# L11n436

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n436 at Knotilus! Brunnian link

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1) (w+1)^2}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $-q^4+2 q^3-q^2+2 q+1+ q^{-2} - q^{-3} +2 q^{-4} - q^{-5}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^4 z^2-z^4 a^{-2} -a^2 z^2-3 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +z^6+5 z^4+5 z^2-2 z^{-2}$ (db) Kauffman polynomial $a^5 z^5-3 a^5 z^3+2 a^4 z^6-8 a^4 z^4+4 a^4 z^2+a^3 z^7+z^7 a^{-3} -5 a^3 z^5-5 z^5 a^{-3} +4 a^3 z^3+5 z^3 a^{-3} +a^2 z^8+2 z^8 a^{-2} -6 a^2 z^6-12 z^6 a^{-2} +8 a^2 z^4+20 z^4 a^{-2} -4 a^2 z^2-12 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +a z^9+z^9 a^{-1} -5 a z^7-5 z^7 a^{-1} +2 a z^5+3 z^5 a^{-1} +6 a z^3+4 z^3 a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-20 z^6+36 z^4-20 z^2+2 z^{-2} +1$ (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 \
-5-4-3-2-1012345χ
9          1-1
7         1 1
5       111 1
3       21  1
1     421   3
-1    252    1
-3   122     1
-5  121      0
-7 111       1
-9 1         1
-111          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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