# L11a319

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a319 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(2 u^2 v+2 u v^2+u v+2 u+2 v\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $\frac{8}{q^{9/2}}-\frac{9}{q^{7/2}}-q^{5/2}+\frac{8}{q^{5/2}}+q^{3/2}-\frac{7}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{7}{q^{11/2}}-3 \sqrt{q}+\frac{5}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 \left(-z^3\right)-2 a^7 z+a^5 z^5+2 a^5 z^3+2 a^3 z^5+6 a^3 z^3+3 a^3 z+a z^5+3 a z^3-z^3 a^{-1} +2 a z+a z^{-1} -3 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^4 a^{10}+2 z^2 a^{10}-2 z^5 a^9+3 z^3 a^9-3 z^6 a^8+5 z^4 a^8-3 z^2 a^8-4 z^7 a^7+11 z^5 a^7-15 z^3 a^7+4 z a^7-3 z^8 a^6+7 z^6 a^6-8 z^4 a^6-2 z^9 a^5+5 z^7 a^5-7 z^5 a^5+3 z^3 a^5-z^{10} a^4+3 z^8 a^4-8 z^6 a^4+11 z^4 a^4-3 z^2 a^4-3 z^9 a^3+13 z^7 a^3-26 z^5 a^3+27 z^3 a^3-6 z a^3-z^{10} a^2+5 z^8 a^2-14 z^6 a^2+22 z^4 a^2-10 z^2 a^2-z^9 a+3 z^7 a-5 z^3 a+4 z a-a z^{-1} -z^8+4 z^6-3 z^4-2 z^2+1-z^7 a^{-1} +6 z^5 a^{-1} -11 z^3 a^{-1} +6 z a^{-1} - a^{-1} z^{-1}$ (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            0
2         31 2
0        2   -2
-2       53   2
-4      54    -1
-6     43     1
-8    45      1
-10   34       -1
-12  14        3
-14 13         -2
-16 1          1
-181           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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