# L11n54

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n54 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 t(2)^5-2 t(2)^4+t(1) t(2)^3+t(2)^2-2 t(1) t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{9/2}-q^{7/2}+2 q^{5/2}-\frac{1}{q^{5/2}}-3 q^{3/2}+\frac{1}{q^{3/2}}+q^{15/2}-q^{13/2}+2 \sqrt{q}-\frac{3}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z^5 a^{-1} -z^5 a^{-3} +a z^3-4 z^3 a^{-1} -4 z^3 a^{-3} +3 a z-4 z a^{-1} -3 z a^{-3} +z a^{-5} +z a^{-7} +2 a z^{-1} -2 a^{-1} z^{-1} - a^{-3} z^{-1} + a^{-5} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-1} -z^9 a^{-3} -3 z^8 a^{-2} -2 z^8 a^{-4} -z^8-a z^7+4 z^7 a^{-1} +5 z^7 a^{-3} -z^7 a^{-5} -z^7 a^{-7} +16 z^6 a^{-2} +13 z^6 a^{-4} -z^6 a^{-8} +4 z^6+6 a z^5-z^5 a^{-1} -6 z^5 a^{-3} +7 z^5 a^{-5} +6 z^5 a^{-7} -22 z^4 a^{-2} -24 z^4 a^{-4} +2 z^4 a^{-6} +5 z^4 a^{-8} -z^4-11 a z^3-5 z^3 a^{-1} +5 z^3 a^{-3} -9 z^3 a^{-5} -8 z^3 a^{-7} +9 z^2 a^{-2} +17 z^2 a^{-4} -2 z^2 a^{-6} -5 z^2 a^{-8} -5 z^2+8 a z+5 z a^{-1} -4 z a^{-3} +z a^{-5} +2 z a^{-7} -3 a^{-4} + a^{-8} +3-2 a z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} + a^{-5} z^{-1}$ (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
16           1-1
14            0
12        111 1
10       21   -1
8      121   0
6     331    -1
4    111     1
2   241      1
0  111       1
-2  2         2
-411          0
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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