# L6a3

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a3 at Knotilus! The link L6a3 is $6^2_1$ in the Rolfsen table of links. It is often seen in "Magen David" (star of David) necklaces.  Ruberman, Cochran, Melvin, Akbulut, Gompf, Kirby  Triangle interlaced with a circle, a traditional symbol of the Christian Trinity (less used in recent centuries)  An architectural trefoil (the outline of three overlapping circles) interlaced with an equilateral triangle, another old Christian Trinitarian symbol.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1)^2 t(2)^2-t(1) t(2)-1}{t(1) t(2)}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^7 z^3+3 a^7 z+a^7 z^{-1} -a^5 z^5-5 a^5 z^3-6 a^5 z-a^5 z^{-1}$ (db) Kauffman polynomial $-z a^{11}-z^2 a^{10}-z^3 a^9+z a^9-z^4 a^8+2 z^2 a^8-z^5 a^7+4 z^3 a^7-4 z a^7+a^7 z^{-1} -z^4 a^6+3 z^2 a^6-a^6-z^5 a^5+5 z^3 a^5-6 z a^5+a^5 z^{-1}$ (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 \
-6-5-4-3-2-10χ
-4      11
-6      11
-8    1  1
-10       0
-12  11   0
-14       0
-1611     0
-181      1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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