# L11n193

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n193 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^4-t(2)^4+t(1)^2 t(2)^3-5 t(1) t(2)^3+3 t(2)^3-4 t(1)^2 t(2)^2+7 t(1) t(2)^2-4 t(2)^2+3 t(1)^2 t(2)-5 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1) t(2)^2}$ (db) Jones polynomial $-\frac{2}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{10}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^9+z^5 a^7+z^3 a^7+z a^7+a^7 z^{-1} +z^5 a^5-2 z a^5-a^5 z^{-1} -2 z^3 a^3-2 z a^3$ (db) Kauffman polynomial $a^{12} z^6-2 a^{12} z^4+a^{12} z^2+4 a^{11} z^7-11 a^{11} z^5+7 a^{11} z^3+5 a^{10} z^8-12 a^{10} z^6+4 a^{10} z^4+2 a^{10} z^2+2 a^9 z^9+4 a^9 z^7-20 a^9 z^5+12 a^9 z^3-2 a^9 z+9 a^8 z^8-19 a^8 z^6+8 a^8 z^4-a^8 z^2+2 a^7 z^9+3 a^7 z^7-9 a^7 z^5+a^7 z^3+3 a^7 z-a^7 z^{-1} +4 a^6 z^8-5 a^6 z^6+6 a^6 z^4-4 a^6 z^2+a^6+3 a^5 z^7-a^5 z^3+3 a^5 z-a^5 z^{-1} +a^4 z^6+4 a^4 z^4-2 a^4 z^2+3 a^3 z^3-2 a^3 z$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         22
-4        41-3
-6       51 4
-8      64  -2
-10     75   2
-12    56    1
-14   57     -2
-16  36      3
-18 14       -3
-20 3        3
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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