# L11n315

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n315 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^2-t(1) t(3) t(2)^2-t(3) t(2)^2+t(2)^2+t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)-t(1) t(3) t(2)+t(3) t(2)-t(2)-t(1) t(3)^2-t(3)^2+t(1) t(3)+t(3)}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $- q^{-7} +2 q^{-6} -2 q^{-5} +2 q^{-4} +q^3- q^{-3} -q^2+ q^{-2} +2 q+ q^{-1}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6 \left(-z^2\right)-a^6+a^4 z^4+3 a^4 z^2+2 a^4+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +a^2+2 a^{-2} -z^4-4 z^2-2 z^{-2} -4$ (db) Kauffman polynomial $a^5 z^9+a^3 z^9+2 a^6 z^8+3 a^4 z^8+a^2 z^8+a^7 z^7-4 a^5 z^7-6 a^3 z^7+z^7 a^{-1} -11 a^6 z^6-19 a^4 z^6-7 a^2 z^6+z^6 a^{-2} +2 z^6-5 a^7 z^5-a^5 z^5+8 a^3 z^5-4 z^5 a^{-1} +16 a^6 z^4+33 a^4 z^4+11 a^2 z^4-5 z^4 a^{-2} -11 z^4+6 a^7 z^3+9 a^5 z^3-3 a^3 z^3-5 a z^3+z^3 a^{-1} -8 a^6 z^2-20 a^4 z^2-5 a^2 z^2+7 z^2 a^{-2} +14 z^2-2 a^7 z-4 a^5 z+6 a z+4 z a^{-1} +2 a^6+4 a^4-2 a^2-4 a^{-2} -7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-7-6-5-4-3-2-101234χ
7           11
5            0
3         21 1
1       211  2
-1      131   1
-3     222    2
-5    231     0
-7   111      1
-9  121       0
-11 11         0
-13 1          1
-151           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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