# L11n426

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n426 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-t(2))^2 (t(3)-1)}{t(1) t(2) \sqrt{t(3)}}$ (db) Jones polynomial $q^{-5} - q^{-4} -q^3+ q^{-3} +q^2+ q^{-1} +2$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^4 z^2+a^4 z^{-2} +2 a^4-a^2 z^4-5 a^2 z^2-2 a^2 z^{-2} -z^2 a^{-2} -7 a^2-2 a^{-2} +z^4+5 z^2+ z^{-2} +7$ (db) Kauffman polynomial $a^4 z^8-7 a^4 z^6+15 a^4 z^4-12 a^4 z^2-a^4 z^{-2} +5 a^4+a^3 z^9-7 a^3 z^7+14 a^3 z^5-7 a^3 z^3+z^3 a^{-3} -3 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +2 a^2 z^8-15 a^2 z^6+35 a^2 z^4+z^4 a^{-2} -32 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+4 a^{-2} +a z^9-7 a z^7+13 a z^5-z^5 a^{-1} -3 a z^3+5 z^3 a^{-1} -7 a z-6 z a^{-1} +2 a z^{-1} +z^8-8 z^6+21 z^4-24 z^2- z^{-2} +13$ (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 \
-6-5-4-3-2-10123χ
7         1-1
5          0
3      111 1
1      2   2
-1    113   3
-3   12     1
-5   12     1
-7 11       0
-9          0
-111         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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