# L11a87

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a87 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^5-5 u v^4+10 u v^3-12 u v^2+7 u v-2 u-2 v^5+7 v^4-12 v^3+10 v^2-5 v+1}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-\frac{14}{q^{9/2}}-q^{7/2}+\frac{19}{q^{7/2}}+5 q^{5/2}-\frac{24}{q^{5/2}}-11 q^{3/2}+\frac{24}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{7}{q^{11/2}}+17 \sqrt{q}-\frac{22}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -3 a^3 z^5-8 a^3 z^3-9 a^3 z-4 a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +4 a z^3-z^3 a^{-1} +2 a z+a z^{-1}$ (db) Kauffman polynomial $-2 a^4 z^{10}-2 a^2 z^{10}-4 a^5 z^9-12 a^3 z^9-8 a z^9-4 a^6 z^8-9 a^4 z^8-18 a^2 z^8-13 z^8-3 a^7 z^7-a^5 z^7+12 a^3 z^7-a z^7-11 z^7 a^{-1} -a^8 z^6+4 a^6 z^6+19 a^4 z^6+37 a^2 z^6-5 z^6 a^{-2} +18 z^6+8 a^7 z^5+18 a^5 z^5+14 a^3 z^5+20 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4+7 a^6 z^4-a^4 z^4-13 a^2 z^4+4 z^4 a^{-2} -4 z^4-8 a^7 z^3-25 a^5 z^3-25 a^3 z^3-12 a z^3-4 z^3 a^{-1} -3 a^8 z^2-11 a^6 z^2-15 a^4 z^2-8 a^2 z^2-z^2+4 a^7 z+16 a^5 z+15 a^3 z+3 a z+a^8+4 a^6+7 a^4+4 a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -4 a^3 z^{-1} -a z^{-1}$ (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
8           11
6          4 -4
4         71 6
2        104  -6
0       127   5
-2      1311    -2
-4     1111     0
-6    813      5
-8   611       -5
-10  29        7
-12 15         -4
-14 2          2
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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