# L11a312

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a312 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)-2 t(1)-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-q^{5/2}+2 q^{3/2}-5 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^7-2 z a^7+2 z^5 a^5+6 z^3 a^5+3 z a^5-z^7 a^3-4 z^5 a^3-5 z^3 a^3-3 z a^3+2 z^5 a+7 z^3 a+5 z a+a z^{-1} -z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^{10} z^4-a^{10} z^2+3 a^9 z^5-3 a^9 z^3+5 a^8 z^6-6 a^8 z^4+2 a^8 z^2+6 a^7 z^7-10 a^7 z^5+9 a^7 z^3-3 a^7 z+5 a^6 z^8-8 a^6 z^6+7 a^6 z^4-3 a^6 z^2+3 a^5 z^9-2 a^5 z^7-5 a^5 z^5+6 a^5 z^3-2 a^5 z+a^4 z^{10}+4 a^4 z^8-16 a^4 z^6+15 a^4 z^4-5 a^4 z^2+5 a^3 z^9-14 a^3 z^7+10 a^3 z^5-3 a^3 z^3+a^2 z^{10}+a^2 z^8-11 a^2 z^6+9 a^2 z^4+2 a z^9-5 a z^7+z^7 a^{-1} -3 a z^5-5 z^5 a^{-1} +11 a z^3+8 z^3 a^{-1} -6 a z+a z^{-1} -5 z a^{-1} + a^{-1} z^{-1} +2 z^8-8 z^6+8 z^4-z^2-1$ (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 \
-7-6-5-4-3-2-101234χ
6           11
4          1 -1
2         41 3
0        31  -2
-2       74   3
-4      75    -2
-6     65     1
-8    57      2
-10   46       -2
-12  25        3
-14 14         -3
-16 2          2
-181           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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