8 19: Difference between revisions

Revision as of 20:10, 28 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-]

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

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-30}+q^{-32}+3q^{-34}+5q^{-36}+8q^{-38}+9q^{-40}+12q^{-42}+10q^{-44}+8q^{-46}+2q^{-48}-4q^{-50}-11q^{-52}-15q^{-54}-17q^{-56}-15q^{-58}-10q^{-60}-5q^{-62}+2q^{-64}+5q^{-66}+8q^{-68}+7q^{-70}+6q^{-72}+3q^{-74}+q^{-76}-q^{-78}-q^{-80}-q^{-82}-q^{-84}$
1,0,0,0	$q^{-20}+q^{-22}+2q^{-24}+3q^{-26}+4q^{-28}+3q^{-30}+3q^{-32}+q^{-34}-q^{-36}-3q^{-38}-4q^{-40}-4q^{-42}-3q^{-44}-q^{-46}+q^{-50}+q^{-52}+q^{-54}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-20}+q^{-22}+q^{-24}+2q^{-26}+2q^{-28}+q^{-30}+q^{-32}+q^{-34}-q^{-40}-q^{-42}-q^{-44}-q^{-46}-q^{-48}-q^{-50}$
1,0	$q^{-30}+q^{-34}+q^{-36}+2q^{-38}+q^{-40}+3q^{-42}+2q^{-44}+3q^{-46}+2q^{-48}+2q^{-50}+q^{-52}+q^{-54}-q^{-56}-q^{-58}-2q^{-60}-3q^{-62}-3q^{-64}-3q^{-66}-3q^{-68}-3q^{-70}-q^{-72}-q^{-74}+2q^{-80}+q^{-82}+q^{-84}+q^{-86}+q^{-88}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-30}+q^{-32}+2q^{-34}+4q^{-36}+5q^{-38}+6q^{-40}+7q^{-42}+6q^{-44}+4q^{-46}+2q^{-48}-2q^{-50}-5q^{-52}-7q^{-54}-8q^{-56}-8q^{-58}-6q^{-60}-4q^{-62}-q^{-64}+q^{-66}+2q^{-68}+3q^{-70}+2q^{-72}+2q^{-74}+q^{-76}$

G2 Invariants.

Weight	Invariant
1,0	$q^{-50}+q^{-52}+q^{-54}+q^{-56}+q^{-58}+q^{-60}+2q^{-62}+2q^{-64}+q^{-66}+q^{-68}+2q^{-70}+2q^{-72}+2q^{-74}+q^{-76}+q^{-80}+2q^{-82}-q^{-94}-2q^{-96}-q^{-98}-q^{-100}-2q^{-102}-2q^{-104}-2q^{-106}-q^{-108}-q^{-110}-2q^{-112}-2q^{-114}-q^{-116}-q^{-122}-q^{-124}+q^{-126}+q^{-128}+q^{-136}+q^{-138}+q^{-144}$

.

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

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

20

80

200

{\frac {1270}{3}}

{\frac {170}{3}}

1600

{\frac {7520}{3}}

{\frac {1280}{3}}

272

{\frac {4000}{3}}

3200

{\frac {25400}{3}}

{\frac {3400}{3}}

{\frac {91951}{6}}

{\frac {2818}{3}}

{\frac {44990}{9}}

{\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

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j-2r=s+1

or

j-2r=s-1

, where

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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=3$	$i=5$	$i=7$
$r=0$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=1$
$r=2$		${\mathbb {Z} }$
$r=3$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=5$		${\mathbb {Z} }$	${\mathbb {Z} }$

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[8, 19]]
Out[2]=	8
In[3]:=	PD[Knot[8, 19]]
Out[3]=	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[4]:=	GaussCode[Knot[8, 19]]
Out[4]=	GaussCode[1, -8, 2, -1, -4, 5, 8, -2, -3, 7, -6, 4, -5, 3, -7, 6]
In[5]:=	BR[Knot[8, 19]]
Out[5]=	BR[3, {1, 1, 1, 2, 1, 1, 1, 2}]
In[6]:=	alex = Alexander[Knot[8, 19]][t]
Out[6]=	-3 -2 2 3 1 + t - t - t + t
In[7]:=	Conway[Knot[8, 19]][z]
Out[7]=	2 4 6 1 + 5 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 19]}
In[9]:=	{KnotDet[Knot[8, 19]], KnotSignature[Knot[8, 19]]}
Out[9]=	{3, 6}
In[10]:=	J=Jones[Knot[8, 19]][q]
Out[10]=	3 5 8 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 19]}
In[12]:=	A2Invariant[Knot[8, 19]][q]
Out[12]=	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[13]:=	Kauffman[Knot[8, 19]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[8, 19]], Vassiliev[3][Knot[8, 19]]}
Out[14]=	{0, 10}
In[15]:=	Kh[Knot[8, 19]][q, t]
Out[15]=	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

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 {{Vassiliev Invariants}}
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+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
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 <tr align=center><td>5</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
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 q  + q  + q  t  + q   t  + q   t  + q   t  + q   t  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

8 19: Difference between revisions

Revision as of 20:10, 28 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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