# L11a289

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a289 at Knotilus!  Celtic or pseudo-Celtic linear decorative knot  Decorative variant with big loops at ends

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+2 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $16 q^{9/2}-18 q^{7/2}+17 q^{5/2}-\frac{1}{q^{5/2}}-15 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-4 q^{15/2}+8 q^{13/2}-12 q^{11/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-7} +z a^{-7} -2 z^5 a^{-5} -5 z^3 a^{-5} -3 z a^{-5} +z^7 a^{-3} +4 z^5 a^{-3} +7 z^3 a^{-3} +5 z a^{-3} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +2 a z-5 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-10} +4 z^5 a^{-9} -2 z^3 a^{-9} +8 z^6 a^{-8} -8 z^4 a^{-8} +3 z^2 a^{-8} +10 z^7 a^{-7} -12 z^5 a^{-7} +6 z^3 a^{-7} -2 z a^{-7} +8 z^8 a^{-6} -4 z^6 a^{-6} -9 z^4 a^{-6} +4 z^2 a^{-6} +4 z^9 a^{-5} +9 z^7 a^{-5} -34 z^5 a^{-5} +24 z^3 a^{-5} -6 z a^{-5} +z^{10} a^{-4} +13 z^8 a^{-4} -33 z^6 a^{-4} +16 z^4 a^{-4} +7 z^9 a^{-3} -7 z^7 a^{-3} -24 z^5 a^{-3} +32 z^3 a^{-3} -10 z a^{-3} +z^{10} a^{-2} +8 z^8 a^{-2} -32 z^6 a^{-2} +28 z^4 a^{-2} -4 z^2 a^{-2} +3 z^9 a^{-1} +a z^7-5 z^7 a^{-1} -4 a z^5-10 z^5 a^{-1} +6 a z^3+22 z^3 a^{-1} -4 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +3 z^8-11 z^6+12 z^4-3 z^2-1$ (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 \
-4-3-2-101234567χ
18           1-1
16          3 3
14         51 -4
12        73  4
10       95   -4
8      97    2
6     89     1
4    79      -2
2   510       5
0  25        -3
-2 15         4
-4 2          -2
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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