# L11n368

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n368 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)+1) (t(2)-1) (t(3)-1)^2 (t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $1- q^{-1} +3 q^{-2} - q^{-3} + q^{-4} +2 q^{-7} -2 q^{-8} +2 q^{-9} - q^{-10}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{10}+2 a^8 z^2+2 a^8+a^6 z^2+a^6 z^{-2} +a^6-a^4 z^6-5 a^4 z^4-7 a^4 z^2-2 a^4 z^{-2} -6 a^4+a^2 z^4+4 a^2 z^2+a^2 z^{-2} +4 a^2$ (db) Kauffman polynomial $a^{11} z^7-5 a^{11} z^5+6 a^{11} z^3-2 a^{11} z+2 a^{10} z^8-11 a^{10} z^6+16 a^{10} z^4-9 a^{10} z^2+2 a^{10}+a^9 z^9-4 a^9 z^7-2 a^9 z^5+11 a^9 z^3-4 a^9 z+3 a^8 z^8-20 a^8 z^6+36 a^8 z^4-22 a^8 z^2+5 a^8+a^7 z^9-6 a^7 z^7+7 a^7 z^5-a^7 z^3+a^6 z^8-7 a^6 z^6+11 a^6 z^4-6 a^6 z^2+a^6 z^{-2} -a^6+a^5 z^5-8 a^5 z^3+8 a^5 z-2 a^5 z^{-1} +3 a^4 z^6-14 a^4 z^4+15 a^4 z^2+2 a^4 z^{-2} -8 a^4+a^3 z^7-3 a^3 z^5-2 a^3 z^3+6 a^3 z-2 a^3 z^{-1} +a^2 z^6-5 a^2 z^4+8 a^2 z^2+a^2 z^{-2} -5 a^2$ (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 \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1            0
-3         31 2
-5       112  2
-7      121   0
-9     221    1
-11    242     0
-13   112      2
-15  121       0
-17 11         0
-19 1          1
-211           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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