# L11n67

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n67 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{5 t(1) t(2)^3-t(2)^3-9 t(1) t(2)^2+6 t(2)^2+6 t(1) t(2)-9 t(2)-t(1)+5}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{14}{q^{13/2}}+\frac{14}{q^{15/2}}-\frac{14}{q^{17/2}}+\frac{10}{q^{19/2}}-\frac{6}{q^{21/2}}+\frac{3}{q^{23/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-a^{13} z^{-1} +z^3 a^{11}+2 z a^{11}+a^{11} z^{-1} -z^5 a^9+4 z a^9+2 a^9 z^{-1} -3 z^5 a^7-9 z^3 a^7-7 z a^7-2 a^7 z^{-1} -z^5 a^5-2 z^3 a^5-z a^5$ (db) Kauffman polynomial $-6 z^4 a^{14}+11 z^2 a^{14}-4 a^{14}-3 z^7 a^{13}+3 z^5 a^{13}-z^3 a^{13}-z a^{13}+a^{13} z^{-1} -5 z^8 a^{12}+13 z^6 a^{12}-27 z^4 a^{12}+28 z^2 a^{12}-9 a^{12}-2 z^9 a^{11}-5 z^7 a^{11}+14 z^5 a^{11}-11 z^3 a^{11}+z a^{11}+a^{11} z^{-1} -10 z^8 a^{10}+19 z^6 a^{10}-17 z^4 a^{10}+11 z^2 a^{10}-4 a^{10}-2 z^9 a^9-8 z^7 a^9+24 z^5 a^9-24 z^3 a^9+12 z a^9-2 a^9 z^{-1} -5 z^8 a^8+3 z^6 a^8+8 z^4 a^8-7 z^2 a^8+2 a^8-6 z^7 a^7+12 z^5 a^7-12 z^3 a^7+9 z a^7-2 a^7 z^{-1} -3 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+2 z^3 a^5-z a^5$ (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
-4         11
-6        31-2
-8       5  5
-10      63  -3
-12     85   3
-14    77    0
-16   77     0
-18  37      4
-20 37       -4
-22 3        3
-243         -3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}^{3}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-7$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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