# L8a14

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a14 at Knotilus! L8a14 is $8^2_{1}$ in the Rolfsen table of links.  Represented as two interlaced squares  Coat of arms of Jaroměř, Czech Republic  Coat of arms of St. Savior, Jersey, Channel Islands, depicting Crown of Thorns religious symbol  Decorative motif of interlaced squares, San Pancrazio, Florence  Arma Christi carving in monastery in Portugal  Handewitt, Schleswig-Holstein oak leaves and acorns  Link on a church of Murato,Corsica

### Polynomial invariants

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

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-8-7-6-5-4-3-2-10χ
-6        11
-8        11
-10      1  1
-12         0
-14    11   0
-16         0
-18  11     0
-20         0
-2211       0
-241        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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