# L11n372

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n372 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(3)-1) \left(-t(1) t(3)^3+t(1) t(2) t(3)^3+t(1) t(2)^2 t(3)^2+t(1) t(3)^2-2 t(1) t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)+t(2)^2 t(3)-2 t(2) t(3)+t(3)-t(2)^2+t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $-q^4+3 q^3-5 q^2+8 q-8+10 q^{-1} -8 q^{-2} +7 q^{-3} -4 q^{-4} +2 q^{-5}$ (db) Signature -2 (db) HOMFLY-PT polynomial $a^6-a^4 z^4-3 a^4 z^2+a^4 z^{-2} -a^4+a^2 z^6+3 a^2 z^4-z^4 a^{-2} +a^2 z^2-2 a^2 z^{-2} -2 z^2 a^{-2} -2 a^2+z^6+3 z^4+2 z^2+ z^{-2} +2$ (db) Kauffman polynomial $3 a^6 z^2-a^6+a^5 z^5+3 a^5 z^3-a^5 z+3 a^4 z^6-a^4 z^4-3 a^4 z^2-a^4 z^{-2} +4 a^4+6 a^3 z^7+z^7 a^{-3} -16 a^3 z^5-4 z^5 a^{-3} +17 a^3 z^3+5 z^3 a^{-3} -10 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +6 a^2 z^8+3 z^8 a^{-2} -20 a^2 z^6-13 z^6 a^{-2} +27 a^2 z^4+17 z^4 a^{-2} -26 a^2 z^2-8 z^2 a^{-2} -2 a^2 z^{-2} +12 a^2+2 a^{-2} +2 a z^9+2 z^9 a^{-1} +2 a z^7-3 z^7 a^{-1} -25 a z^5-12 z^5 a^{-1} +30 a z^3+21 z^3 a^{-1} -14 a z-7 z a^{-1} +2 a z^{-1} +9 z^8-36 z^6+45 z^4-28 z^2- z^{-2} +10$ (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 \
-4-3-2-1012345χ
9         1-1
7        2 2
5       31 -2
3      52  3
1     44   0
-1    64    2
-3   35     2
-5  45      -1
-7 14       3
-913        -2
-112         2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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