L8a14

	Visit L8a14's page at Knotilus! Visit L8a14's page at the original Knot Atlas!
	L8a14 is 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 8^2_{1}} in the Rolfsen table of links.

Floor of the Stock Exchange [1]	Represented as two interlaced squares	A mosaic seen on an Istanbul floor	Coat of arms of Jaroměř, Czech Republic
Coat of arms of St. Savior, Jersey, Channel Islands, depicting Crown of Thorns religious symbol	Cathedral in Tbilisi, Georgia	Coat of arms of Mozota, Spain	Decorative motif of interlaced squares, San Pancrazio, Florence
Macedonian cross	3D depiction	Arma Christi carving in monastery in Portugal	Handewitt, Schleswig-Holstein oak leaves and acorns
Flower carpet, south India	Flower carpet, south India	Islamic art tile	Moroccan craft
Link on a church of Murato,Corsica	Logo of a spanish hiking trail

Knot presentations

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_14,5,15,6 X_16,7,9,8 X_8,9,1,10 X_4,13,5,14 X_6,15,7,16
Gauss code	{1, -2, 3, -7, 4, -8, 5, -6}, {6, -1, 2, -3, 7, -4, 8, -5}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , 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 v} , 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 w} , ...)	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{(u v+1) \left(u^2 v^2+1\right)}{u^{3/2} v^{3/2}}} (db)
Jones polynomial	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{1}{q^{7/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{11/2}}} (db)
Signature	-7 (db)
HOMFLY-PT polynomial	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 z^5 a^9+5 z^3 a^9+6 z a^9+a^9 z^{-1} -z^7 a^7-7 z^5 a^7-15 z^3 a^7-10 z a^7-a^7 z^{-1} } (db)
Kauffman polynomial	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 -z a^{15}-z^2 a^{14}-z^3 a^{13}+z a^{13}-z^4 a^{12}+2 z^2 a^{12}-z^5 a^{11}+3 z^3 a^{11}-z a^{11}-z^6 a^{10}+4 z^4 a^{10}-3 z^2 a^{10}-z^7 a^9+6 z^5 a^9-11 z^3 a^9+7 z a^9-a^9 z^{-1} -z^6 a^8+5 z^4 a^8-6 z^2 a^8+a^8-z^7 a^7+7 z^5 a^7-15 z^3 a^7+10 z a^7-a^7 z^{-1} } (db)

Vassiliev invariants

V₂ and V₃:

(0, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{317}{12}} )

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

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=

-7 is the signature of L8a14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

-10

1

-12

0

-14

1

0

-16

0

-18

1

0

-20

0

-22

1

0

-24

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-8$	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 i=-6}
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
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 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}	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}}
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}	${\mathbb {Z} }_{2}$	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 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}}

Computer Talk

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

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Link[8, Alternating, 14]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 14]]
Out[3]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], X[16, 7, 9, 8], X[8, 9, 1, 10], X[4, 13, 5, 14], X[6, 15, 7, 16]]
In[4]:=	GaussCode[Link[8, Alternating, 14]]
Out[4]=	GaussCode[{1, -2, 3, -7, 4, -8, 5, -6}, {6, -1, 2, -3, 7, -4, 8, -5}]
In[5]:=	BR[Link[8, Alternating, 14]]
Out[5]=	BR[Link[8, Alternating, 14]]
In[6]:=	alex = Alexander[Link[8, Alternating, 14]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 14]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 14]], KnotSignature[Link[8, Alternating, 14]]}
Out[9]=	{Infinity, -7}
In[10]:=	J=Jones[Link[8, Alternating, 14]][q]
Out[10]=	-(23/2) -(21/2) -(19/2) -(17/2) -(15/2) -(13/2) -q + q - q + q - q + q - -(11/2) -(7/2) q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 14]][q]
Out[12]=	-34 -32 -30 -20 -18 2 -14 -12 q + q + q + q + q + --- + q + q 16 q
In[13]:=	Kauffman[Link[8, Alternating, 14]][a, z]
Out[13]=	7 9 8 a a 7 9 11 13 15 8 2 a - -- - -- + 10 a z + 7 a z - a z + a z - a z - 6 a z - z z 10 2 12 2 14 2 7 3 9 3 11 3 3 a z + 2 a z - a z - 15 a z - 11 a z + 3 a z - 13 3 8 4 10 4 12 4 7 5 9 5 11 5 a z + 5 a z + 4 a z - a z + 7 a z + 6 a z - a z - 8 6 10 6 7 7 9 7 a z - a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 14]], Vassiliev[3][Link[8, Alternating, 14]]}
Out[14]=	317 {0, -(---)} 12
In[15]:=	Kh[Link[8, Alternating, 14]][q, t]
Out[15]=	-8 -6 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 24 8 22 8 22 7 18 6 18 5 14 4 q t q t q t q t q t q t 1 1 ------ + ------ 14 3 10 2 q t q t

L8a14

Contents

Knot presentations

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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