# L11a42

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a42 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(v^4-4 v^3+4 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $q^{9/2}-\frac{7}{q^{9/2}}-3 q^{7/2}+\frac{12}{q^{7/2}}+6 q^{5/2}-\frac{16}{q^{5/2}}-12 q^{3/2}+\frac{18}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+15 \sqrt{q}-\frac{18}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-2 a^3 z^5+4 a z^5-2 z^5 a^{-1} +a^5 z^3-6 a^3 z^3+8 a z^3-6 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-7 a^3 z+9 a z-6 z a^{-1} +2 z a^{-3} +a^5 z^{-1} -3 a^3 z^{-1} +4 a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^2 z^{10}-z^{10}-3 a^3 z^9-6 a z^9-3 z^9 a^{-1} -5 a^4 z^8-9 a^2 z^8-4 z^8 a^{-2} -8 z^8-5 a^5 z^7-7 a^3 z^7-z^7 a^{-1} -3 z^7 a^{-3} -3 a^6 z^6+2 a^4 z^6+14 a^2 z^6+8 z^6 a^{-2} -z^6 a^{-4} +18 z^6-a^7 z^5+7 a^5 z^5+23 a^3 z^5+24 a z^5+18 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4+7 a^4 z^4-2 z^4 a^{-2} +3 z^4 a^{-4} -7 z^4+2 a^7 z^3-4 a^5 z^3-24 a^3 z^3-33 a z^3-24 z^3 a^{-1} -9 z^3 a^{-3} -3 a^6 z^2-8 a^4 z^2-8 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} -3 z^2-a^7 z+2 a^5 z+13 a^3 z+19 a z+13 z a^{-1} +4 z a^{-3} +a^6+3 a^4+3 a^2+2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          2 2
6         41 -3
4        82  6
2       74   -3
0      118    3
-2     99     0
-4    79      -2
-6   59       4
-8  27        -5
-10 15         4
-12 2          -2
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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