# L8a13

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a13 at Knotilus! L8a13 is $8^2_{4}$ in the Rolfsen table of links. Contains two L4a1 configurations.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)+t(2)) \left(t(2) t(1)^2+t(2)^2 t(1)-2 t(2) t(1)+t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $\frac{4}{q^{9/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{4}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z a^9+a^9 z^{-1} -z^3 a^7-z a^7-a^7 z^{-1} -2 z^3 a^5-3 z a^5-z^3 a^3-z a^3$ (db) Kauffman polynomial $-z^5 a^{11}+4 z^3 a^{11}-4 z a^{11}-z^6 a^{10}+2 z^4 a^{10}-z^7 a^9+2 z^5 a^9-2 z^3 a^9+3 z a^9-a^9 z^{-1} -3 z^6 a^8+5 z^4 a^8-z^2 a^8+a^8-z^7 a^7-z^3 a^7+3 z a^7-a^7 z^{-1} -2 z^6 a^6+z^4 a^6-3 z^5 a^5+4 z^3 a^5-3 z a^5-2 z^4 a^4+z^2 a^4-z^3 a^3+z a^3$ (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
-2        11
-4       21-1
-6      2  2
-8     22  0
-10    22   0
-12   22    0
-14  12     -1
-16  2      2
-1811       0
-201        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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