# L11n332

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n332 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(v+w-1) (v w-v-w) (u v w-1)}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $- q^{-9} +2 q^{-8} -3 q^{-7} +7 q^{-6} -5 q^{-5} +6 q^{-4} -5 q^{-3} +4 q^{-2} -2 q^{-1} +1$ (db) Signature -4 (db) HOMFLY-PT polynomial $-a^{10}+z^4 a^8+4 z^2 a^8+a^8 z^{-2} +2 a^8-z^6 a^6-4 z^4 a^6-4 z^2 a^6-2 a^6 z^{-2} -3 a^6-z^6 a^4-3 z^4 a^4+a^4 z^{-2} +a^4+z^4 a^2+3 z^2 a^2+a^2$ (db) Kauffman polynomial $z^3 a^{11}-2 z a^{11}+2 z^4 a^{10}-4 z^2 a^{10}+3 a^{10}+z^7 a^9-4 z^5 a^9+10 z^3 a^9-7 z a^9+2 z^8 a^8-10 z^6 a^8+22 z^4 a^8-19 z^2 a^8-a^8 z^{-2} +11 a^8+z^9 a^7-2 z^7 a^7-3 z^5 a^7+14 z^3 a^7-12 z a^7+2 a^7 z^{-1} +4 z^8 a^6-17 z^6 a^6+27 z^4 a^6-22 z^2 a^6-2 a^6 z^{-2} +11 a^6+z^9 a^5-z^7 a^5-6 z^5 a^5+10 z^3 a^5-8 z a^5+2 a^5 z^{-1} +2 z^8 a^4-6 z^6 a^4+3 z^4 a^4-3 z^2 a^4-a^4 z^{-2} +3 a^4+2 z^7 a^3-7 z^5 a^3+5 z^3 a^3-z a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^2-a^2$ (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 \
-7-6-5-4-3-2-1012χ
1         11
-1        1 -1
-3       31 2
-5      32  -1
-7     32   1
-9   133    1
-11   53     2
-13  15      4
-15 12       -1
-17 1        1
-191         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{5}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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