# 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 [1] Rich Schwartz' "72" [2] 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.

 Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X12,5,7,6 X6718 X4,11,5,12 Gauss code {1, -2, 3, -6, 4, -5}, {5, -1, 2, -3, 6, -4}

### 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.