# L11n215

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n215 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^2 t(1)^3-t(2) t(1)^3+t(2)^3 t(1)^2-4 t(2)^2 t(1)^2+4 t(2) t(1)^2-2 t(1)^2-2 t(2)^3 t(1)+4 t(2)^2 t(1)-4 t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{2}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^9-2 z a^9-a^9 z^{-1} +z^5 a^7+3 z^3 a^7+5 z a^7+3 a^7 z^{-1} +z^5 a^5+z^3 a^5-2 z a^5-2 a^5 z^{-1} -2 z^3 a^3-3 z a^3$ (db) Kauffman polynomial $-z^6 a^{12}+4 z^4 a^{12}-4 z^2 a^{12}-2 z^7 a^{11}+7 z^5 a^{11}-7 z^3 a^{11}+2 z a^{11}-2 z^8 a^{10}+5 z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}-a^{10}-z^9 a^9+2 z^5 a^9+3 z^3 a^9-2 z a^9+a^9 z^{-1} -4 z^8 a^8+9 z^6 a^8-10 z^4 a^8+11 z^2 a^8-3 a^8-z^9 a^7-5 z^5 a^7+15 z^3 a^7-12 z a^7+3 a^7 z^{-1} -2 z^8 a^6+2 z^6 a^6-5 z^4 a^6+6 z^2 a^6-3 a^6-2 z^7 a^5+2 z^3 a^5-5 z a^5+2 a^5 z^{-1} -z^6 a^4-2 z^4 a^4+z^2 a^4-3 z^3 a^3+3 z a^3$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         22
-4        31-2
-6       41 3
-8      43  -1
-10     54   1
-12    45    1
-14   34     -1
-16  14      3
-18 13       -2
-20 1        1
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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