# L11n267

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n267 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v w^3-u v w^2+u v+u w^5-u w^4+u w-u+v w^5-v w^4+v w-v-w^5+w^3-w^2}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db) Jones polynomial $-q^5+q^4-q^3-q^2+q-1+4 q^{-1} -2 q^{-2} +4 q^{-3} - q^{-4} + q^{-5}$ (db) Signature 2 (db) HOMFLY-PT polynomial $2 z^6-3 a^2 z^4-z^4 a^{-2} +12 z^4+a^4 z^2-14 a^2 z^2-6 z^2 a^{-2} -z^2 a^{-4} +23 z^2+4 a^4-18 a^2-7 a^{-2} +21+3 a^4 z^{-2} -8 a^2 z^{-2} -2 a^{-2} z^{-2} +7 z^{-2}$ (db) Kauffman polynomial $a^3 z^9+a z^9+a^4 z^8+5 a^2 z^8+4 z^8-4 a^3 z^7-a z^7+3 z^7 a^{-1} -7 a^4 z^6-31 a^2 z^6-z^6 a^{-2} +z^6 a^{-4} -26 z^6-2 a^3 z^5-21 a z^5-20 z^5 a^{-1} +z^5 a^{-5} +18 a^4 z^4+61 a^2 z^4+3 z^4 a^{-2} -4 z^4 a^{-4} +50 z^4+21 a^3 z^3+53 a z^3+33 z^3 a^{-1} -3 z^3 a^{-3} -4 z^3 a^{-5} -22 a^4 z^2-54 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -42 z^2-24 a^3 z-45 a z-21 z a^{-1} +3 z a^{-3} +3 z a^{-5} +13 a^4+28 a^2+7 a^{-2} + a^{-4} +22+8 a^3 z^{-1} +15 a z^{-1} +7 a^{-1} z^{-1} - a^{-3} z^{-1} - a^{-5} z^{-1} -3 a^4 z^{-2} -8 a^2 z^{-2} -2 a^{-2} z^{-2} -7 z^{-2}$ (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 \
-6-5-4-3-2-1012345χ
11           1-1
9            0
7         11 0
5       31   -2
3      121   0
1     341    0
-1    113     3
-3   13       2
-5  31        2
-7 14         3
-9            0
-111           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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