# L11n43

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n43 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{2}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^9 z^{-1} -z^3 a^7+z a^7+2 a^7 z^{-1} +z^5 a^5+z^3 a^5-z a^5-a^5 z^{-1} +z^5 a^3+2 z^3 a^3+2 z a^3+a^3 z^{-1} -z^3 a-2 z a-a z^{-1}$ (db) Kauffman polynomial $-3 z^4 a^{10}+6 z^2 a^{10}-3 a^{10}-z^7 a^9-z^5 a^9+3 z^3 a^9-z a^9+a^9 z^{-1} -2 z^8 a^8+5 z^6 a^8-14 z^4 a^8+18 z^2 a^8-7 a^8-z^9 a^7-3 z^5 a^7+5 z^3 a^7-4 z a^7+2 a^7 z^{-1} -4 z^8 a^6+8 z^6 a^6-14 z^4 a^6+13 z^2 a^6-4 a^6-z^9 a^5-z^7 a^5-z^5 a^5+6 z^3 a^5-5 z a^5+a^5 z^{-1} -2 z^8 a^4+z^6 a^4+z^4 a^4-2 z^7 a^3+7 z^3 a^3-5 z a^3+a^3 z^{-1} -2 z^6 a^2+4 z^4 a^2-z^2 a^2-a^2-z^5 a+3 z^3 a-3 z a+a z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
2         11
0        1 -1
-2       41 3
-4      53  -2
-6     52   3
-8    45    1
-10   55     0
-12  24      2
-14 25       -3
-16 2        2
-182         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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