# L11n256

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n256 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 t(1) t(3)^3-2 t(1) t(3)^2-t(1) t(2) t(3)^2-t(2) t(3)^2-t(3)^2+t(1) t(3)+t(1) t(2) t(3)+2 t(2) t(3)+t(3)-2 t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $-q^7+q^6-q^5-q^4+2 q^3+ q^{-3} -q^2- q^{-2} +4 q+3 q^{-1} -2$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-2} +z^4 a^{-4} -z^4+a^2 z^2-3 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} -3 z^2+2 a^2-3 a^{-2} +6 a^{-4} -2 a^{-6} -3+a^2 z^{-2} -2 a^{-2} z^{-2} +3 a^{-4} z^{-2} - a^{-6} z^{-2} - z^{-2}$ (db) Kauffman polynomial $z^7 a^{-7} -6 z^5 a^{-7} +10 z^3 a^{-7} -7 z a^{-7} +2 a^{-7} z^{-1} +z^8 a^{-6} -6 z^6 a^{-6} +8 z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} z^{-2} + a^{-6} +2 z^7 a^{-5} -16 z^5 a^{-5} +35 z^3 a^{-5} -27 z a^{-5} +8 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +12 z^4 a^{-4} -7 z^2 a^{-4} -3 a^{-4} z^{-2} +5 a^{-4} +3 z^7 a^{-3} -21 z^5 a^{-3} +44 z^3 a^{-3} -34 z a^{-3} +10 a^{-3} z^{-1} +z^8 a^{-2} +a^2 z^6-5 z^6 a^{-2} -5 a^2 z^4+8 z^4 a^{-2} +7 a^2 z^2-8 z^2 a^{-2} +a^2 z^{-2} -2 a^{-2} z^{-2} -4 a^2+4 a^{-2} +a z^7+3 z^7 a^{-1} -3 a z^5-14 z^5 a^{-1} -a z^3+18 z^3 a^{-1} +4 a z-10 z a^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} +z^8-3 z^6-z^4+4 z^2+ z^{-2} -3$ (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 \
-4-3-2-101234567χ
15           1-1
13            0
11         11 0
9       31   -2
7      111   1
5     241    1
3    311     3
1   251      2
-1  111       1
-3  2         2
-511          0
-71           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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