# L11n174

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n174 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{\left(t(2)^2-3 t(2)+1\right) \left(t(1)^2 t(2)^2+t(1) t(2)+1\right)}{t(1) t(2)^2}$ (db) Jones polynomial $-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^9 z+a^9 z^{-1} -a^7 z^5-4 a^7 z^3-4 a^7 z-2 a^7 z^{-1} +a^5 z^7+5 a^5 z^5+7 a^5 z^3+4 a^5 z+2 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-4 a^3 z-a^3 z^{-1}$ (db) Kauffman polynomial $2 a^{11} z+a^{10} z^4+a^9 z^7-3 a^9 z^5+4 a^9 z^3-3 a^9 z+a^9 z^{-1} +2 a^8 z^8-8 a^8 z^6+10 a^8 z^4-6 a^8 z^2+a^7 z^9-a^7 z^7-9 a^7 z^5+15 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +4 a^6 z^8-17 a^6 z^6+19 a^6 z^4-8 a^6 z^2+a^6+a^5 z^9-a^5 z^7-11 a^5 z^5+19 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+10 a^4 z^4-2 a^4 z^2+a^3 z^7-5 a^3 z^5+8 a^3 z^3-5 a^3 z+a^3 z^{-1}$ (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 \
-6-5-4-3-2-1012χ
0        11
-2       1 -1
-4      31 2
-6     22  0
-8    32   1
-10   22    0
-12  23     -1
-14 12      1
-1612       -1
-182        2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-6$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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