# L11n408

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n408 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1)}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db) Jones polynomial $q^3-q^2+q+2- q^{-1} +3 q^{-2} -3 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6 \left(-z^2\right)-a^6+a^4 z^4+2 a^4 z^2+a^4+a^2 z^4+3 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^2+2 a^{-2} -z^4-5 z^2-2 z^{-2} -5$ (db) Kauffman polynomial $a^5 z^9+a^3 z^9+2 a^6 z^8+3 a^4 z^8+a^2 z^8+a^7 z^7-3 a^5 z^7-5 a^3 z^7+z^7 a^{-1} -10 a^6 z^6-15 a^4 z^6-5 a^2 z^6+z^6 a^{-2} +z^6-5 a^7 z^5-2 a^5 z^5+7 a^3 z^5-a z^5-5 z^5 a^{-1} +14 a^6 z^4+21 a^4 z^4+2 a^2 z^4-5 z^4 a^{-2} -10 z^4+7 a^7 z^3+6 a^5 z^3-7 a^3 z^3-3 a z^3+3 z^3 a^{-1} -7 a^6 z^2-13 a^4 z^2+7 a^2 z^2+6 z^2 a^{-2} +19 z^2-2 a^7 z-3 a^5 z+3 a^3 z+7 a z+3 z a^{-1} +2 a^6+2 a^4-6 a^2-4 a^{-2} -9-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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
7           11
5            0
3         11 0
1       41   3
-1      241   1
-3     222    2
-5    22      0
-7   221      1
-9  12        1
-11 12         -1
-13 1          1
-151           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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