# L11n328

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n328 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1) (v w+1)}{\sqrt{u} v w}$ (db) Jones polynomial $q^5-2 q^4- q^{-4} +4 q^3+2 q^{-3} -4 q^2-3 q^{-2} +6 q+5 q^{-1} -4$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -a^2 z^4-2 z^4 a^{-2} -3 a^2 z^2-7 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2-7 a^{-2} +z^6+5 z^4+9 z^2+ z^{-2} +7$ (db) Kauffman polynomial $a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+a^3 z^7-2 a z^7-z^7 a^{-1} +2 z^7 a^{-3} -10 a^2 z^6-7 z^6 a^{-2} +z^6 a^{-4} -18 z^6-5 a^3 z^5-6 a z^5-8 z^5 a^{-1} -7 z^5 a^{-3} +15 a^2 z^4+9 z^4 a^{-2} -2 z^4 a^{-4} +26 z^4+7 a^3 z^3+13 a z^3+14 z^3 a^{-1} +10 z^3 a^{-3} +2 z^3 a^{-5} -9 a^2 z^2-10 z^2 a^{-2} +3 z^2 a^{-4} +z^2 a^{-6} -21 z^2-2 a^3 z-7 a z-10 z a^{-1} -6 z a^{-3} -z a^{-5} +3 a^2+7 a^{-2} + a^{-4} - a^{-6} +9+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
-5-4-3-2-101234χ
11         11
9        1 -1
7       31 2
5      33  0
3     321  2
1    35    2
-1   221    1
-3  13      2
-5 12       -1
-7 1        1
-91         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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