L8a14

L8a13

L8a15

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a14 at Knotilus!
	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

Link Presentations

[edit Notes on L8a14's Link 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}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

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)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   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.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

L8a13 L8a15