# L11n347

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n347 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^2 w^3-u v^2 w^2+u v^2 w-u v^2-u v w+u v+u w^2+v^3 (-w)-v^2 w^3+v^2 w^2+v w^3-v w^2+v w-v}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $1-2 q^{-1} +4 q^{-2} -3 q^{-3} +4 q^{-4} -3 q^{-5} +3 q^{-6} - q^{-7} + q^{-9}$ (db) Signature -4 (db) HOMFLY-PT polynomial $a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^2-2 a^6 z^{-2} -5 a^6-a^4 z^6-4 a^4 z^4-3 a^4 z^2+a^4 z^{-2} +a^4+a^2 z^4+3 a^2 z^2+2 a^2$ (db) Kauffman polynomial $a^{10} z^6-6 a^{10} z^4+8 a^{10} z^2-2 a^{10}-a^9 z^5+2 a^9 z^3+a^9 z-a^8 z^6+4 a^8 z^4-3 a^8 z^2-a^8 z^{-2} +3 a^8+a^7 z^7-5 a^7 z^5+10 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+8 a^6 z^4-13 a^6 z^2-2 a^6 z^{-2} +9 a^6+3 a^5 z^7-11 a^5 z^5+12 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +a^4 z^8-a^4 z^6-6 a^4 z^4+3 a^4 z^2-a^4 z^{-2} +3 a^4+2 a^3 z^7-7 a^3 z^5+4 a^3 z^3+a^3 z+a^2 z^6-4 a^2 z^4+5 a^2 z^2-2 a^2$ (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 \
-8-7-6-5-4-3-2-1012χ
1          11
-1         1 -1
-3        31 2
-5       23  1
-7     131   1
-9     12    1
-11   143     0
-13    2      2
-15  12       -1
-171          1
-191          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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