# L11n195

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n195 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^4-t(1) t(2)^4-2 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-5 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-2 t(2)-t(1)+1}{t(1) t(2)^2}$ (db) Jones polynomial $\frac{1}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z^5 a^7+3 z^3 a^7+3 z a^7+a^7 z^{-1} -z^7 a^5-5 z^5 a^5-9 z^3 a^5-6 z a^5-a^5 z^{-1} +z^5 a^3+2 z^3 a^3$ (db) Kauffman polynomial $a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-6 a^{10} z^4+2 a^{10} z^2+4 a^9 z^7-7 a^9 z^5+2 a^9 z^3-a^9 z+3 a^8 z^8-3 a^8 z^6-a^8 z^4+a^7 z^9+3 a^7 z^7-5 a^7 z^5-a^7 z^3+4 a^7 z-a^7 z^{-1} +4 a^6 z^8-6 a^6 z^6+6 a^6 z^4-3 a^6 z^2+a^6+a^5 z^9-a^5 z^7+7 a^5 z^5-10 a^5 z^3+6 a^5 z-a^5 z^{-1} +a^4 z^8+2 a^4 z^4-2 a^4 z^2+4 a^3 z^5-5 a^3 z^3+a^2 z^4-a^2 z^2$ (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 \
-8-7-6-5-4-3-2-101χ
0         1-1
-2        3 3
-4       42 -2
-6      52  3
-8     54   -1
-10    55    0
-12   35     2
-14  35      -2
-16 14       3
-18 2        -2
-201         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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