# L11a379

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a379 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^4 v^2-u^4 v-u^3 v^4+4 u^3 v^3-6 u^3 v^2+4 u^3 v-u^3+2 u^2 v^4-7 u^2 v^3+9 u^2 v^2-7 u^2 v+2 u^2-u v^4+4 u v^3-6 u v^2+4 u v-u-v^3+v^2}{u^2 v^2}$ (db) Jones polynomial $-q^{5/2}+3 q^{3/2}-7 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{19}{q^{5/2}}-\frac{21}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 \left(-z^3\right)-a^7 z+2 a^5 z^5+4 a^5 z^3+a^5 z-a^3 z^7-3 a^3 z^5-3 a^3 z^3-a^3 z+a^3 z^{-1} +2 a z^5+5 a z^3-z^3 a^{-1} +2 a z-a z^{-1} -2 z a^{-1}$ (db) Kauffman polynomial $-z^4 a^{10}-4 z^5 a^9+z^3 a^9-9 z^6 a^8+8 z^4 a^8-3 z^2 a^8-13 z^7 a^7+18 z^5 a^7-10 z^3 a^7+2 z a^7-12 z^8 a^6+15 z^6 a^6-2 z^4 a^6-7 z^9 a^5+21 z^5 a^5-11 z^3 a^5-2 z^{10} a^4-13 z^8 a^4+39 z^6 a^4-25 z^4 a^4+6 z^2 a^4-11 z^9 a^3+25 z^7 a^3-11 z^5 a^3+3 z^3 a^3-5 z a^3+a^3 z^{-1} -2 z^{10} a^2-4 z^8 a^2+26 z^6 a^2-27 z^4 a^2+8 z^2 a^2-a^2-4 z^9 a+11 z^7 a-6 z^5 a-2 z^3 a-z a+a z^{-1} -3 z^8+11 z^6-13 z^4+5 z^2-z^7 a^{-1} +4 z^5 a^{-1} -5 z^3 a^{-1} +2 z a^{-1}$ (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 \
-7-6-5-4-3-2-101234χ
6           11
4          2 -2
2         51 4
0        72  -5
-2       105   5
-4      108    -2
-6     119     2
-8    811      3
-10   610       -4
-12  38        5
-14 16         -5
-16 3          3
-181           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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