# L11n4

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n4 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^5-4 t(1) t(2)^4+6 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+6 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z^{-1} -a^7 z^3-4 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+8 a^5 z^3+9 a^5 z+4 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-9 a^3 z^3-8 a^3 z-2 a^3 z^{-1} +a z^5+3 a z^3+a z$ (db) Kauffman polynomial $-3 z^3 a^9+4 z a^9-a^9 z^{-1} -z^6 a^8-2 z^4 a^8+4 z^2 a^8-a^8-3 z^7 a^7+9 z^5 a^7-19 z^3 a^7+15 z a^7-3 a^7 z^{-1} -3 z^8 a^6+9 z^6 a^6-16 z^4 a^6+11 z^2 a^6-3 a^6-z^9 a^5-4 z^7 a^5+23 z^5 a^5-39 z^3 a^5+24 z a^5-4 a^5 z^{-1} -6 z^8 a^4+20 z^6 a^4-22 z^4 a^4+9 z^2 a^4-2 a^4-z^9 a^3-4 z^7 a^3+25 z^5 a^3-33 z^3 a^3+16 z a^3-2 a^3 z^{-1} -3 z^8 a^2+9 z^6 a^2-5 z^4 a^2+z^2 a^2-a^2-3 z^7 a+11 z^5 a-10 z^3 a+3 z a-z^6+3 z^4-z^2$ (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 \
-6-5-4-3-2-10123χ
4         1-1
2        2 2
0       21 -1
-2      52  3
-4     43   -1
-6    54    1
-8   34     1
-10  35      -2
-12 24       2
-14 2        -2
-162         2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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