# L11a519

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a519 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(2)-1)^2 (t(3)-1) (t(3) t(1)-t(1)+1) (t(1) t(3)-t(3)+1)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $q^{-6} -q^5-4 q^{-5} +4 q^4+9 q^{-4} -9 q^3-14 q^{-3} +15 q^2+21 q^{-2} -20 q-22 q^{-1} +24$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^4 z^4+2 a^4 z^2+a^4 z^{-2} +a^4-2 a^2 z^6-z^6 a^{-2} -7 a^2 z^4-3 z^4 a^{-2} -8 a^2 z^2-3 z^2 a^{-2} -2 a^2 z^{-2} -4 a^2- a^{-2} +z^8+5 z^6+10 z^4+9 z^2+ z^{-2} +4$ (db) Kauffman polynomial $2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+13 a z^9+7 z^9 a^{-1} +7 a^4 z^8+14 a^2 z^8+10 z^8 a^{-2} +17 z^8+4 a^5 z^7-6 a^3 z^7-20 a z^7-2 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-16 a^4 z^6-44 a^2 z^6-16 z^6 a^{-2} +4 z^6 a^{-4} -47 z^6-9 a^5 z^5-7 a^3 z^5+3 a z^5-12 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+12 a^4 z^4+45 a^2 z^4+12 z^4 a^{-2} -5 z^4 a^{-4} +48 z^4+5 a^5 z^3+6 a^3 z^3+6 a z^3+12 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-6 a^4 z^2-25 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -25 z^2-2 a^3 z-4 a z-3 z a^{-1} -z a^{-3} +3 a^4+7 a^2+2 a^{-2} +7+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2}$ (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 \
-6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         61 -5
5        93  6
3       116   -5
1      139    4
-1     1113     2
-3    1011      -1
-5   613       7
-7  38        -5
-9 16         5
-11 3          -3
-131           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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