# L10a141

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a141 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1) (v+w) (v w+1)}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $-q^6+3 q^5-6 q^4+9 q^3-9 q^2+11 q-9+8 q^{-1} -4 q^{-2} +3 q^{-3} - q^{-4}$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} +z^6 a^{-2} -a^2 z^4+3 z^4 a^{-2} -2 a^2 z^2+3 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +a^2+3 a^{-2} +z^6+3 z^4+z^2-2 z^{-2} -3$ (db) Kauffman polynomial $2 a z^9+2 z^9 a^{-1} +3 a^2 z^8+5 z^8 a^{-2} +8 z^8+a^3 z^7-5 a z^7+2 z^7 a^{-1} +8 z^7 a^{-3} -14 a^2 z^6-5 z^6 a^{-2} +9 z^6 a^{-4} -28 z^6-4 a^3 z^5-a z^5-16 z^5 a^{-1} -13 z^5 a^{-3} +6 z^5 a^{-5} +20 a^2 z^4-12 z^4 a^{-2} -15 z^4 a^{-4} +3 z^4 a^{-6} +26 z^4+4 a^3 z^3+5 a z^3+7 z^3 a^{-1} +z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} -7 a^2 z^2+13 z^2 a^{-2} +9 z^2 a^{-4} -3 z^2+a z+3 z a^{-1} +3 z a^{-3} +z a^{-5} -2 a^2-6 a^{-2} -2 a^{-4} -5-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-5-4-3-2-1012345χ
13          1-1
11         2 2
9        41 -3
7       52  3
5      44   0
3     75    2
1    68     2
-1   23      -1
-3  26       4
-5 12        -1
-7 2         2
-91          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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