# L11a298

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a298 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(2 t(2) t(1)^2+2 t(2)^2 t(1)-t(2) t(1)+2 t(1)+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-6 q^{5/2}+8 q^{3/2}-10 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^5-2 z a^5+z^5 a^3+2 z^3 a^3+2 z^5 a+6 z^3 a+5 z a+a z^{-1} +z^5 a^{-1} +z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} -z^3 a^{-3} -z a^{-3}$ (db) Kauffman polynomial $-a^4 z^{10}-a^2 z^{10}-2 a^5 z^9-5 a^3 z^9-3 a z^9-a^6 z^8+a^4 z^8-4 a^2 z^8-6 z^8+11 a^5 z^7+19 a^3 z^7-8 z^7 a^{-1} +6 a^6 z^6+11 a^4 z^6+23 a^2 z^6-8 z^6 a^{-2} +10 z^6-20 a^5 z^5-17 a^3 z^5+23 a z^5+14 z^5 a^{-1} -6 z^5 a^{-3} -12 a^6 z^4-22 a^4 z^4-22 a^2 z^4+11 z^4 a^{-2} -3 z^4 a^{-4} +2 z^4+14 a^5 z^3-26 a z^3-6 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +8 a^6 z^2+9 a^4 z^2+4 a^2 z^2-4 z^2 a^{-2} -z^2-4 a^5 z+10 a z+4 z a^{-1} -2 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-7-6-5-4-3-2-101234χ
10           11
8          2 -2
6         41 3
4        42  -2
2       64   2
0      76    -1
-2     44     0
-4    47      3
-6   34       -1
-8  14        3
-10 13         -2
-12 1          1
-141           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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