# L10a156

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a156 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(3)-1) (-t(2) t(1)+t(2) t(3) t(1)+t(1)+t(2)-1) (-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+t(3)-1)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $-q^5- q^{-5} +4 q^4+4 q^{-4} -8 q^3-8 q^{-3} +13 q^2+13 q^{-2} -15 q-15 q^{-1} +18$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^8-a^2 z^6-z^6 a^{-2} +5 z^6-3 a^2 z^4-3 z^4 a^{-2} +8 z^4-2 a^2 z^2-2 z^2 a^{-2} +3 z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db) Kauffman polynomial $a^5 z^5+z^5 a^{-5} -a^5 z^3-z^3 a^{-5} +4 a^4 z^6+4 z^6 a^{-4} -6 a^4 z^4-6 z^4 a^{-4} +2 a^4 z^2+2 z^2 a^{-4} +7 a^3 z^7+7 z^7 a^{-3} -11 a^3 z^5-11 z^5 a^{-3} +4 a^3 z^3+4 z^3 a^{-3} +7 a^2 z^8+7 z^8 a^{-2} -9 a^2 z^6-9 z^6 a^{-2} +2 a^2 z^4+2 z^4 a^{-2} +a^2 z^2+z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +3 a z^9+3 z^9 a^{-1} +7 a z^7+7 z^7 a^{-1} -19 a z^5-19 z^5 a^{-1} +9 a z^3+9 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +14 z^8-26 z^6+16 z^4-2 z^2+2 z^{-2} -3$ (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
11          1-1
9         3 3
7        51 -4
5       83  5
3      86   -2
1     107    3
-1    710     3
-3   68      -2
-5  38       5
-7 15        -4
-9 3         3
-111          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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