# L11n257

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n257 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v w^2-u v w-3 u w^2+4 u w-2 u+2 v w^3-4 v w^2+3 v w+w^2-w}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $2 q^{-1} -2 q^{-2} +6 q^{-3} -6 q^{-4} +8 q^{-5} -6 q^{-6} +6 q^{-7} -5 q^{-8} +2 q^{-9} - q^{-10}$ (db) Signature -2 (db) HOMFLY-PT polynomial $-a^{10} z^{-2} -a^{10}+3 z^2 a^8+3 a^8 z^{-2} +5 a^8-2 z^4 a^6-5 z^2 a^6-2 a^6 z^{-2} -5 a^6-z^4 a^4-z^2 a^4-a^4 z^{-2} -2 a^4+2 z^2 a^2+a^2 z^{-2} +3 a^2$ (db) Kauffman polynomial $a^{11} z^7-5 a^{11} z^5+9 a^{11} z^3-7 a^{11} z+2 a^{11} z^{-1} +2 a^{10} z^8-8 a^{10} z^6+8 a^{10} z^4-2 a^{10} z^2-a^{10} z^{-2} +a^{10}+a^9 z^9+2 a^9 z^7-24 a^9 z^5+40 a^9 z^3-27 a^9 z+8 a^9 z^{-1} +6 a^8 z^8-23 a^8 z^6+22 a^8 z^4-7 a^8 z^2-3 a^8 z^{-2} +5 a^8+a^7 z^9+5 a^7 z^7-33 a^7 z^5+49 a^7 z^3-34 a^7 z+10 a^7 z^{-1} +4 a^6 z^8-13 a^6 z^6+11 a^6 z^4-4 a^6 z^2-2 a^6 z^{-2} +4 a^6+4 a^5 z^7-13 a^5 z^5+17 a^5 z^3-10 a^5 z+2 a^5 z^{-1} +2 a^4 z^6-3 a^4 z^4+4 a^4 z^2+a^4 z^{-2} -3 a^4+a^3 z^5-a^3 z^3+4 a^3 z-2 a^3 z^{-1} +3 a^2 z^2+a^2 z^{-2} -4 a^2$ (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 \
-9-8-7-6-5-4-3-2-10χ
-1         22
-3        330
-5       3 14
-7      44  0
-9     53   2
-11    251   2
-13   44     0
-15  12      1
-17 14       -3
-19 1        1
-211         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$

### Computer Talk

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