# L11n95

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n95 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^5+t(2)^5-2 t(2)^4-t(1) t(2)^3+2 t(2)^3+2 t(1) t(2)^2-t(2)^2-2 t(1) t(2)+t(1)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{1}{q^{7/2}}+\frac{1}{q^{27/2}}-\frac{2}{q^{25/2}}+\frac{3}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{4}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{2}{q^{13/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z a^{13}-a^{13} z^{-1} +2 z^3 a^{11}+4 z a^{11}+a^{11} z^{-1} +2 z^3 a^9+4 z a^9+2 a^9 z^{-1} -z^7 a^7-7 z^5 a^7-14 z^3 a^7-9 z a^7-2 a^7 z^{-1}$ (db) Kauffman polynomial $a^{16} z^6-4 a^{16} z^4+3 a^{16} z^2-a^{16}+2 a^{15} z^7-8 a^{15} z^5+6 a^{15} z^3-a^{15} z+2 a^{14} z^8-8 a^{14} z^6+6 a^{14} z^4+a^{13} z^9-3 a^{13} z^7-a^{13} z^5+3 a^{13} z^3+a^{13} z-a^{13} z^{-1} +3 a^{12} z^8-16 a^{12} z^6+26 a^{12} z^4-15 a^{12} z^2+3 a^{12}+a^{11} z^9-6 a^{11} z^7+12 a^{11} z^5-10 a^{11} z^3+4 a^{11} z-a^{11} z^{-1} +a^{10} z^8-7 a^{10} z^6+14 a^{10} z^4-7 a^{10} z^2-2 a^9 z^5+7 a^9 z^3-7 a^9 z+2 a^9 z^{-1} -2 a^8 z^4+5 a^8 z^2-3 a^8+a^7 z^7-7 a^7 z^5+14 a^7 z^3-9 a^7 z+2 a^7 z^{-1}$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-6           11
-8           11
-10        11  0
-12       2    2
-14      231   0
-16     31     2
-18    231     0
-20   23       -1
-22  121       0
-24 12         -1
-26 1          1
-281           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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