# L11a316

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a316 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^3 v^4-2 u^3 v^3+2 u^3 v^2-u^3 v+u^2 v^5-4 u^2 v^4+5 u^2 v^3-5 u^2 v^2+3 u^2 v-u^2-u v^5+3 u v^4-5 u v^3+5 u v^2-4 u v+u-v^4+2 v^3-2 v^2+v}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $-\frac{14}{q^{9/2}}+\frac{10}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{3}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{10}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{16}{q^{13/2}}+\frac{15}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^9 \left(-z^5\right)-3 a^9 z^3-2 a^9 z-a^9 z^{-1} +a^7 z^7+4 a^7 z^5+6 a^7 z^3+6 a^7 z+3 a^7 z^{-1} +a^5 z^7+3 a^5 z^5-4 a^5 z-2 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-2 a^3 z$ (db) Kauffman polynomial $a^{14} z^4-a^{14} z^2+3 a^{13} z^5-3 a^{13} z^3+5 a^{12} z^6-5 a^{12} z^4+a^{12} z^2+7 a^{11} z^7-11 a^{11} z^5+8 a^{11} z^3-2 a^{11} z+8 a^{10} z^8-18 a^{10} z^6+18 a^{10} z^4-6 a^{10} z^2+a^{10}+6 a^9 z^9-12 a^9 z^7+6 a^9 z^5+a^9 z^3+a^9 z-a^9 z^{-1} +2 a^8 z^{10}+6 a^8 z^8-33 a^8 z^6+39 a^8 z^4-18 a^8 z^2+3 a^8+10 a^7 z^9-33 a^7 z^7+35 a^7 z^5-22 a^7 z^3+11 a^7 z-3 a^7 z^{-1} +2 a^6 z^{10}+a^6 z^8-21 a^6 z^6+26 a^6 z^4-13 a^6 z^2+3 a^6+4 a^5 z^9-13 a^5 z^7+11 a^5 z^5-7 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+11 a^4 z^4-3 a^4 z^2+a^3 z^7-4 a^3 z^5+5 a^3 z^3-2 a^3 z$ (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 \
-9-8-7-6-5-4-3-2-1012χ
0           11
-2          2 -2
-4         51 4
-6        63  -3
-8       84   4
-10      76    -1
-12     98     1
-14    68      2
-16   48       -4
-18  26        4
-20 14         -3
-22 2          2
-241           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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