8 19: Difference between revisions

Revision as of 21:04, 29 August 2005

Visit 8 19's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 19's page at the original Knot Atlas!

8 19 is the first non-obvious torus knot in the table - it is in fact T(4,3). It is also the pretzel knot P(3,3,-2).

8_19 is the first non-homologically thin knot in the Rolfsen table. (That is, it's the first knot whose Khovanov homology has 'off-diagonal' elements.)

Knotscape		Symmetrical form ; (3,4) torus knot		True-lover's knot with sticked free ends		Equal to the previous, from knotilus
Pretzel knot P(2,-3,-3)		French logo		Seen in Singapore

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_9,15,10,14 X_5,13,6,12 X_13,7,14,6 X_11,1,12,16 X_15,11,16,10 X₂₈₃₇
Gauss code	1, -8, 2, -1, -4, 5, 8, -2, -3, 7, -6, 4, -5, 3, -7, 6
Dowker-Thistlethwaite code	4 8 -12 2 -14 -16 -6 -10
Conway Notation	[3,3,2-]

Minimum Braid Representative:

Length is 8, width is 3.

Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	3
3-genus	3
Bridge index	3
Super bridge index	4
Nakanishi index	1
Maximal Thurston-Bennequin number	[5][-12]
Hyperbolic Volume	Not hyperbolic
A-Polynomial	See Data:8 19/A-polynomial

[edit Notes for 8 19'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 3}
Topological 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}
Concordance genus	$3$
Rasmussen s-Invariant	-6

[edit Notes for 8 19's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-t^{2}+1-t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+5z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 3, 6 }
Jones polynomial	$-q^{8}+q^{5}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+6z^{4}a^{-6}-z^{4}a^{-8}+10z^{2}a^{-6}-5z^{2}a^{-8}+5a^{-6}-5a^{-8}+a^{-10}$
Kauffman polynomial (db, data sources)	$z^{6}a^{-6}+z^{6}a^{-8}+z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-6}-6z^{4}a^{-8}-5z^{3}a^{-7}-5z^{3}a^{-9}+10z^{2}a^{-6}+10z^{2}a^{-8}+5za^{-7}+5za^{-9}-5a^{-6}-5a^{-8}-a^{-10}$
The A2 invariant	$q^{-10}+q^{-12}+2q^{-14}+2q^{-16}+2q^{-18}-q^{-22}-2q^{-24}-2q^{-26}-q^{-28}+q^{-32}$
The G2 invariant	$q^{-50}+q^{-52}+q^{-54}+q^{-56}+q^{-58}+q^{-60}+2q^{-62}+2q^{-64}+q^{-66}+q^{-68}+2q^{-70}+2q^{-72}+2q^{-74}+q^{-76}+q^{-80}+2q^{-82}-q^{-94}-2q^{-96}-q^{-98}-q^{-100}-2q^{-102}-2q^{-104}-2q^{-106}-q^{-108}-q^{-110}-2q^{-112}-2q^{-114}-q^{-116}-q^{-122}-q^{-124}+q^{-126}+q^{-128}+q^{-136}+q^{-138}+q^{-144}$

Further Quantum Invariants

Further quantum knot invariants for 8_19.

A1 Invariants.

Weight	Invariant
1	$q^{-5}+q^{-7}+q^{-9}+q^{-11}-q^{-15}-q^{-17}$
2	$q^{-10}+q^{-12}+q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}-q^{-28}-q^{-30}-q^{-32}-q^{-34}-q^{-36}+q^{-48}$
3	$q^{-15}+q^{-17}+q^{-19}+q^{-21}+q^{-23}+q^{-25}+q^{-27}+q^{-29}+q^{-31}+q^{-33}-q^{-41}-q^{-43}-q^{-45}-q^{-47}-q^{-49}-q^{-51}-q^{-53}-q^{-55}+q^{-77}+q^{-79}+q^{-81}+q^{-83}-q^{-87}-q^{-89}$
4	$q^{-20}+q^{-22}+q^{-24}+q^{-26}+q^{-28}+q^{-30}+q^{-32}+q^{-34}+q^{-36}+q^{-38}+q^{-40}+q^{-42}+q^{-44}-q^{-54}-q^{-56}-q^{-58}-q^{-60}-q^{-62}-q^{-64}-q^{-66}-q^{-68}-q^{-70}-q^{-72}-q^{-74}+q^{-106}+q^{-108}+q^{-110}+q^{-112}+q^{-114}+q^{-116}+q^{-118}-q^{-124}-q^{-126}-q^{-128}-q^{-130}-q^{-132}+q^{-144}$
5	$q^{-25}+q^{-27}+q^{-29}+q^{-31}+q^{-33}+q^{-35}+q^{-37}+q^{-39}+q^{-41}+q^{-43}+q^{-45}+q^{-47}+q^{-49}+q^{-51}+q^{-53}+q^{-55}-q^{-67}-q^{-69}-q^{-71}-q^{-73}-q^{-75}-q^{-77}-q^{-79}-q^{-81}-q^{-83}-q^{-85}-q^{-87}-q^{-89}-q^{-91}-q^{-93}+q^{-135}+q^{-137}+q^{-139}+q^{-141}+q^{-143}+q^{-145}+q^{-147}+q^{-149}+q^{-151}+q^{-153}-q^{-161}-q^{-163}-q^{-165}-q^{-167}-q^{-169}-q^{-171}-q^{-173}-q^{-175}+q^{-197}+q^{-199}+q^{-201}+q^{-203}-q^{-207}-q^{-209}$
6	$q^{-30}+q^{-32}+q^{-34}+q^{-36}+q^{-38}+q^{-40}+q^{-42}+q^{-44}+q^{-46}+q^{-48}+q^{-50}+q^{-52}+q^{-54}+q^{-56}+q^{-58}+q^{-60}+q^{-62}+q^{-64}+q^{-66}-q^{-80}-q^{-82}-q^{-84}-q^{-86}-q^{-88}-q^{-90}-q^{-92}-q^{-94}-q^{-96}-q^{-98}-q^{-100}-q^{-102}-q^{-104}-q^{-106}-q^{-108}-q^{-110}-q^{-112}+q^{-164}+q^{-166}+q^{-168}+q^{-170}+q^{-172}+q^{-174}+q^{-176}+q^{-178}+q^{-180}+q^{-182}+q^{-184}+q^{-186}+q^{-188}-q^{-198}-q^{-200}-q^{-202}-q^{-204}-q^{-206}-q^{-208}-q^{-210}-q^{-212}-q^{-214}-q^{-216}-q^{-218}+q^{-250}+q^{-252}+q^{-254}+q^{-256}+q^{-258}+q^{-260}+q^{-262}-q^{-268}-q^{-270}-q^{-272}-q^{-274}-q^{-276}+q^{-288}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-10}+q^{-12}+2q^{-14}+2q^{-16}+2q^{-18}-q^{-22}-2q^{-24}-2q^{-26}-q^{-28}+q^{-32}$
1,1	$q^{-20}+2q^{-22}+4q^{-24}+6q^{-26}+7q^{-28}+6q^{-30}+4q^{-32}+2q^{-34}-q^{-36}-4q^{-38}-6q^{-40}-8q^{-42}-7q^{-44}-6q^{-46}-4q^{-48}+2q^{-52}+4q^{-54}+4q^{-56}+4q^{-58}+q^{-60}-2q^{-64}-2q^{-66}-q^{-68}+2q^{-72}$
2,0	$q^{-20}+q^{-22}+2q^{-24}+2q^{-26}+3q^{-28}+3q^{-30}+4q^{-32}+3q^{-34}+3q^{-36}+q^{-38}-2q^{-42}-3q^{-44}-5q^{-46}-5q^{-48}-5q^{-50}-4q^{-52}-3q^{-54}-2q^{-56}+q^{-58}+2q^{-60}+4q^{-62}+4q^{-64}+4q^{-66}+q^{-68}-2q^{-72}-2q^{-74}-q^{-76}+q^{-80}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-20}+q^{-22}+3q^{-24}+4q^{-26}+6q^{-28}+5q^{-30}+5q^{-32}+q^{-34}-2q^{-36}-6q^{-38}-7q^{-40}-7q^{-42}-5q^{-44}-q^{-46}+q^{-48}+3q^{-50}+2q^{-52}+2q^{-54}$
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^{-15} + q^{-17} +2 q^{-19} +3 q^{-21} +3 q^{-23} +2 q^{-25} + q^{-27} - q^{-29} -3 q^{-31} -3 q^{-33} -3 q^{-35} - q^{-37} + q^{-41} + q^{-43} }
1,0,1	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^{-30} +2 q^{-32} +5 q^{-34} +9 q^{-36} +14 q^{-38} +17 q^{-40} +19 q^{-42} +16 q^{-44} +9 q^{-46} -10 q^{-50} -19 q^{-52} -25 q^{-54} -27 q^{-56} -23 q^{-58} -16 q^{-60} -7 q^{-62} +3 q^{-64} +9 q^{-66} +15 q^{-68} +15 q^{-70} +11 q^{-72} +7 q^{-74} +2 q^{-76} -2 q^{-78} -4 q^{-80} -3 q^{-82} -3 q^{-84} - q^{-86} - q^{-88} +2 q^{-96} }

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^{-30} + q^{-32} +3 q^{-34} +5 q^{-36} +8 q^{-38} +9 q^{-40} +12 q^{-42} +10 q^{-44} +8 q^{-46} +2 q^{-48} -4 q^{-50} -11 q^{-52} -15 q^{-54} -17 q^{-56} -15 q^{-58} -10 q^{-60} -5 q^{-62} +2 q^{-64} +5 q^{-66} +8 q^{-68} +7 q^{-70} +6 q^{-72} +3 q^{-74} + q^{-76} - q^{-78} - q^{-80} - q^{-82} - q^{-84} }

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^{-20} + q^{-22} +2 q^{-24} +3 q^{-26} +4 q^{-28} +3 q^{-30} +3 q^{-32} + q^{-34} - q^{-36} -3 q^{-38} -4 q^{-40} -4 q^{-42} -3 q^{-44} - q^{-46} + q^{-50} + q^{-52} + q^{-54} }

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^{-20} + q^{-22} + q^{-24} +2 q^{-26} +2 q^{-28} + q^{-30} + q^{-32} + q^{-34} - q^{-40} - q^{-42} - q^{-44} - q^{-46} - q^{-48} - 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^{-30} + q^{-34} + q^{-36} +2 q^{-38} + q^{-40} +3 q^{-42} +2 q^{-44} +3 q^{-46} +2 q^{-48} +2 q^{-50} + q^{-52} + q^{-54} - q^{-56} - q^{-58} -2 q^{-60} -3 q^{-62} -3 q^{-64} -3 q^{-66} -3 q^{-68} -3 q^{-70} - q^{-72} - q^{-74} +2 q^{-80} + q^{-82} + q^{-84} + q^{-86} + q^{-88} }

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^{-30} + q^{-32} +2 q^{-34} +4 q^{-36} +5 q^{-38} +6 q^{-40} +7 q^{-42} +6 q^{-44} +4 q^{-46} +2 q^{-48} -2 q^{-50} -5 q^{-52} -7 q^{-54} -8 q^{-56} -8 q^{-58} -6 q^{-60} -4 q^{-62} - q^{-64} + q^{-66} +2 q^{-68} +3 q^{-70} +2 q^{-72} +2 q^{-74} + q^{-76} }

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^{-50} + q^{-52} + q^{-54} + q^{-56} + q^{-58} + q^{-60} +2 q^{-62} +2 q^{-64} + q^{-66} + q^{-68} +2 q^{-70} +2 q^{-72} +2 q^{-74} + q^{-76} + q^{-80} +2 q^{-82} - q^{-94} -2 q^{-96} - q^{-98} - q^{-100} -2 q^{-102} -2 q^{-104} -2 q^{-106} - q^{-108} - q^{-110} -2 q^{-112} -2 q^{-114} - q^{-116} - q^{-122} - q^{-124} + q^{-126} + q^{-128} + q^{-136} + q^{-138} + q^{-144} }

.

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.

In[3]:= K = Knot["8 19"];

In[4]:= Alexander[K][t]

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

Out[4]= $t^{3}-t^{2}+1-t^{-2}+t^{-3}$

In[5]:= Conway[K][z]

Out[5]= $z^{6}+5z^{4}+5z^{2}+1$

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]= { 3, 6 }

In[8]:= Jones[K][q]

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

Out[8]= $-q^{8}+q^{5}+q^{3}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $z^{6}a^{-6}+6z^{4}a^{-6}-z^{4}a^{-8}+10z^{2}a^{-6}-5z^{2}a^{-8}+5a^{-6}-5a^{-8}+a^{-10}$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $z^{6}a^{-6}+z^{6}a^{-8}+z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-6}-6z^{4}a^{-8}-5z^{3}a^{-7}-5z^{3}a^{-9}+10z^{2}a^{-6}+10z^{2}a^{-8}+5za^{-7}+5za^{-9}-5a^{-6}-5a^{-8}-a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V₂ and V₃:

(5, 10)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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 20}

Failed to parse (SVG (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}

Failed to parse (SVG (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 \frac{1270}{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{170}{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 1600}

Failed to parse (SVG (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{7520}{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{1280}{3}}

272

Failed to parse (SVG (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{4000}{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 3200}

Failed to parse (SVG (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{25400}{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{3400}{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{91951}{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{2818}{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{44990}{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 \frac{1429}{18}}

{\frac {3631}{6}}

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=} 6 is the signature of 8 19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

0

1

2

3

4

5

χ

17

1

-1

15

1

-1

13

1

0

11

1

9

1

7

1

5

1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=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 i=5}	$i=7$
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 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}}	Failed to parse (SVG (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 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=3}		Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (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}}	Failed to parse (SVG (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}}	Failed to parse (SVG (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^{23}-q^{22}+q^{20}-q^{19}-q^{16}-q^{13}+q^{12}+q^9+q^6}
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^{43}+q^{41}+q^{40}-q^{39}+q^{37}-q^{35}+q^{33}-q^{31}+q^{29}-q^{27}-q^{26}+q^{25}-q^{23}-q^{22}+q^{21}-q^{19}+q^{17}+q^{13}+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^{70}-q^{69}+q^{65}-2 q^{64}+q^{60}-q^{59}+q^{57}+q^{55}-q^{54}+q^{52}-q^{49}+q^{47}-q^{44}+q^{42}-q^{39}+q^{37}-q^{35}-q^{34}+q^{32}-q^{30}-q^{29}+q^{27}-q^{25}+q^{22}+q^{17}+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^{102}+q^{100}+q^{99}-q^{96}-q^{95}+q^{94}+q^{93}-q^{90}-q^{89}+q^{88}+q^{87}-q^{85}-q^{84}-q^{83}+q^{82}+q^{81}-q^{79}-q^{78}+q^{76}+q^{75}+q^{74}-q^{73}-q^{72}+q^{70}+q^{69}+q^{68}-q^{67}-q^{66}+q^{63}+q^{62}-q^{61}-q^{60}+q^{57}+q^{56}-q^{55}-q^{54}+q^{51}+q^{50}-q^{49}-q^{48}+q^{45}-q^{43}-q^{42}+q^{39}-q^{37}-q^{36}+q^{33}-q^{31}+q^{27}+q^{21}+q^{15}}
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^{141}-q^{140}-q^{135}+q^{134}-q^{133}+q^{130}+q^{127}-q^{126}+q^{123}-q^{121}+q^{120}-q^{119}+q^{116}-q^{114}+q^{113}-q^{112}+q^{109}-q^{107}-q^{105}+q^{102}-q^{100}-q^{98}+2 q^{95}-q^{93}+2 q^{88}-q^{86}+2 q^{81}-q^{79}-q^{78}+2 q^{74}-q^{72}-q^{71}+2 q^{67}-q^{65}-q^{64}+2 q^{60}-q^{58}-q^{57}+q^{53}-q^{51}-q^{50}+q^{46}-q^{44}-q^{43}+q^{39}-q^{37}+q^{32}+q^{25}+q^{18}}
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^{185}+q^{183}+q^{182}-q^{178}-q^{177}+q^{175}+q^{174}-q^{170}-q^{169}-q^{168}+q^{167}+q^{166}-q^{162}-q^{161}+q^{159}+q^{158}+q^{157}-q^{154}-q^{153}+q^{151}+q^{150}+q^{149}-q^{147}-q^{146}-q^{145}+q^{143}+q^{142}+q^{141}-q^{139}-q^{138}-q^{137}+q^{135}+q^{134}+q^{133}-q^{131}-q^{130}-q^{129}+q^{126}+q^{125}-q^{123}-q^{122}-q^{121}+q^{118}+q^{117}-q^{115}-q^{114}+q^{110}+q^{109}+q^{108}-q^{107}-q^{106}+q^{102}+q^{101}+q^{100}-q^{99}-q^{98}+q^{94}+q^{93}-q^{91}-q^{90}+q^{86}+q^{85}-q^{83}-q^{82}+q^{78}+q^{77}-q^{75}-q^{74}+q^{70}+q^{69}-q^{67}-q^{66}+q^{61}-q^{59}-q^{58}+q^{53}-q^{51}-q^{50}+q^{45}-q^{43}+q^{37}+q^{29}+q^{21}}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[8, 19]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[9, 15, 10, 14], X[5, 13, 6, 12], X[13, 7, 14, 6], X[11, 1, 12, 16], X[15, 11, 16, 10], X[2, 8, 3, 7]]
In[3]:=	GaussCode[Knot[8, 19]]
Out[3]=	GaussCode[1, -8, 2, -1, -4, 5, 8, -2, -3, 7, -6, 4, -5, 3, -7, 6]
In[4]:=	DTCode[Knot[8, 19]]
Out[4]=	DTCode[4, 8, -12, 2, -14, -16, -6, -10]
In[5]:=	br = BR[Knot[8, 19]]
Out[5]=	BR[3, {1, 1, 1, 2, 1, 1, 1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 8}
In[7]:=	BraidIndex[Knot[8, 19]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[8, 19]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[8, 19]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 3, 4, 1}
In[10]:=	alex = Alexander[Knot[8, 19]][t]
Out[10]=	-3 -2 2 3 1 + t - t - t + t
In[11]:=	Conway[Knot[8, 19]][z]
Out[11]=	2 4 6 1 + 5 z + 5 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 19]}
In[13]:=	{KnotDet[Knot[8, 19]], KnotSignature[Knot[8, 19]]}
Out[13]=	{3, 6}
In[14]:=	Jones[Knot[8, 19]][q]
Out[14]=	3 5 8 q + q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 19]}
In[16]:=	A2Invariant[Knot[8, 19]][q]
Out[16]=	10 12 14 16 18 22 24 26 28 32 q + q + 2 q + 2 q + 2 q - q - 2 q - 2 q - q + q
In[17]:=	HOMFLYPT[Knot[8, 19]][a, z]
Out[17]=	2 2 4 4 6 -10 5 5 5 z 10 z z 6 z z a - -- + -- - ---- + ----- - -- + ---- + -- 8 6 8 6 8 6 6 a a a a a a a
In[18]:=	Kauffman[Knot[8, 19]][a, z]
Out[18]=	2 2 3 3 4 -10 5 5 5 z 5 z 10 z 10 z 5 z 5 z 6 z -a - -- - -- + --- + --- + ----- + ----- - ---- - ---- - ---- - 8 6 9 7 8 6 9 7 8 a a a a a a a a a 4 5 5 6 6 6 z z z z z ---- + -- + -- + -- + -- 6 9 7 8 6 a a a a a
In[19]:=	{Vassiliev[2][Knot[8, 19]], Vassiliev[3][Knot[8, 19]]}
Out[19]=	{5, 10}
In[20]:=	Kh[Knot[8, 19]][q, t]
Out[20]=	5 7 9 2 13 3 11 4 13 4 15 5 17 5 q + q + q t + q t + q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[8, 19], 2][q]
Out[21]=	6 9 12 13 16 19 20 22 23 q + q + q - q - q - q + q - q + q

See/edit the Rolfsen_Splice_Template.

Back to the top.

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8_20

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+{| width=100%
-See/edit the [[Rolfsen_Splice_Template]].
+|align=left|See/edit the [[Rolfsen_Splice_Template]].
+Back to the [[#top|top]].
+|align=right|{{Knot Navigation Links|ext=gif}}
+|}
  [[Category:Knot Page]]

8 19: Difference between revisions

Revision as of 21:04, 29 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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