15}+q^{-17}-2q^{-19}-2q^{-21}+2q^{-23}+2q^{-25}-q^{-27}-2q^{-29}+q^{-31}+2q^{-33}-q^{-35}-2q^{-37}+q^{-39}+q^{-41}-2q^{-43}-2q^{-45}+q^{-49}+q^{-51}-2q^{-55}-q^{-57}+3q^{-59}+3q^{-61}-2q^{-63}-4q^{-65}+2q^{-67}+3q^{-69}-2q^{-73}-q^{-75}+q^{-77}$
4	$q^{28}-q^{24}-q^{22}-q^{20}+2q^{18}+2q^{16}+q^{14}-q^{12}-4q^{10}-q^{8}+q^{6}+3q^{4}+3q^{2}-q^{-2}-3q^{-4}-2q^{-6}+2q^{-8}+4q^{-10}+4q^{-12}-5q^{-16}-4q^{-18}+q^{-20}+6q^{-22}+6q^{-24}-2q^{-26}-5q^{-28}-4q^{-30}+2q^{-32}+8q^{-34}+3q^{-36}-3q^{-38}-6q^{-40}-3q^{-42}+5q^{-44}+4q^{-46}-q^{-48}-5q^{-50}-4q^{-52}+3q^{-54}+3q^{-56}-q^{-58}-3q^{-60}-2q^{-62}+3q^{-64}+2q^{-66}-q^{-68}-3q^{-70}-2q^{-72}+3q^{-74}+3q^{-76}+q^{-78}-q^{-80}-4q^{-82}-3q^{-84}+3q^{-86}+7q^{-88}+5q^{-90}-4q^{-92}-10q^{-94}-3q^{-96}+6q^{-98}+10q^{-100}+2q^{-102}-10q^{-104}-6q^{-106}+6q^{-110}+4q^{-112}-3q^{-114}-2q^{-116}-q^{-118}+2q^{-120}+q^{-122}-q^{-124}$
5	$q^{45}-q^{41}-q^{39}-q^{37}+2q^{33}+3q^{31}+q^{29}-q^{27}-3q^{25}-4q^{23}-2q^{21}+2q^{19}+5q^{17}+4q^{15}+2q^{13}-q^{11}-4q^{9}-5q^{7}-3q^{5}+4q+7q^{-1}+6q^{-3}+q^{-5}-6q^{-7}-9q^{-9}-7q^{-11}+9q^{-15}+13q^{-17}+8q^{-19}-3q^{-21}-11q^{-23}-12q^{-25}-4q^{-27}+8q^{-29}+16q^{-31}+12q^{-33}-12q^{-37}-16q^{-39}-8q^{-41}+6q^{-43}+16q^{-45}+13q^{-47}-13q^{-51}-16q^{-53}-5q^{-55}+9q^{-57}+16q^{-59}+9q^{-61}-6q^{-63}-15q^{-65}-12q^{-67}+2q^{-69}+12q^{-71}+11q^{-73}-11q^{-77}-11q^{-79}-q^{-81}+8q^{-83}+8q^{-85}-7q^{-89}-6q^{-91}+3q^{-93}+7q^{-95}+4q^{-97}-4q^{-99}-7q^{-101}-2q^{-103}+5q^{-105}+8q^{-107}+4q^{-109}-3q^{-111}-7q^{-113}-7q^{-115}-2q^{-117}+6q^{-119}+11q^{-121}+9q^{-123}+q^{-125}-10q^{-127}-17q^{-129}-10q^{-131}+6q^{-133}+18q^{-135}+17q^{-137}+2q^{-139}-17q^{-141}-24q^{-143}-11q^{-145}+12q^{-147}+22q^{-149}+16q^{-151}-2q^{-153}-16q^{-155}-15q^{-157}-2q^{-159}+9q^{-161}+9q^{-163}+3q^{-165}-3q^{-167}-4q^{-169}-2q^{-171}+q^{-173}+q^{-175}$

A2 Invariants.

Weight	Invariant
1,0	$1+q^{-2}+q^{-4}+q^{-6}-2q^{-12}+q^{-18}+q^{-20}-q^{-26}$
1,1	$q^{4}+4-2q^{-2}+6q^{-4}-4q^{-6}+4q^{-8}-4q^{-10}-q^{-12}+2q^{-14}-4q^{-16}+6q^{-18}-4q^{-20}+10q^{-22}-4q^{-24}+8q^{-26}-4q^{-28}+2q^{-30}-4q^{-32}-4q^{-34}-6q^{-38}+4q^{-40}-2q^{-42}+4q^{-44}+2q^{-48}-q^{-52}-2q^{-54}+q^{-60}$
2,0	$q^{4}+q^{2}+1+q^{-4}+q^{-6}-q^{-10}+q^{-20}+q^{-22}+q^{-24}+q^{-26}+2q^{-28}+2q^{-30}+q^{-32}-3q^{-36}-3q^{-38}-3q^{-40}-2q^{-42}-q^{-44}+2q^{-48}+3q^{-50}+2q^{-52}-q^{-58}-q^{-60}-q^{-62}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$1+2q^{-4}+2q^{-6}+2q^{-8}+2q^{-10}+q^{-12}-q^{-16}-2q^{-18}-2q^{-20}-q^{-22}-q^{-24}+q^{-28}+q^{-30}+2q^{-32}+q^{-34}+q^{-36}-q^{-40}-q^{-42}-q^{-44}-q^{-46}+q^{-50}$
1,0,0	$q^{-1}+q^{-3}+2q^{-5}+q^{-7}+2q^{-9}-q^{-13}-2q^{-15}-2q^{-17}-q^{-19}+2q^{-23}+q^{-25}+2q^{-27}-q^{-33}-q^{-35}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-4}+2q^{-6}+3q^{-8}+4q^{-10}+4q^{-12}+3q^{-14}+q^{-16}-2q^{-20}-4q^{-22}-4q^{-24}-2q^{-26}-q^{-28}+2q^{-34}+q^{-36}-q^{-38}+q^{-42}+2q^{-46}+3q^{-48}+q^{-50}-q^{-56}-3q^{-58}-2q^{-60}-q^{-66}+q^{-68}+q^{-70}$
1,0,0,0	$q^{-2}+q^{-4}+2q^{-6}+2q^{-8}+2q^{-10}+2q^{-12}-q^{-16}-3q^{-18}-2q^{-20}-3q^{-22}-q^{-24}+2q^{-28}+2q^{-30}+2q^{-32}+2q^{-34}-q^{-40}-q^{-42}-q^{-44}$

B2 Invariants.

Weight	Invariant
0,1	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 1+2 q^{-4} +2 q^{-8} + q^{-12} - q^{-16} -2 q^{-20} + q^{-22} -3 q^{-24} +2 q^{-26} - q^{-28} + q^{-30} + q^{-34} + q^{-36} + q^{-40} - q^{-42} + q^{-44} - q^{-46} - q^{-50} }
1,0	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 q^2+2 q^{-6} + q^{-8} +2 q^{-14} + q^{-16} - q^{-20} + q^{-24} - q^{-28} - q^{-30} - q^{-38} + q^{-42} - q^{-46} + q^{-50} + q^{-52} - q^{-56} + q^{-58} + q^{-60} - q^{-64} - q^{-72} - q^{-74} + q^{-80} }

D4 Invariants.

Weight

Invariant

1,0,0,0

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 q^{-2} +2 q^{-6} + q^{-8} +4 q^{-10} +2 q^{-12} +3 q^{-14} +2 q^{-16} +2 q^{-18} -2 q^{-22} -2 q^{-24} -4 q^{-26} -2 q^{-28} -4 q^{-30} - q^{-32} -3 q^{-34} +2 q^{-36} +3 q^{-40} +2 q^{-42} +3 q^{-44} +2 q^{-46} +2 q^{-48} + q^{-50} - q^{-52} -2 q^{-56} - q^{-58} -2 q^{-60} - q^{-64} + q^{-70} }

G2 Invariants.

Weight

Invariant

1,0

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 q^{-2} +2 q^{-6} - q^{-8} + q^{-10} + q^{-12} - q^{-14} +4 q^{-16} - q^{-18} +2 q^{-20} + q^{-22} - q^{-24} +3 q^{-26} - q^{-30} +2 q^{-32} - q^{-34} +2 q^{-38} -3 q^{-40} +2 q^{-42} -2 q^{-44} - q^{-48} -3 q^{-50} + q^{-52} -3 q^{-54} + q^{-56} -2 q^{-58} - q^{-64} + q^{-66} - q^{-68} +2 q^{-72} +3 q^{-78} -2 q^{-80} +4 q^{-82} +2 q^{-88} -2 q^{-90} +2 q^{-92} - q^{-100} - q^{-102} + q^{-104} - q^{-106} - q^{-108} - q^{-112} - q^{-116} + q^{-120} }

.

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 43"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+3t^{2}-2t+1-2t^{-1}+3t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 13, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{7}+2q^{6}-2q^{5}+2q^{4}-2q^{3}+2q^{2}-q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}+z^{4}a^{-2}-5z^{4}a^{-4}+z^{4}a^{-6}+4z^{2}a^{-2}-7z^{2}a^{-4}+4z^{2}a^{-6}+3a^{-2}-4a^{-4}+3a^{-6}-a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-3}+z^{7}a^{-5}+z^{6}a^{-2}+3z^{6}a^{-4}+2z^{6}a^{-6}-4z^{5}a^{-3}-3z^{5}a^{-5}+z^{5}a^{-7}-5z^{4}a^{-2}-13z^{4}a^{-4}-8z^{4}a^{-6}+3z^{3}a^{-3}+z^{3}a^{-5}-2z^{3}a^{-7}+7z^{2}a^{-2}+14z^{2}a^{-4}+9z^{2}a^{-6}+2z^{2}a^{-8}+za^{-7}+za^{-9}-3a^{-2}-4a^{-4}-3a^{-6}-a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, 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\leftrightarrow q^{-1}} ): {K11n12,}

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 43"];

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]=

{ 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^3+3 t^2-2 t+1-2 t^{-1} +3 t^{-2} - t^{-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^7+2 q^6-2 q^5+2 q^4-2 q^3+2 q^2-q+1} }

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]=

{K11n12,}

Vassiliev invariants

V2 and V3:

(1, 2)

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 $4$ 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 16} 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 \frac{254}{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{82}{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 64} 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{1024}{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{256}{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 80} ${\frac {32}{3}}$ 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 128} 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{1016}{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{328}{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{37231}{30}} 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{154}{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{31502}{45}} 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{593}{18}} 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{2671}{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 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=} 4 is the signature of 9 43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 χ 15               1 -1 13             1   1 11           1 1   0 9         1 1     0 7       1 1       0 5     1 1         0 3   1 2           1 1                 0 -1 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}} $i=3$ $i=5$ 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=-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}_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=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}} 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} 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}\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=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=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}_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^{17}-q^{16}-q^{15}+2 q^{14}-q^{13}-2 q^{12}+2 q^{11}+q^{10}-3 q^9+q^8+3 q^7-3 q^6+4 q^4-3 q^3-q^2+3 q-1- q^{-1} + q^{-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 q^{37}-2 q^{36}-q^{35}+2 q^{34}+4 q^{33}-3 q^{32}-7 q^{31}+4 q^{30}+9 q^{29}-3 q^{28}-11 q^{27}+3 q^{26}+11 q^{25}-2 q^{24}-11 q^{23}+2 q^{22}+9 q^{21}-2 q^{20}-8 q^{19}+2 q^{18}+6 q^{17}-q^{16}-5 q^{15}+q^{14}+3 q^{13}-2 q^{11}+q^{10}-q^9+q^7+3 q^6-3 q^5-2 q^4+2 q^3+4 q^2-2 q-3+3 q^{-2} - q^{-4} - q^{-5} + q^{-6} }
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^{60}+2 q^{59}+q^{58}-3 q^{57}-q^{56}-2 q^{55}+9 q^{54}+3 q^{53}-9 q^{52}-7 q^{51}-6 q^{50}+21 q^{49}+11 q^{48}-13 q^{47}-16 q^{46}-13 q^{45}+27 q^{44}+20 q^{43}-11 q^{42}-20 q^{41}-19 q^{40}+26 q^{39}+23 q^{38}-9 q^{37}-18 q^{36}-19 q^{35}+21 q^{34}+22 q^{33}-7 q^{32}-15 q^{31}-18 q^{30}+16 q^{29}+21 q^{28}-5 q^{27}-11 q^{26}-18 q^{25}+9 q^{24}+20 q^{23}-q^{22}-6 q^{21}-17 q^{20}+q^{19}+17 q^{18}+2 q^{17}-12 q^{15}-5 q^{14}+11 q^{13}+q^{12}+3 q^{11}-4 q^{10}-5 q^9+6 q^8-4 q^7+2 q^6+q^5-q^4+6 q^3-6 q^2-2 q+ q^{-1} +7 q^{-2} -3 q^{-3} -2 q^{-4} -2 q^{-5} - q^{-6} +4 q^{-7} - q^{-10} - q^{-11} + q^{-12} }
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^{85}-3 q^{83}-2 q^{82}+q^{81}+6 q^{80}+7 q^{79}-14 q^{77}-15 q^{76}+20 q^{74}+25 q^{73}+6 q^{72}-24 q^{71}-38 q^{70}-13 q^{69}+27 q^{68}+44 q^{67}+21 q^{66}-23 q^{65}-50 q^{64}-29 q^{63}+20 q^{62}+51 q^{61}+32 q^{60}-15 q^{59}-48 q^{58}-34 q^{57}+12 q^{56}+46 q^{55}+32 q^{54}-11 q^{53}-41 q^{52}-30 q^{51}+9 q^{50}+39 q^{49}+27 q^{48}-8 q^{47}-33 q^{46}-27 q^{45}+5 q^{44}+30 q^{43}+26 q^{42}-q^{41}-25 q^{40}-27 q^{39}-4 q^{38}+20 q^{37}+26 q^{36}+10 q^{35}-14 q^{34}-26 q^{33}-14 q^{32}+6 q^{31}+23 q^{30}+19 q^{29}+q^{28}-19 q^{27}-21 q^{26}-8 q^{25}+12 q^{24}+22 q^{23}+14 q^{22}-6 q^{21}-18 q^{20}-18 q^{19}-2 q^{18}+14 q^{17}+18 q^{16}+6 q^{15}-6 q^{14}-14 q^{13}-10 q^{12}+2 q^{11}+10 q^{10}+7 q^9+2 q^8-3 q^7-5 q^6-2 q^5+q^4+2+3 q^{-1} +2 q^{-2} -3 q^{-3} -4 q^{-4} -3 q^{-5} +4 q^{-7} +5 q^{-8} -2 q^{-10} -2 q^{-11} -3 q^{-12} +3 q^{-14} + q^{-15} - q^{-18} - q^{-19} + q^{-20} }
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^{122}-2 q^{121}-q^{120}+2 q^{119}+q^{118}+2 q^{117}-q^{116}+2 q^{115}-9 q^{114}-8 q^{113}+6 q^{112}+9 q^{111}+14 q^{110}+6 q^{109}+q^{108}-35 q^{107}-34 q^{106}+q^{105}+25 q^{104}+49 q^{103}+38 q^{102}+16 q^{101}-70 q^{100}-85 q^{99}-31 q^{98}+24 q^{97}+86 q^{96}+87 q^{95}+55 q^{94}-81 q^{93}-124 q^{92}-74 q^{91}-3 q^{90}+92 q^{89}+117 q^{88}+96 q^{87}-64 q^{86}-128 q^{85}-95 q^{84}-28 q^{83}+77 q^{82}+115 q^{81}+110 q^{80}-50 q^{79}-116 q^{78}-91 q^{77}-32 q^{76}+68 q^{75}+103 q^{74}+103 q^{73}-49 q^{72}-106 q^{71}-80 q^{70}-26 q^{69}+65 q^{68}+93 q^{67}+93 q^{66}-48 q^{65}-95 q^{64}-71 q^{63}-27 q^{62}+55 q^{61}+81 q^{60}+89 q^{59}-36 q^{58}-76 q^{57}-64 q^{56}-36 q^{55}+35 q^{54}+64 q^{53}+86 q^{52}-15 q^{51}-48 q^{50}-54 q^{49}-47 q^{48}+8 q^{47}+40 q^{46}+79 q^{45}+8 q^{44}-15 q^{43}-36 q^{42}-51 q^{41}-18 q^{40}+8 q^{39}+59 q^{38}+21 q^{37}+17 q^{36}-7 q^{35}-38 q^{34}-30 q^{33}-23 q^{32}+24 q^{31}+14 q^{30}+32 q^{29}+22 q^{28}-8 q^{27}-17 q^{26}-35 q^{25}-10 q^{24}-11 q^{23}+20 q^{22}+30 q^{21}+17 q^{20}+12 q^{19}-20 q^{18}-19 q^{17}-27 q^{16}-6 q^{15}+12 q^{14}+16 q^{13}+28 q^{12}+2 q^{11}-4 q^{10}-17 q^9-15 q^8-6 q^7-q^6+18 q^5+4 q^4+6 q^3-4 q-4-6 q^{-1} +7 q^{-2} -7 q^{-3} + q^{-5} +2 q^{-6} +3 q^{-7} + q^{-8} +9 q^{-9} -7 q^{-10} -4 q^{-11} -4 q^{-12} -2 q^{-13} +2 q^{-15} +9 q^{-16} - q^{-17} -2 q^{-19} -2 q^{-20} -3 q^{-21} - q^{-22} +4 q^{-23} + q^{-25} - q^{-28} - q^{-29} + q^{-30} }
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^{161}+2 q^{160}+q^{159}-2 q^{158}-2 q^{157}-2 q^{156}+4 q^{155}+2 q^{154}+q^{153}+5 q^{152}-q^{151}-11 q^{150}-13 q^{149}-9 q^{148}+13 q^{147}+24 q^{146}+21 q^{145}+22 q^{144}-12 q^{143}-45 q^{142}-57 q^{141}-46 q^{140}+16 q^{139}+74 q^{138}+96 q^{137}+85 q^{136}-3 q^{135}-97 q^{134}-142 q^{133}-145 q^{132}-30 q^{131}+111 q^{130}+194 q^{129}+206 q^{128}+68 q^{127}-101 q^{126}-220 q^{125}-264 q^{124}-129 q^{123}+77 q^{122}+236 q^{121}+307 q^{120}+172 q^{119}-43 q^{118}-221 q^{117}-325 q^{116}-212 q^{115}+6 q^{114}+204 q^{113}+329 q^{112}+228 q^{111}+18 q^{110}-181 q^{109}-318 q^{108}-232 q^{107}-31 q^{106}+164 q^{105}+305 q^{104}+226 q^{103}+32 q^{102}-156 q^{101}-294 q^{100}-215 q^{99}-27 q^{98}+154 q^{97}+283 q^{96}+206 q^{95}+22 q^{94}-153 q^{93}-277 q^{92}-198 q^{91}-17 q^{90}+150 q^{89}+264 q^{88}+194 q^{87}+20 q^{86}-141 q^{85}-255 q^{84}-189 q^{83}-25 q^{82}+127 q^{81}+238 q^{80}+187 q^{79}+37 q^{78}-109 q^{77}-220 q^{76}-184 q^{75}-52 q^{74}+86 q^{73}+199 q^{72}+182 q^{71}+68 q^{70}-57 q^{69}-174 q^{68}-179 q^{67}-87 q^{66}+29 q^{65}+146 q^{64}+168 q^{63}+105 q^{62}+5 q^{61}-112 q^{60}-158 q^{59}-117 q^{58}-34 q^{57}+71 q^{56}+134 q^{55}+125 q^{54}+67 q^{53}-33 q^{52}-108 q^{51}-118 q^{50}-85 q^{49}-10 q^{48}+66 q^{47}+105 q^{46}+99 q^{45}+43 q^{44}-33 q^{43}-74 q^{42}-88 q^{41}-67 q^{40}-12 q^{39}+40 q^{38}+75 q^{37}+73 q^{36}+32 q^{35}-4 q^{34}-36 q^{33}-62 q^{32}-52 q^{31}-25 q^{30}+7 q^{29}+40 q^{28}+39 q^{27}+40 q^{26}+27 q^{25}-9 q^{24}-26 q^{23}-35 q^{22}-39 q^{21}-17 q^{20}-3 q^{19}+21 q^{18}+41 q^{17}+27 q^{16}+21 q^{15}+6 q^{14}-25 q^{13}-29 q^{12}-33 q^{11}-19 q^{10}+9 q^9+13 q^8+25 q^7+31 q^6+8 q^5-q^4-18 q^3-23 q^2-8 q-10+15 q^{-2} +8 q^{-3} +10 q^{-4} +3 q^{-5} -6 q^{-6} +3 q^{-7} -4 q^{-8} -5 q^{-9} + q^{-10} -6 q^{-11} - q^{-12} -4 q^{-14} +6 q^{-15} +5 q^{-16} +4 q^{-17} +7 q^{-18} -4 q^{-19} -4 q^{-20} -3 q^{-21} -7 q^{-22} -2 q^{-23} +3 q^{-25} +8 q^{-26} + q^{-27} + q^{-29} -3 q^{-30} -2 q^{-31} -3 q^{-32} - q^{-33} +3 q^{-34} + q^{-35} + q^{-37} - q^{-40} - q^{-41} + q^{-42} }

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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