# L11a416

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a416 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+2 t(1) t(2)^2-3 t(1) t(3) t(2)^2+4 t(3) t(2)^2-2 t(2)^2-3 t(1) t(3)^2 t(2)+4 t(3)^2 t(2)-4 t(1) t(2)+8 t(1) t(3) t(2)-8 t(3) t(2)+3 t(2)+2 t(1) t(3)^2-2 t(3)^2+2 t(1)-4 t(1) t(3)+3 t(3)-1}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $- q^{-8} +3 q^{-7} -6 q^{-6} +12 q^{-5} -15 q^{-4} +q^3+19 q^{-3} -4 q^2-18 q^{-2} +8 q+17 q^{-1} -12$ (db) Signature -2 (db) HOMFLY-PT polynomial $-a^8+3 a^6 z^2+a^6 z^{-2} +3 a^6-3 a^4 z^4-5 a^4 z^2-2 a^4 z^{-2} -5 a^4+a^2 z^6+2 a^2 z^4+4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +3 a^2-2 z^4-2 z^2$ (db) Kauffman polynomial $a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-6 a^8 z^4+5 a^8 z^2-2 a^8+4 a^7 z^7-3 a^7 z^5-3 a^7 z^3+3 a^7 z+4 a^6 z^8+2 a^6 z^6-15 a^6 z^4+18 a^6 z^2+a^6 z^{-2} -8 a^6+3 a^5 z^9+4 a^5 z^7-9 a^5 z^5+a^5 z^3+5 a^5 z-2 a^5 z^{-1} +a^4 z^{10}+9 a^4 z^8-13 a^4 z^6-4 a^4 z^4+16 a^4 z^2+2 a^4 z^{-2} -9 a^4+7 a^3 z^9-5 a^3 z^7-8 a^3 z^5+4 a^3 z^3+3 a^3 z-2 a^3 z^{-1} +a^2 z^{10}+11 a^2 z^8-27 a^2 z^6+z^6 a^{-2} +15 a^2 z^4-2 z^4 a^{-2} +a^2 z^2+z^2 a^{-2} +a^2 z^{-2} -4 a^2+4 a z^9-a z^7+4 z^7 a^{-1} -13 a z^5-10 z^5 a^{-1} +8 a z^3+6 z^3 a^{-1} +6 z^8-14 z^6+8 z^4-z^2$ (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 \
-7-6-5-4-3-2-101234χ
7           11
5          3 -3
3         51 4
1        73  -4
-1       105   5
-3      98    -1
-5     109     1
-7    711      4
-9   58       -3
-11  28        6
-13 14         -3
-15 2          2
-171           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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