# L11n106

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n106 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^5-2 u v^4-u v^3+3 u v^2-2 u v-2 v^4+3 v^3-v^2-2 v+1}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{11} z^{-1} +4 z a^9+3 a^9 z^{-1} -z^5 a^7-7 z^3 a^7-9 z a^7-3 a^7 z^{-1} +z^7 a^5+6 z^5 a^5+10 z^3 a^5+8 z a^5+2 a^5 z^{-1} -z^5 a^3-5 z^3 a^3-5 z a^3-a^3 z^{-1}$ (db) Kauffman polynomial $a^{12} z^6-4 a^{12} z^4+3 a^{12} z^2-a^{12}+2 a^{11} z^7-8 a^{11} z^5+6 a^{11} z^3-3 a^{11} z+a^{11} z^{-1} +2 a^{10} z^8-8 a^{10} z^6+4 a^{10} z^4+3 a^{10} z^2-2 a^{10}+a^9 z^9-3 a^9 z^7-5 a^9 z^5+15 a^9 z^3-11 a^9 z+3 a^9 z^{-1} +3 a^8 z^8-17 a^8 z^6+24 a^8 z^4-9 a^8 z^2+a^7 z^9-4 a^7 z^7-5 a^7 z^5+22 a^7 z^3-14 a^7 z+3 a^7 z^{-1} +2 a^6 z^8-14 a^6 z^6+25 a^6 z^4-14 a^6 z^2+2 a^6+2 a^5 z^7-14 a^5 z^5+23 a^5 z^3-12 a^5 z+2 a^5 z^{-1} +a^4 z^8-6 a^4 z^6+9 a^4 z^4-5 a^4 z^2+a^3 z^7-6 a^3 z^5+10 a^3 z^3-6 a^3 z+a^3 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-1012χ
0           11
-2            0
-4        121 0
-6       111  1
-8      221   -1
-10     321    2
-12    241     1
-14   221      1
-16  121       0
-18 12         -1
-20 1          1
-221           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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