# L11n444

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n444 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1) t(4)^2 t(3)^2-t(2) t(4)^2 t(3)^2+t(4)^2 t(3)^2-t(2) t(3)^2+t(1) t(4) t(3)^2+2 t(2) t(4) t(3)^2-t(4) t(3)^2+2 t(1) t(4)^2 t(3)+t(2) t(4)^2 t(3)-t(4)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-2 t(1) t(4) t(3)-2 t(2) t(4) t(3)-t(1) t(4)^2-t(1)+t(1) t(2)-t(2)+2 t(1) t(4)-t(1) t(2) t(4)+t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $2 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^9 z^{-1} +a^9 z^{-3} -4 z a^7-7 a^7 z^{-1} -3 a^7 z^{-3} +5 z^3 a^5+14 z a^5+12 a^5 z^{-1} +3 a^5 z^{-3} -2 z^5 a^3-8 z^3 a^3-13 z a^3-7 a^3 z^{-1} -a^3 z^{-3} +2 z^3 a+3 z a+a z^{-1}$ (db) Kauffman polynomial $-z^7 a^9+5 z^5 a^9-10 z^3 a^9+10 z a^9-5 a^9 z^{-1} +a^9 z^{-3} -2 z^8 a^8+7 z^6 a^8-4 z^4 a^8-7 z^2 a^8-3 a^8 z^{-2} +9 a^8-z^9 a^7-4 z^7 a^7+31 z^5 a^7-50 z^3 a^7+35 z a^7-14 a^7 z^{-1} +3 a^7 z^{-3} -7 z^8 a^6+21 z^6 a^6-4 z^4 a^6-24 z^2 a^6-6 a^6 z^{-2} +21 a^6-z^9 a^5-11 z^7 a^5+54 z^5 a^5-74 z^3 a^5+49 z a^5-18 a^5 z^{-1} +3 a^5 z^{-3} -5 z^8 a^4+9 z^6 a^4+11 z^4 a^4-28 z^2 a^4-3 a^4 z^{-2} +18 a^4-8 z^7 a^3+27 z^5 a^3-39 z^3 a^3+30 z a^3-11 a^3 z^{-1} +a^3 z^{-3} -5 z^6 a^2+11 z^4 a^2-14 z^2 a^2+6 a^2-z^5 a-5 z^3 a+6 z a-2 a z^{-1} -3 z^2+1$ (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 \
-8-7-6-5-4-3-2-101χ
2         2-2
0        3 3
-2       43 -1
-4      631 4
-6     57   2
-8    541   2
-10   48     4
-12  22      0
-14 15       4
-16 1        -1
-181         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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