9 48

9_47

9_49

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 48's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 48 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X12,8,13,7 X3,11,4,10 X11,3,12,2 X14,6,15,5 X6,14,7,13 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	[21,21,21-]

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

[edit Notes on presentations of 9 48]

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 48"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X12,8,13,7 X3,11,4,10 X11,3,12,2 X14,6,15,5 X6,14,7,13 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]=

[21,21,21-]

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}(4,\{1,1,2,-1,2,1,-3,2,-1,2,-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, 11, 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, 4}, {5, 3}, {4, 7}, {2, 5}, {8, 6}, {7, 1}, {3, 8}, {9, 2}, {6, 10}, {1, 9}]

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 $\{4,6\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-1][-8] Hyperbolic Volume 9.53188 A-Polynomial See Data:9 48/A-polynomial

[edit Notes for 9 48's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 48's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+7t-11+7t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 27, 2 }
Jones polynomial	$-2q^{6}+3q^{5}-4q^{4}+6q^{3}-4q^{2}+4q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-z^{2}a^{-2}+3z^{2}a^{-4}+z^{2}+3a^{-4}-2a^{-6}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-3}+z^{7}a^{-5}+3z^{6}a^{-2}+4z^{6}a^{-4}+z^{6}a^{-6}+3z^{5}a^{-1}+2z^{5}a^{-3}-z^{5}a^{-5}-5z^{4}a^{-2}-6z^{4}a^{-4}+z^{4}-5z^{3}a^{-1}-3z^{3}a^{-3}+5z^{3}a^{-5}+3z^{3}a^{-7}+2z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-z^{2}-za^{-3}-5za^{-5}-4za^{-7}+3a^{-4}+2a^{-6}$
The A2 invariant	$q^{4}-q^{2}-1+q^{-2}-q^{-4}+2q^{-6}+q^{-8}+2q^{-10}+2q^{-12}+q^{-16}-2q^{-18}-2q^{-20}$
The G2 invariant	$q^{18}-2q^{16}+4q^{14}-6q^{12}+3q^{10}+q^{8}-6q^{6}+14q^{4}-14q^{2}+15-8q^{-2}-7q^{-4}+16q^{-6}-23q^{-8}+18q^{-10}-9q^{-12}-5q^{-14}+13q^{-16}-13q^{-18}+10q^{-20}+q^{-22}-13q^{-24}+16q^{-26}-13q^{-28}+q^{-30}+15q^{-32}-24q^{-34}+27q^{-36}-14q^{-38}+11q^{-40}+7q^{-42}-22q^{-44}+29q^{-46}-22q^{-48}+19q^{-50}-2q^{-52}-14q^{-54}+21q^{-56}-8q^{-58}+6q^{-60}+q^{-62}-15q^{-64}+14q^{-66}-5q^{-68}-8q^{-70}+14q^{-72}-24q^{-74}+21q^{-76}-5q^{-78}-10q^{-80}+11q^{-82}-18q^{-84}+16q^{-86}-9q^{-88}-2q^{-90}+3q^{-92}-7q^{-94}+7q^{-96}-2q^{-98}+q^{-100}+q^{-102}$

Further Quantum Invariants

Further quantum knot invariants for 9_48.

A1 Invariants.

Weight	Invariant
1	$q^{3}-2q+q^{-1}+2q^{-5}+2q^{-7}-q^{-9}+q^{-11}-2q^{-13}$
2	$q^{10}-2q^{8}-2q^{6}+6q^{4}-6+5q^{-2}+2q^{-4}-7q^{-6}+2q^{-8}+4q^{-10}-2q^{-12}+q^{-14}+4q^{-16}+4q^{-18}-5q^{-20}+6q^{-24}-7q^{-26}-4q^{-28}+5q^{-30}-3q^{-32}-3q^{-34}+3q^{-36}+q^{-38}$
3	$q^{21}-2q^{19}-2q^{17}+3q^{15}+6q^{13}-13q^{9}-4q^{7}+14q^{5}+11q^{3}-13q-19q^{-1}+12q^{-3}+26q^{-5}-4q^{-7}-27q^{-9}-q^{-11}+23q^{-13}+7q^{-15}-22q^{-17}-7q^{-19}+14q^{-21}+11q^{-23}-7q^{-25}-9q^{-27}+2q^{-29}+14q^{-31}+9q^{-33}-15q^{-35}-12q^{-37}+12q^{-39}+21q^{-41}-15q^{-43}-27q^{-45}+4q^{-47}+25q^{-49}-2q^{-51}-24q^{-53}-6q^{-55}+19q^{-57}+9q^{-59}-9q^{-61}-7q^{-63}+5q^{-65}+8q^{-67}-q^{-69}-2q^{-71}-2q^{-73}$
4	$q^{36}-2q^{34}-2q^{32}+3q^{30}+3q^{28}+6q^{26}-7q^{24}-13q^{22}-4q^{20}+5q^{18}+32q^{16}+8q^{14}-25q^{12}-34q^{10}-19q^{8}+52q^{6}+51q^{4}-63-81q^{-2}+30q^{-4}+93q^{-6}+64q^{-8}-44q^{-10}-120q^{-12}-27q^{-14}+78q^{-16}+104q^{-18}+2q^{-20}-106q^{-22}-62q^{-24}+33q^{-26}+90q^{-28}+31q^{-30}-60q^{-32}-57q^{-34}-q^{-36}+53q^{-38}+38q^{-40}-14q^{-42}-43q^{-44}-28q^{-46}+20q^{-48}+50q^{-50}+35q^{-52}-42q^{-54}-62q^{-56}-14q^{-58}+60q^{-60}+82q^{-62}-29q^{-64}-94q^{-66}-63q^{-68}+48q^{-70}+120q^{-72}+14q^{-74}-85q^{-76}-97q^{-78}+5q^{-80}+110q^{-82}+57q^{-84}-27q^{-86}-84q^{-88}-39q^{-90}+52q^{-92}+55q^{-94}+21q^{-96}-34q^{-98}-40q^{-100}+q^{-102}+17q^{-104}+20q^{-106}-q^{-108}-13q^{-110}-7q^{-112}-3q^{-114}+3q^{-116}+3q^{-118}+q^{-120}$
5	$q^{55}-2q^{53}-2q^{51}+3q^{49}+3q^{47}+3q^{45}-q^{43}-7q^{41}-13q^{39}-4q^{37}+14q^{35}+23q^{33}+20q^{31}-4q^{29}-37q^{27}-57q^{25}-18q^{23}+48q^{21}+88q^{19}+69q^{17}-24q^{15}-128q^{13}-149q^{11}-29q^{9}+144q^{7}+229q^{5}+139q^{3}-105q-305q^{-1}-269q^{-3}+18q^{-5}+328q^{-7}+385q^{-9}+115q^{-11}-291q^{-13}-471q^{-15}-249q^{-17}+206q^{-19}+491q^{-21}+361q^{-23}-85q^{-25}-460q^{-27}-411q^{-29}-18q^{-31}+372q^{-33}+424q^{-35}+98q^{-37}-283q^{-39}-372q^{-41}-141q^{-43}+180q^{-45}+312q^{-47}+159q^{-49}-111q^{-51}-235q^{-53}-155q^{-55}+37q^{-57}+179q^{-59}+157q^{-61}+10q^{-63}-131q^{-65}-166q^{-67}-64q^{-69}+96q^{-71}+196q^{-73}+133q^{-75}-74q^{-77}-229q^{-79}-196q^{-81}+25q^{-83}+265q^{-85}+295q^{-87}+19q^{-89}-300q^{-91}-372q^{-93}-115q^{-95}+286q^{-97}+454q^{-99}+203q^{-101}-245q^{-103}-491q^{-105}-306q^{-107}+150q^{-109}+483q^{-111}+382q^{-113}-30q^{-115}-399q^{-117}-420q^{-119}-84q^{-121}+288q^{-123}+395q^{-125}+182q^{-127}-139q^{-129}-317q^{-131}-223q^{-133}+23q^{-135}+206q^{-137}+204q^{-139}+57q^{-141}-98q^{-143}-153q^{-145}-90q^{-147}+23q^{-149}+79q^{-151}+67q^{-153}+19q^{-155}-27q^{-157}-45q^{-159}-21q^{-161}+3q^{-163}+12q^{-165}+17q^{-167}+8q^{-169}-q^{-171}-4q^{-173}-2q^{-175}-2q^{-177}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}-q^{2}-1+q^{-2}-q^{-4}+2q^{-6}+q^{-8}+2q^{-10}+2q^{-12}+q^{-16}-2q^{-18}-2q^{-20}$
1,1	$q^{12}-4q^{10}+10q^{8}-20q^{6}+30q^{4}-42q^{2}+58-62q^{-2}+59q^{-4}-48q^{-6}+26q^{-8}-2q^{-10}-34q^{-12}+68q^{-14}-86q^{-16}+112q^{-18}-108q^{-20}+114q^{-22}-94q^{-24}+70q^{-26}-40q^{-28}+2q^{-30}+22q^{-32}-46q^{-34}+61q^{-36}-68q^{-38}+56q^{-40}-48q^{-42}+31q^{-44}-22q^{-46}+10q^{-48}-2q^{-50}+2q^{-52}+2q^{-54}$
2,0	$q^{12}-q^{10}-2q^{8}+3q^{4}+3q^{2}-4-q^{-2}+4q^{-4}+q^{-6}-4q^{-8}-2q^{-10}+2q^{-12}-q^{-14}+2q^{-18}+5q^{-20}+2q^{-22}+6q^{-24}+4q^{-26}-q^{-28}+q^{-30}+3q^{-32}-2q^{-34}-7q^{-36}-4q^{-38}-q^{-40}-4q^{-42}-6q^{-44}+4q^{-48}+4q^{-50}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	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^8-2 q^6+3 q^2-4+3 q^{-2} +4 q^{-4} -6 q^{-6} + q^{-10} -4 q^{-12} +6 q^{-16} +6 q^{-18} +7 q^{-20} +4 q^{-22} +6 q^{-24} -3 q^{-26} -9 q^{-28} - q^{-30} -6 q^{-32} -7 q^{-34} +4 q^{-36} + q^{-38} - q^{-40} +3 q^{-42} }
1,0,0	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^5-q^3- q^{-1} + q^{-3} - q^{-5} + q^{-7} + q^{-9} + q^{-11} +2 q^{-13} +2 q^{-15} +3 q^{-17} + q^{-21} -2 q^{-23} -2 q^{-25} -2 q^{-27} }

A4 Invariants.

Weight	Invariant
0,1,0,0	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^{10}-q^8-2 q^6+2 q^4+q^2-2+5 q^{-4} + q^{-6} -5 q^{-8} +3 q^{-12} -5 q^{-14} -7 q^{-16} +3 q^{-18} + q^{-20} + q^{-22} +9 q^{-24} +14 q^{-26} +9 q^{-28} +9 q^{-30} +10 q^{-32} + q^{-34} -10 q^{-36} -8 q^{-38} -7 q^{-40} -13 q^{-42} -10 q^{-44} - q^{-46} +2 q^{-48} +2 q^{-52} +4 q^{-54} +3 q^{-56} }
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^6-q^4- q^{-2} + q^{-4} - q^{-6} + q^{-8} + q^{-12} + q^{-14} +2 q^{-16} +2 q^{-18} +3 q^{-20} +3 q^{-22} + q^{-26} -2 q^{-28} -2 q^{-30} -2 q^{-32} -2 q^{-34} }

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 q^8-2 q^6+4 q^4-5 q^2+4-5 q^{-2} +4 q^{-4} -2 q^{-6} +3 q^{-10} -4 q^{-12} +8 q^{-14} -8 q^{-16} +10 q^{-18} -7 q^{-20} +8 q^{-22} -4 q^{-24} +3 q^{-26} + q^{-28} -3 q^{-30} +4 q^{-32} -5 q^{-34} +4 q^{-36} -5 q^{-38} +3 q^{-40} -3 q^{-42} }
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^{14}-2 q^{10}-2 q^8+2 q^6+4 q^4-q^2-4- q^{-2} +6 q^{-4} +4 q^{-6} -4 q^{-8} -5 q^{-10} +3 q^{-14} - q^{-16} -4 q^{-18} -2 q^{-20} +4 q^{-22} +3 q^{-24} +6 q^{-30} +7 q^{-32} +3 q^{-34} -2 q^{-36} +2 q^{-38} +4 q^{-40} -7 q^{-44} -5 q^{-46} + q^{-48} + q^{-50} -5 q^{-52} -7 q^{-54} - q^{-56} +4 q^{-58} +3 q^{-60} -2 q^{-62} -2 q^{-64} + q^{-66} +3 q^{-68} }

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^{10}-2 q^8+2 q^6-3 q^4+4 q^2-4+4 q^{-2} -3 q^{-4} +4 q^{-6} -2 q^{-8} - q^{-10} -2 q^{-14} +2 q^{-16} -6 q^{-18} +6 q^{-20} -3 q^{-22} +12 q^{-24} - q^{-26} +12 q^{-28} +10 q^{-32} - q^{-34} -5 q^{-38} -6 q^{-40} -2 q^{-42} -8 q^{-44} - q^{-46} -6 q^{-48} +4 q^{-50} -2 q^{-52} +4 q^{-54} -2 q^{-56} +3 q^{-58} }

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^{18}-2 q^{16}+4 q^{14}-6 q^{12}+3 q^{10}+q^8-6 q^6+14 q^4-14 q^2+15-8 q^{-2} -7 q^{-4} +16 q^{-6} -23 q^{-8} +18 q^{-10} -9 q^{-12} -5 q^{-14} +13 q^{-16} -13 q^{-18} +10 q^{-20} + q^{-22} -13 q^{-24} +16 q^{-26} -13 q^{-28} + q^{-30} +15 q^{-32} -24 q^{-34} +27 q^{-36} -14 q^{-38} +11 q^{-40} +7 q^{-42} -22 q^{-44} +29 q^{-46} -22 q^{-48} +19 q^{-50} -2 q^{-52} -14 q^{-54} +21 q^{-56} -8 q^{-58} +6 q^{-60} + q^{-62} -15 q^{-64} +14 q^{-66} -5 q^{-68} -8 q^{-70} +14 q^{-72} -24 q^{-74} +21 q^{-76} -5 q^{-78} -10 q^{-80} +11 q^{-82} -18 q^{-84} +16 q^{-86} -9 q^{-88} -2 q^{-90} +3 q^{-92} -7 q^{-94} +7 q^{-96} -2 q^{-98} + q^{-100} + q^{-102} }

.

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 48"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}+7t-11+7t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n1,}

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 48"];

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^2+7 t-11+7 t^{-1} - t^{-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 -2 q^6+3 q^5-4 q^4+6 q^3-4 q^2+4 q-3+ 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]=

{K11n1,}

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, 5)

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 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 12} 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 40} 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 190} 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 42} 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 480} 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{2800}{3}} ${\frac {448}{3}}$ 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 200} 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} 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 800} 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 2280} 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 504} 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{46031}{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{6226}{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{38422}{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{337}{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 \frac{4111}{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 $\chi$ (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=} 2 is the signature of 9 48. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 χ 13               2 -2 11             1   1 9           3 2   -1 7         3 1     2 5       1 3       2 3     3 3         0 1   1 2           1 -1   2             -2 -3 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}} $i=1$ $i=3$ 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}} 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}\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=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^{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}^{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=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} 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}^{3}\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=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^{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=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=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^{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}}

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^{18}+2 q^{17}-6 q^{16}+q^{15}+10 q^{14}-15 q^{13}-2 q^{12}+23 q^{11}-21 q^{10}-7 q^9+32 q^8-21 q^7-10 q^6+29 q^5-15 q^4-12 q^3+20 q^2-6 q-9+9 q^{-1} -3 q^{-3} + q^{-4} }
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 -2 q^{35}+q^{33}+9 q^{32}-5 q^{31}-12 q^{30}-q^{29}+27 q^{28}+5 q^{27}-37 q^{26}-19 q^{25}+49 q^{24}+32 q^{23}-58 q^{22}-50 q^{21}+61 q^{20}+68 q^{19}-67 q^{18}-74 q^{17}+58 q^{16}+92 q^{15}-62 q^{14}-86 q^{13}+47 q^{12}+94 q^{11}-44 q^{10}-83 q^9+26 q^8+79 q^7-15 q^6-67 q^5+2 q^4+53 q^3+8 q^2-37 q-12+22 q^{-1} +14 q^{-2} -13 q^{-3} -9 q^{-4} +4 q^{-5} +5 q^{-6} -3 q^{-8} + q^{-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 q^{58}+2 q^{57}-6 q^{55}-4 q^{54}-5 q^{53}+14 q^{52}+21 q^{51}-9 q^{50}-20 q^{49}-46 q^{48}+20 q^{47}+76 q^{46}+25 q^{45}-23 q^{44}-137 q^{43}-25 q^{42}+133 q^{41}+109 q^{40}+30 q^{39}-242 q^{38}-127 q^{37}+145 q^{36}+208 q^{35}+136 q^{34}-314 q^{33}-238 q^{32}+114 q^{31}+273 q^{30}+247 q^{29}-336 q^{28}-312 q^{27}+66 q^{26}+293 q^{25}+324 q^{24}-321 q^{23}-342 q^{22}+18 q^{21}+278 q^{20}+353 q^{19}-269 q^{18}-327 q^{17}-36 q^{16}+222 q^{15}+350 q^{14}-178 q^{13}-268 q^{12}-93 q^{11}+127 q^{10}+306 q^9-70 q^8-166 q^7-119 q^6+22 q^5+213 q^4+6 q^3-58 q^2-90 q-41+102 q^{-1} +24 q^{-2} +5 q^{-3} -39 q^{-4} -40 q^{-5} +31 q^{-6} +9 q^{-7} +14 q^{-8} -6 q^{-9} -16 q^{-10} +4 q^{-11} +5 q^{-13} -3 q^{-15} + q^{-16} }

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_47 9_49