# L11n287

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n287 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1)}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db) Jones polynomial $q^2+2- q^{-1} +3 q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} +2 q^{-6} - q^{-7}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6 \left(-z^2\right)-a^6+a^4 z^4+3 a^4 z^2+2 a^4-a^2 z^2+a^2 z^{-2} + a^{-2} z^{-2} + a^{-2} -z^2-2 z^{-2} -2$ (db) Kauffman polynomial $a^5 z^9+a^3 z^9+2 a^6 z^8+3 a^4 z^8+a^2 z^8+a^7 z^7-4 a^5 z^7-5 a^3 z^7-11 a^6 z^6-17 a^4 z^6-6 a^2 z^6-5 a^7 z^5+6 a^3 z^5+a z^5+16 a^6 z^4+28 a^4 z^4+12 a^2 z^4+6 a^7 z^3+8 a^5 z^3-2 a^3 z^3-4 a z^3-7 a^6 z^2-18 a^4 z^2-10 a^2 z^2+z^2 a^{-2} +2 z^2-2 a^7 z-4 a^5 z+4 a z+2 z a^{-1} +2 a^6+4 a^4-2 a^{-2} -3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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
5         11
3         11
1       31 2
-1      241 1
-3     222  2
-5    23    1
-7   121    0
-9  121     0
-11 11       0
-13 1        1
-151         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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