# L11n176

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n176 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u^2 v^4-u^2 v^3-u v^4+u v^3+u v^2+u v-u-v+2}{u v^2}$ (db) Jones polynomial $\frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{23/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $a^{13} (-z)+a^{11} z^5+5 a^{11} z^3+4 a^{11} z-a^{11} z^{-1} -a^9 z^7-5 a^9 z^5-5 a^9 z^3+2 a^9 z+3 a^9 z^{-1} -a^7 z^7-6 a^7 z^5-11 a^7 z^3-8 a^7 z-2 a^7 z^{-1}$ (db) Kauffman polynomial $-z^7 a^{13}+5 z^5 a^{13}-6 z^3 a^{13}+z a^{13}-2 z^8 a^{12}+11 z^6 a^{12}-17 z^4 a^{12}+8 z^2 a^{12}+a^{12}-z^9 a^{11}+4 z^7 a^{11}-2 z^5 a^{11}-z^3 a^{11}-a^{11} z^{-1} -3 z^8 a^{10}+15 z^6 a^{10}-18 z^4 a^{10}+3 z^2 a^{10}+3 a^{10}-z^9 a^9+4 z^7 a^9-z^5 a^9-6 z^3 a^9+7 z a^9-3 a^9 z^{-1} -z^8 a^8+4 z^6 a^8-z^4 a^8-5 z^2 a^8+3 a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+8 z a^7-2 a^7 z^{-1}$ (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 \
-8-7-6-5-4-3-2-10χ
-6        11
-8       110
-10      2  2
-12    111  1
-14    32   1
-16  121    0
-18  23     -1
-20 11      0
-22 1       -1
-241        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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