# L11a444

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a444 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^2 t(2)^3-2 t(3)^2 t(2)^3-t(1) t(3) t(2)^3+t(3) t(2)^3+t(1) t(3)^3 t(2)^2-2 t(3)^3 t(2)^2-4 t(1) t(3)^2 t(2)^2+5 t(3)^2 t(2)^2-t(1) t(2)^2+4 t(1) t(3) t(2)^2-4 t(3) t(2)^2+t(2)^2-t(1) t(3)^3 t(2)+t(3)^3 t(2)+4 t(1) t(3)^2 t(2)-4 t(3)^2 t(2)+2 t(1) t(2)-5 t(1) t(3) t(2)+4 t(3) t(2)-t(2)-t(1) t(3)^2+t(3)^2+2 t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $-q^8+4 q^7-8 q^6+13 q^5-15 q^4+18 q^3+ q^{-3} -17 q^2-2 q^{-2} +14 q+6 q^{-1} -9$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2-5 a^{-2} +3 a^{-4} -2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2}$ (db) Kauffman polynomial $z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -6 z^4 a^{-8} +2 z^2 a^{-8} +7 z^7 a^{-7} -11 z^5 a^{-7} +3 z^3 a^{-7} +7 z^8 a^{-6} -9 z^6 a^{-6} +z^4 a^{-6} +2 z^2 a^{-6} -2 a^{-6} +4 z^9 a^{-5} +z^7 a^{-5} -8 z^5 a^{-5} +3 z^3 a^{-5} +z a^{-5} +z^{10} a^{-4} +9 z^8 a^{-4} -18 z^6 a^{-4} +12 z^4 a^{-4} -5 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +6 z^9 a^{-3} -6 z^7 a^{-3} -5 z^5 a^{-3} +13 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +5 z^8 a^{-2} +a^2 z^6-13 z^6 a^{-2} -4 a^2 z^4+15 z^4 a^{-2} +5 a^2 z^2-15 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2+9 a^{-2} +2 z^9 a^{-1} +2 a z^7+2 z^7 a^{-1} -5 a z^5-14 z^5 a^{-1} +2 a z^3+16 z^3 a^{-1} +a z-8 z a^{-1} +2 a^{-1} z^{-1} +3 z^8-7 z^6+6 z^4-5 z^2- z^{-2} +3$ (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
17           1-1
15          3 3
13         51 -4
11        83  5
9       97   -2
7      96    3
5     89     1
3    69      -3
1   49       5
-1  25        -3
-3  4         4
-512          -1
-71           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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