# L6a1

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a1 at Knotilus! L6a1 is $6^2_3$ in the Rolfsen table of links.
 A kolam with two cycles/components[1] Depiction with two eights interlaced Mongolian ornament ; the two eights are horizontal Another one, sum of two L6a1 Another depiction

 Planar diagram presentation X6172 X10,3,11,4 X12,8,5,7 X8,12,9,11 X2536 X4,9,1,10 Gauss code {1, -5, 2, -6}, {5, -1, 3, -4, 6, -2, 4, -3}

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v-2 u-2 v+1}{\sqrt{u} \sqrt{v}}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{5/2}}-q^{3/2}+\frac{2}{q^{3/2}}+2 \sqrt{q}-\frac{2}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^{-1} -2 z a^3-a^3 z^{-1} +z^3 a+z a-z a^{-1}$ (db) Kauffman polynomial $a^5 z^3-2 a^5 z+a^5 z^{-1} +a^4 z^4-a^4+a^3 z^5-a^3 z+a^3 z^{-1} +3 a^2 z^4-3 a^2 z^2+a z^5+z^3 a^{-1} -z a^{-1} +2 z^4-3 z^2$ (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 \
-4-3-2-1012χ
4      11
2     1 -1
0    11 0
-2   22  0
-4  1    1
-6  2    2
-811     0
-101      1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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