# L4a1

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L4a1 at Knotilus! L4a1 is $4^2_1$ in the Rolfsen table of links. It frequently occurs in late Roman mosaics and some medieval decorations. In this context, it is called the "Solomon's knot" (sigillum Salomonis) or "guilloche knot". It is also the "Kramo-bone" symbol (meaning "one being bad makes all appear to be bad") of the Adinkra symbol system. Link L10a101 contains multiple L4a1 configurations.  Carving above door of church in Italy  Linked hearts used as symbol of Vendée region of France  Decorative fitting closely within square.  Made of two (impossible) Penrose rectangles.  Array of "Solomon's knots" forming overall circular patterns.  Rotated knotwork cross with eight L4a1 sub-configurations  Knotopologynn-diagram for "Solomon's knots"

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-u-v}{\sqrt{u} \sqrt{v}}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{3/2}}-\frac{1}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^{-1} -a^3 z-a^3 z^{-1} -a z$ (db) Kauffman polynomial $a^5 z^3-3 a^5 z+a^5 z^{-1} +a^4 z^2-a^4+a^3 z^3-2 a^3 z+a^3 z^{-1} +a^2 z^2+a z$ (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 \
-4-3-2-10χ
0    11
-2   110
-4     0
-6  1  1
-81    1
-101    1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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