# L10a127

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a127 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1) (2 v w-v-w+2)}{\sqrt{u} v w}$ (db) Jones polynomial $-q^8+4 q^7-8 q^6+12 q^5-15 q^4+17 q^3-14 q^2+13 q-7+4 q^{-1} - q^{-2}$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -z^4+z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} -z^2- a^{-2} +1-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2}$ (db) Kauffman polynomial $2 z^9 a^{-3} +2 z^9 a^{-5} +5 z^8 a^{-2} +11 z^8 a^{-4} +6 z^8 a^{-6} +6 z^7 a^{-1} +10 z^7 a^{-3} +11 z^7 a^{-5} +7 z^7 a^{-7} -13 z^6 a^{-4} -5 z^6 a^{-6} +4 z^6 a^{-8} +4 z^6+a z^5-8 z^5 a^{-1} -20 z^5 a^{-3} -24 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -11 z^4 a^{-2} -z^4 a^{-4} -3 z^4 a^{-6} -6 z^4 a^{-8} -7 z^4-a z^3+2 z^3 a^{-1} +7 z^3 a^{-3} +11 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} +6 z^2 a^{-2} +2 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2 a^{-8} +4 z^2-z a^{-1} -z a^{-3} + a^{-2} + a^{-4} +1+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
-3-2-101234567χ
17          1-1
15         3 3
13        51 -4
11       73  4
9      85   -3
7     97    2
5    710     3
3   67      -1
1  39       6
-1 14        -3
-3 3         3
-51          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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