# L11n225

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n225 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2) t(1)^2+2 t(2)^2 t(1)-t(2) t(1)+2 t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 \left(-z^3\right)-a^7 z+a^5 z^5+a^5 z^3-2 a^5 z-a^5 z^{-1} +2 a^3 z^5+7 a^3 z^3+8 a^3 z+3 a^3 z^{-1} -2 a z^3-5 a z-2 a z^{-1}$ (db) Kauffman polynomial $-z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-5 z^6 a^8+7 z^4 a^8-3 z^2 a^8-5 z^7 a^7+8 z^5 a^7-7 z^3 a^7+2 z a^7-3 z^8 a^6+3 z^6 a^6-4 z^4 a^6+4 z^2 a^6-a^6-z^9 a^5-2 z^7 a^5+2 z^5 a^5+4 z^3 a^5-4 z a^5+a^5 z^{-1} -4 z^8 a^4+10 z^6 a^4-14 z^4 a^4+12 z^2 a^4-3 a^4-z^9 a^3+3 z^7 a^3-12 z^5 a^3+24 z^3 a^3-15 z a^3+3 a^3 z^{-1} -z^8 a^2+2 z^6 a^2-2 z^4 a^2+4 z^2 a^2-3 a^2-3 z^5 a+10 z^3 a-9 z a+2 a z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
2         22
0        1 -1
-2       52 3
-4      53  -2
-6     53   2
-8    45    1
-10   45     -1
-12  24      2
-14 14       -3
-16 2        2
-181         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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