# L11a380

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a380 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^4 v^2-u^4 v-u^3 v^4+4 u^3 v^3-5 u^3 v^2+3 u^3 v-u^3+2 u^2 v^4-6 u^2 v^3+7 u^2 v^2-6 u^2 v+2 u^2-u v^4+3 u v^3-5 u v^2+4 u v-u-v^3+v^2}{u^2 v^2}$ (db) Jones polynomial $-q^{5/2}+3 q^{3/2}-7 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{15}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^7-2 z a^7+2 z^5 a^5+5 z^3 a^5+2 z a^5-z^7 a^3-3 z^5 a^3-2 z^3 a^3+a^3 z^{-1} +2 z^5 a+5 z^3 a+z a-a z^{-1} -z^3 a^{-1} -2 z a^{-1}$ (db) Kauffman polynomial $a^{10} z^4-a^{10} z^2+3 a^9 z^5-2 a^9 z^3+6 a^8 z^6-6 a^8 z^4+3 a^8 z^2+9 a^7 z^7-15 a^7 z^5+14 a^7 z^3-5 a^7 z+9 a^6 z^8-15 a^6 z^6+9 a^6 z^4-a^6 z^2+6 a^5 z^9-5 a^5 z^7-11 a^5 z^5+14 a^5 z^3-5 a^5 z+2 a^4 z^{10}+8 a^4 z^8-31 a^4 z^6+25 a^4 z^4-7 a^4 z^2+10 a^3 z^9-27 a^3 z^7+19 a^3 z^5-8 a^3 z^3+4 a^3 z-a^3 z^{-1} +2 a^2 z^{10}+2 a^2 z^8-21 a^2 z^6+21 a^2 z^4-6 a^2 z^2+a^2+4 a z^9-12 a z^7+z^7 a^{-1} +8 a z^5-4 z^5 a^{-1} -a z^3+5 z^3 a^{-1} +2 a z-a z^{-1} -2 z a^{-1} +3 z^8-11 z^6+12 z^4-4 z^2$ (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 \
-7-6-5-4-3-2-101234χ
6           11
4          2 -2
2         51 4
0        62  -4
-2       95   4
-4      97    -2
-6     98     1
-8    710      3
-10   58       -3
-12  27        5
-14 15         -4
-16 2          2
-181           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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