# L11a254

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a254 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^3 v^5-2 u^3 v^4+2 u^3 v^3-u^3 v^2-u^2 v^5+3 u^2 v^4-3 u^2 v^3+3 u^2 v^2-u^2 v-u v^4+3 u v^3-3 u v^2+3 u v-u-v^3+2 v^2-2 v+1}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $-\frac{10}{q^{9/2}}+\frac{8}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{21/2}}+\frac{3}{q^{19/2}}-\frac{5}{q^{17/2}}+\frac{7}{q^{15/2}}-\frac{9}{q^{13/2}}+\frac{10}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-a^5 z^9+a^7 z^7-7 a^5 z^7+a^3 z^7+5 a^7 z^5-17 a^5 z^5+5 a^3 z^5+7 a^7 z^3-17 a^5 z^3+6 a^3 z^3+3 a^7 z-5 a^5 z+a^5 z^{-1} -a^3 z^{-1}$ (db) Kauffman polynomial $a^{13} z^3+3 a^{12} z^4-a^{12} z^2+5 a^{11} z^5-4 a^{11} z^3+a^{11} z+6 a^{10} z^6-7 a^{10} z^4+a^{10} z^2+6 a^9 z^7-9 a^9 z^5-a^9 z^3+a^9 z+6 a^8 z^8-15 a^8 z^6+8 a^8 z^4-3 a^8 z^2+5 a^7 z^9-17 a^7 z^7+17 a^7 z^5-11 a^7 z^3+4 a^7 z+2 a^6 z^{10}-2 a^6 z^8-17 a^6 z^6+27 a^6 z^4-9 a^6 z^2+8 a^5 z^9-40 a^5 z^7+61 a^5 z^5-33 a^5 z^3+5 a^5 z+a^5 z^{-1} +2 a^4 z^{10}-7 a^4 z^8-a^4 z^6+16 a^4 z^4-7 a^4 z^2-a^4+3 a^3 z^9-17 a^3 z^7+30 a^3 z^5-18 a^3 z^3+a^3 z+a^3 z^{-1} +a^2 z^8-5 a^2 z^6+7 a^2 z^4-3 a^2 z^2$ (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 \
-8-7-6-5-4-3-2-10123χ
2           1-1
0          2 2
-2         21 -1
-4        52  3
-6       43   -1
-8      64    2
-10     44     0
-12    56      -1
-14   35       2
-16  24        -2
-18 13         2
-20 2          -2
-221           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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