# L11a399

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a399 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 (t(1)-1) (t(2)-1)^2 (t(3)-1)^2}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $q^6-4 q^5- q^{-5} +8 q^4+4 q^{-4} -13 q^3-7 q^{-3} +18 q^2+14 q^{-2} -20 q-17 q^{-1} +21$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^4 a^{-4} -a^4 z^2+z^2 a^{-4} +a^4 z^{-2} -z^6 a^{-2} +2 a^2 z^4-2 z^4 a^{-2} +2 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} -a^2-z^6-z^4+ z^{-2} +1$ (db) Kauffman polynomial $z^6 a^{-6} -2 z^4 a^{-6} +4 z^7 a^{-5} +a^5 z^5-10 z^5 a^{-5} -a^5 z^3+5 z^3 a^{-5} +7 z^8 a^{-4} +4 a^4 z^6-19 z^6 a^{-4} -7 a^4 z^4+15 z^4 a^{-4} +4 a^4 z^2-4 z^2 a^{-4} -a^4 z^{-2} +a^4+6 z^9 a^{-3} +6 a^3 z^7-11 z^7 a^{-3} -7 a^3 z^5+2 z^5 a^{-3} +a^3 z^3+3 z^3 a^{-3} -a^3 z+2 a^3 z^{-1} +2 z^{10} a^{-2} +6 a^2 z^8+9 z^8 a^{-2} -a^2 z^6-32 z^6 a^{-2} -12 a^2 z^4+30 z^4 a^{-2} +8 a^2 z^2-8 z^2 a^{-2} -2 a^2 z^{-2} +a^2+5 a z^9+11 z^9 a^{-1} -a z^7-22 z^7 a^{-1} -3 a z^5+17 z^5 a^{-1} -2 a z^3-6 z^3 a^{-1} -a z+2 a z^{-1} +2 z^{10}+8 z^8-17 z^6+8 z^4- z^{-2} +1$ (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 \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         51 4
7        83  -5
5       105   5
3      108    -2
1     1110     1
-1    812      4
-3   69       -3
-5  310        7
-7 14         -3
-9 3          3
-111           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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