# L11n437

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n437 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u w+v) \left(u v^2 w-u v^2+u-v^2 w+w-1\right)}{u v^{3/2} w}$ (db) Jones polynomial $1- q^{-1} + q^{-2} + q^{-4} + q^{-5} + q^{-7} - q^{-8} + q^{-9}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^8 z^2+a^8 z^{-2} +a^8-2 a^6 z^{-2} -3 a^6-a^4 z^6-5 a^4 z^4-5 a^4 z^2+a^4 z^{-2} +a^2 z^4+4 a^2 z^2+2 a^2$ (db) Kauffman polynomial $a^{10} z^6-5 a^{10} z^4+5 a^{10} z^2+a^9 z^7-5 a^9 z^5+5 a^9 z^3+a^8 z^8-7 a^8 z^6+16 a^8 z^4-17 a^8 z^2-a^8 z^{-2} +7 a^8+a^7 z^7-7 a^7 z^5+13 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +a^6 z^8-8 a^6 z^6+21 a^6 z^4-25 a^6 z^2-2 a^6 z^{-2} +11 a^6+a^5 z^7-7 a^5 z^5+13 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +a^4 z^6-5 a^4 z^4+3 a^4 z^2-a^4 z^{-2} +3 a^4+a^3 z^7-5 a^3 z^5+5 a^3 z^3+a^2 z^6-5 a^2 z^4+6 a^2 z^2-2 a^2$ (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           0
-3       121 0
-5      111  1
-7     241   1
-9    112    2
-11   131     1
-13  111      1
-15  11       0
-1711         0
-191          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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