# L10a157

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a157 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(w-1) \left(u^2 v^2 w+u^2 v w^2-u^2 v w-u^2 w^2-u v^2 w+u v^2-u v w^2+u v w-u v+u w^2-u w-v^2-v w+v+w\right)}{u v w^{3/2}}$ (db) Jones polynomial $- q^{-6} +3 q^{-5} -q^4-5 q^{-4} +3 q^3+8 q^{-3} -4 q^2-9 q^{-2} +8 q+10 q^{-1} -8$ (db) Signature -2 (db) HOMFLY-PT polynomial $a^2 z^6+z^6-a^4 z^4+3 a^2 z^4-z^4 a^{-2} +3 z^4-2 a^4 z^2+2 a^2 z^2-2 z^2 a^{-2} +z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db) Kauffman polynomial $a^7 z^3+3 a^6 z^4-a^6 z^2+5 a^5 z^5-3 a^5 z^3+7 a^4 z^6-10 a^4 z^4+4 a^4 z^2+7 a^3 z^7+z^7 a^{-3} -13 a^3 z^5-4 z^5 a^{-3} +5 a^3 z^3+4 z^3 a^{-3} +5 a^2 z^8+3 z^8 a^{-2} -9 a^2 z^6-14 z^6 a^{-2} -a^2 z^4+20 z^4 a^{-2} +3 a^2 z^2-8 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +2 a z^9+2 z^9 a^{-1} +a z^7-5 z^7 a^{-1} -16 a z^5-2 z^5 a^{-1} +11 a z^3+6 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +8 z^8-30 z^6+32 z^4-10 z^2+2 z^{-2} -3$ (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
9          1-1
7         2 2
5        21 -1
3       62  4
1      33   0
-1     75    2
-3    45     1
-5   45      -1
-7  25       3
-9 13        -2
-11 2         2
-131          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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