# L11a520

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a520 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1) \left(u v w^2-u v w+u v-u w^2+2 u w-u-v w^2+2 v w-v+w^2-w+1\right)}{u v w^{3/2}}$ (db) Jones polynomial $-q^5+5 q^4-13 q^3+23 q^2-31 q+37-36 q^{-1} +33 q^{-2} -23 q^{-3} +15 q^{-4} -6 q^{-5} + q^{-6}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^4 z^4+a^4 z^{-2} -2 a^2 z^6-z^6 a^{-2} -4 a^2 z^4-2 z^4 a^{-2} -2 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} -a^2+z^8+4 z^6+7 z^4+4 z^2+ z^{-2} +1$ (db) Kauffman polynomial $7 a^2 z^{10}+7 z^{10}+17 a^3 z^9+36 a z^9+19 z^9 a^{-1} +15 a^4 z^8+22 a^2 z^8+21 z^8 a^{-2} +28 z^8+6 a^5 z^7-27 a^3 z^7-67 a z^7-21 z^7 a^{-1} +13 z^7 a^{-3} +a^6 z^6-27 a^4 z^6-72 a^2 z^6-32 z^6 a^{-2} +5 z^6 a^{-4} -81 z^6-6 a^5 z^5+6 a^3 z^5+26 a z^5-13 z^5 a^{-3} +z^5 a^{-5} +11 a^4 z^4+41 a^2 z^4+21 z^4 a^{-2} -2 z^4 a^{-4} +53 z^4+a^3 z^3-a z^3+3 z^3 a^{-1} +5 z^3 a^{-3} +2 a^4 z^2-2 a^2 z^2-6 z^2 a^{-2} -10 z^2-a^3 z-a z+a^4+a^2+1+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2}$ (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          4 4
7         91 -8
5        144  10
3       179   -8
1      2014    6
-1     1819     1
-3    1518      -3
-5   1020       10
-7  513        -8
-9 110         9
-11 5          -5
-131           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{20}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{15}$ $r=-1$ ${\mathbb Z}^{18}\oplus{\mathbb Z}_2^{18}$ ${\mathbb Z}^{18}$ $r=0$ ${\mathbb Z}^{19}\oplus{\mathbb Z}_2^{18}$ ${\mathbb Z}^{20}$ $r=1$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{17}$ ${\mathbb Z}^{17}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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