# L11n188

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n188 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^4-t(2)^4+2 t(1)^2 t(2)^3-5 t(1) t(2)^3+2 t(2)^3-3 t(1)^2 t(2)^2+7 t(1) t(2)^2-3 t(2)^2+2 t(1)^2 t(2)-5 t(1) t(2)+2 t(2)-t(1)^2+t(1)}{t(1) t(2)^2}$ (db) Jones polynomial $-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{2}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-2 a^7 z^3-3 a^7 z-a^7 z^{-1} +2 a^5 z^5+6 a^5 z^3+7 a^5 z+3 a^5 z^{-1} +a^3 z^5-4 a^3 z-2 a^3 z^{-1} -a z^3-a z$ (db) Kauffman polynomial $3 a^{10} z^4-4 a^{10} z^2+a^9 z^7+3 a^9 z^5-6 a^9 z^3+a^9 z+2 a^8 z^8-2 a^8 z^4+2 a^8 z^2-a^8+a^7 z^9+5 a^7 z^7-12 a^7 z^5+12 a^7 z^3-5 a^7 z+a^7 z^{-1} +6 a^6 z^8-7 a^6 z^6-3 a^6 z^4+10 a^6 z^2-3 a^6+a^5 z^9+9 a^5 z^7-26 a^5 z^5+28 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +4 a^4 z^8-4 a^4 z^6-3 a^4 z^4+5 a^4 z^2-3 a^4+5 a^3 z^7-10 a^3 z^5+8 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +3 a^2 z^6-5 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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 \
-7-6-5-4-3-2-1012χ
2         11
0        2 -2
-2       51 4
-4      53  -2
-6     74   3
-8    66    0
-10   56     -1
-12  36      3
-14 25       -3
-16 3        3
-182         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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