# L11a242

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a242 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{4 t(2)^2 t(1)^2-7 t(2) t(1)^2+3 t(1)^2-7 t(2)^2 t(1)+13 t(2) t(1)-7 t(1)+3 t(2)^2-7 t(2)+4}{t(1) t(2)}$ (db) Jones polynomial $q^{21/2}-4 q^{19/2}+8 q^{17/2}-12 q^{15/2}+16 q^{13/2}-18 q^{11/2}+17 q^{9/2}-15 q^{7/2}+10 q^{5/2}-6 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-9} -z^5 a^{-7} +z a^{-7} -2 z^5 a^{-5} -3 z^3 a^{-5} -z a^{-5} -z^5 a^{-3} -z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-6} -z^{10} a^{-8} -3 z^9 a^{-5} -7 z^9 a^{-7} -4 z^9 a^{-9} -4 z^8 a^{-4} -9 z^8 a^{-6} -11 z^8 a^{-8} -6 z^8 a^{-10} -3 z^7 a^{-3} -z^7 a^{-5} +7 z^7 a^{-7} +z^7 a^{-9} -4 z^7 a^{-11} -2 z^6 a^{-2} +5 z^6 a^{-4} +23 z^6 a^{-6} +31 z^6 a^{-8} +14 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} +2 z^5 a^{-3} +7 z^5 a^{-5} +10 z^5 a^{-7} +16 z^5 a^{-9} +10 z^5 a^{-11} +3 z^4 a^{-2} -6 z^4 a^{-4} -23 z^4 a^{-6} -23 z^4 a^{-8} -7 z^4 a^{-10} +2 z^4 a^{-12} +3 z^3 a^{-1} +3 z^3 a^{-3} -9 z^3 a^{-5} -16 z^3 a^{-7} -13 z^3 a^{-9} -6 z^3 a^{-11} +4 z^2 a^{-4} +7 z^2 a^{-6} +5 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} -3 z a^{-1} -3 z a^{-3} +3 z a^{-5} +5 z a^{-7} +3 z a^{-9} +z a^{-11} - a^{-2} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-2-10123456789χ
22           1-1
20          3 3
18         51 -4
16        73  4
14       95   -4
12      97    2
10     89     1
8    79      -2
6   49       5
4  26        -4
2 15         4
0 1          -1
-21           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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