# L11a442

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a442 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^3 w^3-u v^3 w^2-u v^2 w^3+2 u v^2 w^2-u v^2 w-u v w^2+2 u v w-u v-u w+2 u-2 v^3 w^3+v^3 w^2+v^2 w^3-2 v^2 w^2+v^2 w+v w^2-2 v w+v+w-1}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $-q^9+3 q^8-4 q^7+6 q^6-7 q^5+8 q^4-7 q^3+6 q^2-4 q+4- q^{-1} + q^{-2}$ (db) Signature 4 (db) HOMFLY-PT polynomial $-z^6 a^{-6} -4 z^4 a^{-6} -3 z^2 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +12 z^4 a^{-4} +11 z^2 a^{-4} + a^{-4} z^{-2} +5 a^{-4} -2 z^6 a^{-2} -11 z^4 a^{-2} -18 z^2 a^{-2} -2 a^{-2} z^{-2} -11 a^{-2} +z^4+5 z^2+ z^{-2} +6$ (db) Kauffman polynomial $z^{10} a^{-2} +z^{10} a^{-4} +z^9 a^{-1} +4 z^9 a^{-3} +3 z^9 a^{-5} -3 z^8 a^{-2} +4 z^8 a^{-6} +z^8-4 z^7 a^{-1} -18 z^7 a^{-3} -10 z^7 a^{-5} +4 z^7 a^{-7} -5 z^6 a^{-2} -13 z^6 a^{-4} -11 z^6 a^{-6} +4 z^6 a^{-8} -7 z^6+18 z^5 a^{-3} +8 z^5 a^{-5} -6 z^5 a^{-7} +4 z^5 a^{-9} +22 z^4 a^{-2} +19 z^4 a^{-4} +8 z^4 a^{-6} -3 z^4 a^{-8} +3 z^4 a^{-10} +17 z^4+12 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} -3 z^3 a^{-9} +z^3 a^{-11} -25 z^2 a^{-2} -10 z^2 a^{-4} -3 z^2 a^{-6} -2 z^2 a^{-8} -2 z^2 a^{-10} -18 z^2-11 z a^{-1} -11 z a^{-3} +13 a^{-2} +5 a^{-4} + a^{-8} +8+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
-4-3-2-101234567χ
19           1-1
17          2 2
15         21 -1
13        42  2
11       54   -1
9      32    1
7     45     1
5    23      -1
3   35       2
1  11        0
-1  3         3
-311          0
-51           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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