# L11n149

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n149 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^4-t(2)^4-3 t(1) t(2)^3+2 t(2)^3-2 t(1)^2 t(2)^2+3 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2+t(1)}{t(1) t(2)^2}$ (db) Jones polynomial $\frac{6}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{7}{q^{13/2}}-\frac{7}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 \left(-z^3\right)-a^9 z+a^7 z^5+3 a^7 z^3+4 a^7 z+a^7 z^{-1} -3 a^5 z^3-5 a^5 z-a^5 z^{-1} -a^3 z^3-a^3 z$ (db) Kauffman polynomial $-z^6 a^{12}+3 z^4 a^{12}-z^2 a^{12}-3 z^7 a^{11}+11 z^5 a^{11}-9 z^3 a^{11}+z a^{11}-3 z^8 a^{10}+10 z^6 a^{10}-7 z^4 a^{10}+z^2 a^{10}-z^9 a^9-z^7 a^9+12 z^5 a^9-13 z^3 a^9+4 z a^9-4 z^8 a^8+13 z^6 a^8-14 z^4 a^8+6 z^2 a^8-z^9 a^7+2 z^7 a^7-3 z^5 a^7+3 z^3 a^7-3 z a^7+a^7 z^{-1} -z^8 a^6+2 z^6 a^6-6 z^4 a^6+5 z^2 a^6-a^6-4 z^5 a^5+6 z^3 a^5-5 z a^5+a^5 z^{-1} -2 z^4 a^4+z^2 a^4-z^3 a^3+z a^3$ (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
-2         11
-4        21-1
-6       3  3
-8      32  -1
-10     43   1
-12    33    0
-14   34     -1
-16  24      2
-18 12       -1
-20 2        2
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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