# L10a163

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a163 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) (w-1) \left(u v w^2-u v w+u w+v w-w+1\right)}{u v w^{3/2}}$ (db) Jones polynomial $q^{-7} -4 q^{-6} +8 q^{-5} -12 q^{-4} +q^3+16 q^{-3} -4 q^2-15 q^{-2} +8 q+16 q^{-1} -11$ (db) Signature -2 (db) HOMFLY-PT polynomial $-a^2 z^8+a^4 z^6-5 a^2 z^6+z^6+3 a^4 z^4-8 a^2 z^4+3 z^4+2 a^4 z^2-4 a^2 z^2+2 z^2-a^2+1+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2}$ (db) Kauffman polynomial $a^8 z^4+4 a^7 z^5-2 a^7 z^3+8 a^6 z^6-8 a^6 z^4+3 a^6 z^2+10 a^5 z^7-12 a^5 z^5+4 a^5 z^3+8 a^4 z^8-7 a^4 z^6-3 a^4 z^4+2 a^4 z^2-a^4 z^{-2} +a^4+3 a^3 z^9+9 a^3 z^7-26 a^3 z^5+12 a^3 z^3-a^3 z+2 a^3 z^{-1} +14 a^2 z^8-31 a^2 z^6+z^6 a^{-2} +18 a^2 z^4-2 z^4 a^{-2} -4 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +a^2+3 a z^9+3 a z^7+4 z^7 a^{-1} -20 a z^5-10 z^5 a^{-1} +12 a z^3+6 z^3 a^{-1} -a z+2 a z^{-1} +6 z^8-15 z^6+10 z^4-2 z^2- z^{-2} +1$ (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 \
-6-5-4-3-2-101234χ
7          11
5         3 -3
3        51 4
1       63  -3
-1      105   5
-3     78    1
-5    98     1
-7   59      4
-9  37       -4
-11 15        4
-13 3         -3
-151          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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