# L11n404

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n404 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $0$ (db) Jones polynomial $q^2-q+2+2 q^{-2} + q^{-3} - q^{-4} + q^{-5} - q^{-6} + q^{-7} - q^{-8}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^6 \left(-z^4\right)-4 a^6 z^2-a^6 z^{-2} -4 a^6+a^4 z^6+7 a^4 z^4+16 a^4 z^2+4 a^4 z^{-2} +13 a^4-a^2 z^6-7 a^2 z^4-16 a^2 z^2-5 a^2 z^{-2} -14 a^2+z^4+4 z^2+2 z^{-2} +5$ (db) Kauffman polynomial $z^5 a^9-4 z^3 a^9+2 z a^9+z^6 a^8-4 z^4 a^8+2 z^2 a^8+z^7 a^7-5 z^5 a^7+6 z^3 a^7-4 z a^7+a^7 z^{-1} +z^8 a^6-6 z^6 a^6+10 z^4 a^6-10 z^2 a^6-a^6 z^{-2} +7 a^6+3 z^7 a^5-21 z^5 a^5+40 z^3 a^5-25 z a^5+5 a^5 z^{-1} +3 z^8 a^4-22 z^6 a^4+50 z^4 a^4-49 z^2 a^4-4 a^4 z^{-2} +22 a^4+z^9 a^3-4 z^7 a^3-7 z^5 a^3+34 z^3 a^3-31 z a^3+9 a^3 z^{-1} +3 z^8 a^2-22 z^6 a^2+52 z^4 a^2-53 z^2 a^2-5 a^2 z^{-2} +23 a^2+z^9 a-6 z^7 a+8 z^5 a+4 z^3 a-12 z a+5 a z^{-1} +z^8-7 z^6+16 z^4-16 z^2-2 z^{-2} +9$ (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 \
-7-6-5-4-3-2-101234χ
5           11
3            0
1         21 1
-1       31   2
-3      152   2
-5     322    3
-7    121     0
-9   242      0
-11   11       0
-13 121        0
-15            0
-171           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{5}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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