# L11a212

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a212 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u^2 v^4-6 u^2 v^3+5 u^2 v^2-2 u^2 v-2 u v^4+7 u v^3-11 u v^2+7 u v-2 u-2 v^3+5 v^2-6 v+2}{u v^2}$ (db) Jones polynomial $-\frac{16}{q^{9/2}}+\frac{11}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{3}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{4}{q^{21/2}}+\frac{8}{q^{19/2}}-\frac{13}{q^{17/2}}+\frac{17}{q^{15/2}}-\frac{19}{q^{13/2}}+\frac{18}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^9 \left(-z^5\right)-2 a^9 z^3+a^7 z^7+3 a^7 z^5+2 a^7 z^3+a^7 z+a^7 z^{-1} +a^5 z^7+3 a^5 z^5+a^5 z^3-2 a^5 z-a^5 z^{-1} -a^3 z^5-3 a^3 z^3-2 a^3 z$ (db) Kauffman polynomial $-z^4 a^{14}-4 z^5 a^{13}+2 z^3 a^{13}-8 z^6 a^{12}+7 z^4 a^{12}-z^2 a^{12}-11 z^7 a^{11}+14 z^5 a^{11}-5 z^3 a^{11}-11 z^8 a^{10}+17 z^6 a^{10}-7 z^4 a^{10}-7 z^9 a^9+6 z^7 a^9+9 z^5 a^9-8 z^3 a^9+2 z a^9-2 z^{10} a^8-11 z^8 a^8+39 z^6 a^8-31 z^4 a^8+7 z^2 a^8-11 z^9 a^7+30 z^7 a^7-22 z^5 a^7+7 z^3 a^7-3 z a^7+a^7 z^{-1} -2 z^{10} a^6-3 z^8 a^6+25 z^6 a^6-28 z^4 a^6+10 z^2 a^6-a^6-4 z^9 a^5+12 z^7 a^5-9 z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1} -3 z^8 a^4+11 z^6 a^4-12 z^4 a^4+4 z^2 a^4-z^7 a^3+4 z^5 a^3-5 z^3 a^3+2 z a^3$ (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 \
-9-8-7-6-5-4-3-2-1012χ
0           11
-2          2 -2
-4         51 4
-6        73  -4
-8       94   5
-10      97    -2
-12     109     1
-14    79      2
-16   610       -4
-18  38        5
-20 15         -4
-22 3          3
-241           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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