# L11a355

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a355 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)+t(2)-1) (t(2) t(1)-t(1)-t(2)) \left(t(1)^2 t(2)^2+t(1) t(2)+1\right)}{t(1)^2 t(2)^2}$ (db) Jones polynomial $-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{7}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z^5 a^9-4 z^3 a^9-3 z a^9+z^7 a^7+5 z^5 a^7+8 z^3 a^7+6 z a^7+a^7 z^{-1} +z^7 a^5+4 z^5 a^5+2 z^3 a^5-3 z a^5-a^5 z^{-1} -z^5 a^3-4 z^3 a^3-3 z a^3$ (db) Kauffman polynomial $-z^4 a^{14}+2 z^2 a^{14}-2 z^5 a^{13}+4 z^3 a^{13}-2 z a^{13}-2 z^6 a^{12}+2 z^4 a^{12}-2 z^7 a^{11}+2 z^5 a^{11}-3 z a^{11}-2 z^8 a^{10}+4 z^6 a^{10}-5 z^4 a^{10}+z^2 a^{10}-2 z^9 a^9+7 z^7 a^9-13 z^5 a^9+10 z^3 a^9-2 z a^9-z^{10} a^8+2 z^8 a^8-3 z^4 a^8+4 z^2 a^8-4 z^9 a^7+18 z^7 a^7-30 z^5 a^7+26 z^3 a^7-9 z a^7+a^7 z^{-1} -z^{10} a^6+2 z^8 a^6+3 z^6 a^6-5 z^4 a^6+3 z^2 a^6-a^6-2 z^9 a^5+8 z^7 a^5-8 z^5 a^5+5 z^3 a^5-5 z a^5+a^5 z^{-1} -2 z^8 a^4+9 z^6 a^4-10 z^4 a^4+2 z^2 a^4-z^7 a^3+5 z^5 a^3-7 z^3 a^3+3 z a^3$ (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          1 -1
-4         31 2
-6        32  -1
-8       42   2
-10      43    -1
-12     44     0
-14    34      1
-16   34       -1
-18  14        3
-20 12         -1
-22 1          1
-241           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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