# L11n277

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n277 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v w^4-u v w^3+u v w^2-u v w+u w-v^2 w^3+v w^3-v w^2+v w-v}{\sqrt{u} v w^2}$ (db) Jones polynomial $2 q^{-2} - q^{-3} +4 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +2 q^{-8} - q^{-9}$ (db) Signature -4 (db) HOMFLY-PT polynomial $-a^{10} z^{-2} -a^{10}+z^4 a^8+5 z^2 a^8+4 a^8 z^{-2} +8 a^8-z^6 a^6-6 z^4 a^6-14 z^2 a^6-5 a^6 z^{-2} -15 a^6+2 z^4 a^4+8 z^2 a^4+2 a^4 z^{-2} +8 a^4$ (db) Kauffman polynomial $z^3 a^{11}-2 z a^{11}+a^{11} z^{-1} +2 z^4 a^{10}-3 z^2 a^{10}-a^{10} z^{-2} +2 a^{10}+z^7 a^9-5 z^5 a^9+14 z^3 a^9-13 z a^9+5 a^9 z^{-1} +z^8 a^8-6 z^6 a^8+18 z^4 a^8-21 z^2 a^8-4 a^8 z^{-2} +13 a^8+2 z^7 a^7-10 z^5 a^7+26 z^3 a^7-27 z a^7+9 a^7 z^{-1} +z^8 a^6-6 z^6 a^6+19 z^4 a^6-29 z^2 a^6-5 a^6 z^{-2} +20 a^6+z^7 a^5-5 z^5 a^5+13 z^3 a^5-16 z a^5+5 a^5 z^{-1} +3 z^4 a^4-11 z^2 a^4-2 a^4 z^{-2} +10 a^4$ (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 \
-7-6-5-4-3-2-10χ
-3       22
-5      121
-7     3  3
-9    12  1
-11   32   1
-13   1    1
-15 23     -1
-17 1      1
-191       -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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