# L10a48

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a48 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{3 t(2) t(1)-4 t(1)-4 t(2)+3}{\sqrt{t(1)} \sqrt{t(2)}}$ (db) Jones polynomial $-q^{11/2}+2 q^{9/2}-2 q^{7/2}+3 q^{5/2}-4 q^{3/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^{-1} -z a^{-5} +z^3 a^{-3} -2 a^3 z-a^3 z^{-1} +z a^{-3} +a z^3+z^3 a^{-1}$ (db) Kauffman polynomial $-z^9 a^{-1} -z^9 a^{-3} -3 z^8 a^{-2} -2 z^8 a^{-4} -z^8-a z^7+4 z^7 a^{-1} +4 z^7 a^{-3} -z^7 a^{-5} -a^2 z^6+14 z^6 a^{-2} +11 z^6 a^{-4} +2 z^6-a^3 z^5+a z^5-6 z^5 a^{-1} -3 z^5 a^{-3} +5 z^5 a^{-5} -a^4 z^4-18 z^4 a^{-2} -17 z^4 a^{-4} -a^5 z^3-a^3 z^3+6 z^3 a^{-1} -6 z^3 a^{-5} +7 z^2 a^{-2} +7 z^2 a^{-4} +2 a^5 z+2 a^3 z-z a^{-1} +z a^{-5} +a^4-a^5 z^{-1} -a^3 z^{-1}$ (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 \
-4-3-2-10123456χ
12          11
10         1 -1
8        11 0
6       21  -1
4      21   1
2     22    0
0    22     0
-2   23      1
-4  11       0
-6  2        2
-811         0
-101          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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