# L11a279

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a279 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(1)^2 t(2)^4-t(1)^2 t(2)^3+2 t(1) t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2+2 t(1) t(2)-t(2)+1\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $\frac{12}{q^{9/2}}-\frac{14}{q^{7/2}}-q^{5/2}+\frac{13}{q^{5/2}}+3 q^{3/2}-\frac{12}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{9}{q^{11/2}}-6 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^5 z^7-5 a^5 z^5-8 a^5 z^3-5 a^5 z-a^5 z^{-1} +a^3 z^9+7 a^3 z^7+18 a^3 z^5+21 a^3 z^3+11 a^3 z+3 a^3 z^{-1} -a z^7-5 a z^5-8 a z^3-6 a z-2 a z^{-1}$ (db) Kauffman polynomial $-z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-5 z^6 a^8+6 z^4 a^8-2 z^2 a^8-6 z^7 a^7+9 z^5 a^7-6 z^3 a^7+z a^7-6 z^8 a^6+12 z^6 a^6-12 z^4 a^6+6 z^2 a^6-a^6-5 z^9 a^5+13 z^7 a^5-17 z^5 a^5+14 z^3 a^5-6 z a^5+a^5 z^{-1} -2 z^{10} a^4-z^8 a^4+18 z^6 a^4-27 z^4 a^4+16 z^2 a^4-3 a^4-9 z^9 a^3+36 z^7 a^3-52 z^5 a^3+39 z^3 a^3-15 z a^3+3 a^3 z^{-1} -2 z^{10} a^2+2 z^8 a^2+13 z^6 a^2-20 z^4 a^2+10 z^2 a^2-3 a^2-4 z^9 a+16 z^7 a-19 z^5 a+12 z^3 a-7 z a+2 a z^{-1} -3 z^8+12 z^6-12 z^4+3 z^2-z^7 a^{-1} +4 z^5 a^{-1} -4 z^3 a^{-1} +z a^{-1}$ (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 \
-7-6-5-4-3-2-101234χ
6           11
4          2 -2
2         41 3
0        42  -2
-2       84   4
-4      76    -1
-6     76     1
-8    57      2
-10   47       -3
-12  25        3
-14 14         -3
-16 2          2
-181           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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