# L8a7

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a7 at Knotilus! L8a7 is $8^2_{14}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(2)^3+4 t(1) t(2)^2-4 t(2)^2-4 t(1) t(2)+4 t(2)+t(1)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $\frac{5}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z-a^9 z^{-1} -a^7 z^3+2 a^7 z+3 a^7 z^{-1} -3 a^5 z^3-5 a^5 z-2 a^5 z^{-1} -a^3 z^3$ (db) Kauffman polynomial $a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-8 a^{10} z^4+5 a^{10} z^2+a^{10}+2 a^9 z^7-a^9 z^5-5 a^9 z^3+2 a^9 z-a^9 z^{-1} +8 a^8 z^6-16 a^8 z^4+4 a^8 z^2+3 a^8+2 a^7 z^7+4 a^7 z^5-12 a^7 z^3+6 a^7 z-3 a^7 z^{-1} +5 a^6 z^6-5 a^6 z^4-a^6 z^2+3 a^6+6 a^5 z^5-8 a^5 z^3+5 a^5 z-2 a^5 z^{-1} +3 a^4 z^4+a^3 z^3$ (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       31-2
-6      3  3
-8     23  1
-10    53   2
-12   23    1
-14  24     -2
-16 12      1
-18 2       -2
-201        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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