# L11n39

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n39 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1)}{\sqrt{u} \sqrt{v}}$ (db) Jones polynomial $-\frac{3}{q^{9/2}}+\frac{2}{q^{7/2}}-q^{5/2}-\frac{2}{q^{5/2}}+2 q^{3/2}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}-2 \sqrt{q}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)-a^7 z^{-1} +2 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -a^3 z^5-5 a^3 z^3-8 a^3 z-3 a^3 z^{-1} +a z^5+5 a z^3-z^3 a^{-1} +6 a z+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^8 z^6-4 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^7-8 a^7 z^5+7 a^7 z^3-3 a^7 z+a^7 z^{-1} +a^6 z^8-2 a^6 z^6-5 a^6 z^4+5 a^6 z^2-2 a^6+3 a^5 z^7-15 a^5 z^5+19 a^5 z^3-12 a^5 z+3 a^5 z^{-1} +2 a^4 z^8-12 a^4 z^6+17 a^4 z^4-7 a^4 z^2+a^3 z^9-4 a^3 z^7-7 a^3 z^5+27 a^3 z^3-18 a^3 z+3 a^3 z^{-1} +3 a^2 z^8-20 a^2 z^6+32 a^2 z^4-13 a^2 z^2+2 a^2+a z^9-4 a z^7+z^7 a^{-1} -5 a z^5-5 z^5 a^{-1} +21 a z^3+6 z^3 a^{-1} -13 a z-4 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +2 z^8-11 z^6+14 z^4-5 z^2$ (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          1 -1
2         11 0
0       321  2
-2      231   0
-4     222    2
-6    231     0
-8   221      1
-10  121       0
-12 12         -1
-14 1          1
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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