# L11a518

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a518 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-2 t(1)^2 t(3)^3+t(1) t(3)^3+t(1)^2 t(2) t(3)^3-t(1) t(2) t(3)^3+t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-t(1) t(2)^2 t(3)^2-2 t(1) t(3)^2-2 t(1)^2 t(2) t(3)^2+2 t(1) t(2) t(3)^2-t(2) t(3)^2+t(3)^2-t(1)^2 t(2)^2 t(3)+2 t(1) t(2)^2 t(3)-t(2)^2 t(3)+t(1) t(3)+t(1)^2 t(2) t(3)-2 t(1) t(2) t(3)+2 t(2) t(3)-t(3)-t(1) t(2)^2+2 t(2)^2+t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $-q^6+3 q^5+ q^{-5} -5 q^4- q^{-4} +8 q^3+4 q^{-3} -9 q^2-5 q^{-2} +10 q+8 q^{-1} -9$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +z^6-2 a^2 z^4+3 z^4 a^{-2} -z^4 a^{-4} +3 z^4+a^4 z^2-7 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2+3 a^4-6 a^2+ a^{-2} +2+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2}$ (db) Kauffman polynomial $z^3 a^{-7} +3 z^4 a^{-6} -z^2 a^{-6} +5 z^5 a^{-5} -3 z^3 a^{-5} +a^4 z^8-7 a^4 z^6+7 z^6 a^{-4} +17 a^4 z^4-10 z^4 a^{-4} -17 a^4 z^2+4 z^2 a^{-4} -a^4 z^{-2} +7 a^4+a^3 z^9-4 a^3 z^7+7 z^7 a^{-3} +a^3 z^5-13 z^5 a^{-3} +10 a^3 z^3+5 z^3 a^{-3} -9 a^3 z+2 a^3 z^{-1} +a^2 z^{10}-3 a^2 z^8+5 z^8 a^{-2} -3 a^2 z^6-8 z^6 a^{-2} +17 a^2 z^4-5 z^4 a^{-2} -20 a^2 z^2+7 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2-2 a^{-2} +4 a z^9+3 z^9 a^{-1} -16 a z^7-5 z^7 a^{-1} +15 a z^5-4 z^5 a^{-1} +4 a z^3+3 z^3 a^{-1} -9 a z+2 a z^{-1} +z^{10}+z^8-11 z^6+8 z^4-z^2- z^{-2} +3$ (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 \
-6-5-4-3-2-1012345χ
13           1-1
11          2 2
9         31 -2
7        52  3
5       54   -1
3      54    1
1     67     1
-1    23      -1
-3   36       3
-5  12        -1
-7  3         3
-911          0
-111           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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