# L11a209

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a209 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{\left(2 v^2-3 v+2\right) (u v-u+1) (u v-v+1)}{u v^2}$ (db) Jones polynomial $-\frac{12}{q^{9/2}}-q^{7/2}+\frac{17}{q^{7/2}}+4 q^{5/2}-\frac{20}{q^{5/2}}-9 q^{3/2}+\frac{20}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{7}{q^{11/2}}+13 \sqrt{q}-\frac{19}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7+a z^7-a^5 z^5+4 a^3 z^5+3 a z^5-z^5 a^{-1} -3 a^5 z^3+7 a^3 z^3+2 a z^3-2 z^3 a^{-1} -3 a^5 z+6 a^3 z-a z-z a^{-1} -a^5 z^{-1} +2 a^3 z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^8 z^6-3 a^8 z^4+2 a^8 z^2+3 a^7 z^7-8 a^7 z^5+5 a^7 z^3+5 a^6 z^8-13 a^6 z^6+12 a^6 z^4-7 a^6 z^2+2 a^6+5 a^5 z^9-11 a^5 z^7+12 a^5 z^5-11 a^5 z^3+4 a^5 z-a^5 z^{-1} +2 a^4 z^{10}+6 a^4 z^8-24 a^4 z^6+31 a^4 z^4-21 a^4 z^2+5 a^4+11 a^3 z^9-25 a^3 z^7+27 a^3 z^5+z^5 a^{-3} -18 a^3 z^3-z^3 a^{-3} +8 a^3 z-2 a^3 z^{-1} +2 a^2 z^{10}+10 a^2 z^8-29 a^2 z^6+4 z^6 a^{-2} +30 a^2 z^4-5 z^4 a^{-2} -14 a^2 z^2+3 a^2+6 a z^9-3 a z^7+8 z^7 a^{-1} -8 a z^5-14 z^5 a^{-1} +7 a z^3+8 z^3 a^{-1} +a z-3 z a^{-1} + a^{-1} z^{-1} +9 z^8-15 z^6+9 z^4-2 z^2-1$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          3 -3
4         61 5
2        84  -4
0       115   6
-2      109    -1
-4     1010     0
-6    710      3
-8   510       -5
-10  27        5
-12 15         -4
-14 2          2
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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