# L11n292

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n292 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1) t(3)^4+t(1) t(2) t(3)^4-t(2)^2 t(3)^3+2 t(1) t(3)^3-2 t(1) t(2) t(3)^3+t(2) t(3)^3+2 t(2)^2 t(3)^2-2 t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2-2 t(2)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)+t(2)^2-t(2)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $q^5-3 q^4- q^{-4} +5 q^3+3 q^{-3} -6 q^2-4 q^{-2} +8 q+7 q^{-1} -6$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^2 a^{-4} + a^{-4} -a^2 z^4-2 z^4 a^{-2} -2 a^2 z^2-5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^{-2} +z^6+4 z^4+5 z^2-2 z^{-2} +1$ (db) Kauffman polynomial $2 a z^9+2 z^9 a^{-1} +3 a^2 z^8+5 z^8 a^{-2} +8 z^8+a^3 z^7-5 a z^7-2 z^7 a^{-1} +4 z^7 a^{-3} -14 a^2 z^6-21 z^6 a^{-2} +z^6 a^{-4} -36 z^6-4 a^3 z^5-6 a z^5-16 z^5 a^{-1} -14 z^5 a^{-3} +19 a^2 z^4+30 z^4 a^{-2} +z^4 a^{-4} +48 z^4+4 a^3 z^3+14 a z^3+24 z^3 a^{-1} +17 z^3 a^{-3} +3 z^3 a^{-5} -9 a^2 z^2-20 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -25 z^2-a^3 z-3 a z-7 z a^{-1} -7 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-5-4-3-2-101234χ
11         11
9        2 -2
7       31 2
5      43  -1
3     431  2
1    46    2
-1   331    1
-3  25      3
-5 12       -1
-7 2        2
-91         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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