# L11n82

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n82 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{4 u v^3-6 u v^2+4 u v-u-v^3+4 v^2-6 v+4}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{4}{q^{21/2}}+\frac{2}{q^{23/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-a^{13} z^{-1} +z^3 a^{11}+3 z a^{11}+2 a^{11} z^{-1} -z^5 a^9-2 z^3 a^9-z a^9-a^9 z^{-1} -2 z^5 a^7-5 z^3 a^7-z a^7+a^7 z^{-1} -z^5 a^5-3 z^3 a^5-3 z a^5-a^5 z^{-1}$ (db) Kauffman polynomial $-3 z^4 a^{14}+6 z^2 a^{14}-3 a^{14}-z^7 a^{13}-z^5 a^{13}+3 z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+5 z^6 a^{12}-16 z^4 a^{12}+20 z^2 a^{12}-7 a^{12}-z^9 a^{11}-7 z^5 a^{11}+14 z^3 a^{11}-9 z a^{11}+2 a^{11} z^{-1} -4 z^8 a^{10}+5 z^6 a^{10}-8 z^4 a^{10}+11 z^2 a^{10}-4 a^{10}-z^9 a^9-2 z^7 a^9-z^5 a^9+8 z^3 a^9-5 z a^9+a^9 z^{-1} -2 z^8 a^8-2 z^6 a^8+8 z^4 a^8-3 z^2 a^8-3 z^7 a^7+4 z^5 a^7-2 z a^7+a^7 z^{-1} -2 z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-4         11
-6        21-1
-8       4  4
-10      42  -2
-12     64   2
-14    55    0
-16   55     0
-18  25      3
-20 25       -3
-22 2        2
-242         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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