L8a14

L8a13

L8a15

	Visit L8a14's page at Knotilus! Visit L8a14's page at the original Knot Atlas!
	L8a14 is $8_{1}^{2}$ 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	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X16,7,9,8 X8,9,1,10 X4,13,5,14 X6,15,7,16
Gauss code	{1, -2, 3, -7, 4, -8, 5, -6}, {6, -1, 2, -3, 7, -4, 8, -5}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, $-{\frac {317}{12}}$)

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 Data:L8a14/V 2,1 Data:L8a14/V 3,1 Data:L8a14/V 4,1 Data:L8a14/V 4,2 Data:L8a14/V 4,3 Data:L8a14/V 5,1 Data:L8a14/V 5,2 Data:L8a14/V 5,3 Data:L8a14/V 5,4 Data:L8a14/V 6,1 Data:L8a14/V 6,2 Data:L8a14/V 6,3 Data:L8a14/V 6,4 Data:L8a14/V 6,5 Data:L8a14/V 6,6 Data:L8a14/V 6,7 Data:L8a14/V 6,8 Data:L8a14/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 1 -8                 1 1 -10             1     1 -12                   0 -14         1 1       0 -16                   0 -18     1 1           0 -20                   0 -22 1 1               0 -24 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-8$ $i=-6$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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[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