# L11n301

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n301 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2) t(3)^4-2 t(1) t(2)^2 t(3)^3+t(2)^2 t(3)^3+2 t(1) t(2) t(3)^3-3 t(2) t(3)^3+t(3)^3+2 t(1) t(2)^2 t(3)^2-t(2)^2 t(3)^2+t(1) t(3)^2-4 t(1) t(2) t(3)^2+4 t(2) t(3)^2-2 t(3)^2-t(1) t(2)^2 t(3)-t(1) t(3)+3 t(1) t(2) t(3)-2 t(2) t(3)+2 t(3)-t(1) t(2)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $-q^{11}+3 q^{10}-6 q^9+9 q^8-11 q^7+12 q^6-10 q^5+9 q^4-4 q^3+3 q^2$ (db) Signature 4 (db) HOMFLY-PT polynomial $-z^2 a^{-10} - a^{-10} z^{-2} -2 a^{-10} +3 z^4 a^{-8} +10 z^2 a^{-8} +4 a^{-8} z^{-2} +11 a^{-8} -2 z^6 a^{-6} -10 z^4 a^{-6} -20 z^2 a^{-6} -5 a^{-6} z^{-2} -18 a^{-6} +3 z^4 a^{-4} +10 z^2 a^{-4} +2 a^{-4} z^{-2} +9 a^{-4}$ (db) Kauffman polynomial $z^5 a^{-13} -2 z^3 a^{-13} +z a^{-13} +3 z^6 a^{-12} -6 z^4 a^{-12} +3 z^2 a^{-12} - a^{-12} +4 z^7 a^{-11} -6 z^5 a^{-11} +z^3 a^{-11} -2 z a^{-11} + a^{-11} z^{-1} +3 z^8 a^{-10} -9 z^4 a^{-10} +6 z^2 a^{-10} - a^{-10} z^{-2} +z^9 a^{-9} +8 z^7 a^{-9} -26 z^5 a^{-9} +33 z^3 a^{-9} -19 z a^{-9} +5 a^{-9} z^{-1} +7 z^8 a^{-8} -17 z^6 a^{-8} +23 z^4 a^{-8} -19 z^2 a^{-8} -4 a^{-8} z^{-2} +13 a^{-8} +z^9 a^{-7} +7 z^7 a^{-7} -28 z^5 a^{-7} +48 z^3 a^{-7} -35 z a^{-7} +9 a^{-7} z^{-1} +4 z^8 a^{-6} -14 z^6 a^{-6} +32 z^4 a^{-6} -38 z^2 a^{-6} -5 a^{-6} z^{-2} +22 a^{-6} +3 z^7 a^{-5} -9 z^5 a^{-5} +18 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +6 z^4 a^{-4} -16 z^2 a^{-4} -2 a^{-4} z^{-2} +11 a^{-4}$ (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 \
0123456789χ
23         1-1
21        2 2
19       41 -3
17      52  3
15     75   -2
13    54    1
11   57     2
9  45      -1
7 16       5
523        -1
33         3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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