# L11n412

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n412 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(3)^3-2 t(1)^2 t(3)^2+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+t(1) t(3)^2+t(1)^2 t(2) t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2+2 t(1)^2 t(3)-t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-t(1) t(3)-t(1)^2 t(2) t(3)+2 t(1) t(2) t(3)-t(2) t(3)-t(2)^2}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $- q^{-10} +2 q^{-9} -4 q^{-8} +6 q^{-7} -6 q^{-6} +8 q^{-5} -6 q^{-4} +6 q^{-3} -3 q^{-2} +2 q^{-1}$ (db) Signature -2 (db) HOMFLY-PT polynomial $-a^{10} z^{-2} -a^{10}+3 z^2 a^8+4 a^8 z^{-2} +6 a^8-2 z^4 a^6-6 z^2 a^6-5 a^6 z^{-2} -9 a^6-z^4 a^4+2 a^4 z^{-2} +2 a^4+2 z^2 a^2+2 a^2$ (db) Kauffman polynomial $a^{11} z^7-5 a^{11} z^5+8 a^{11} z^3-5 a^{11} z+a^{11} z^{-1} +2 a^{10} z^8-9 a^{10} z^6+12 a^{10} z^4-7 a^{10} z^2-a^{10} z^{-2} +4 a^{10}+a^9 z^9+a^9 z^7-20 a^9 z^5+35 a^9 z^3-21 a^9 z+5 a^9 z^{-1} +6 a^8 z^8-26 a^8 z^6+38 a^8 z^4-32 a^8 z^2-4 a^8 z^{-2} +17 a^8+a^7 z^9+4 a^7 z^7-28 a^7 z^5+44 a^7 z^3-33 a^7 z+9 a^7 z^{-1} +4 a^6 z^8-15 a^6 z^6+26 a^6 z^4-32 a^6 z^2-5 a^6 z^{-2} +20 a^6+4 a^5 z^7-12 a^5 z^5+18 a^5 z^3-16 a^5 z+5 a^5 z^{-1} +2 a^4 z^6-4 a^4 z^2-2 a^4 z^{-2} +6 a^4+a^3 z^5+a^3 z^3+a^3 z+3 a^2 z^2-2 a^2$ (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 \
-9-8-7-6-5-4-3-2-10χ
-1         22
-3        32-1
-5       3  3
-7      44  0
-9     42   2
-11    24    2
-13   44     0
-15  13      2
-17 13       -2
-19 1        1
-211         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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