# L8a3

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a3 at Knotilus! L8a3 is $8^2_{9}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u v^2-4 u v+u+v^3-4 v^2+2 v}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-\frac{4}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{5/2}}-q^{3/2}+\frac{5}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 z^{-1} -3 z a^5-2 a^5 z^{-1} +2 z^3 a^3+3 z a^3+2 a^3 z^{-1} +z^3 a-z a-a z^{-1} -z a^{-1}$ (db) Kauffman polynomial $-z^5 a^7+3 z^3 a^7-3 z a^7+a^7 z^{-1} -2 z^6 a^6+5 z^4 a^6-2 z^2 a^6-z^7 a^5-2 z^5 a^5+11 z^3 a^5-8 z a^5+2 a^5 z^{-1} -5 z^6 a^4+11 z^4 a^4-6 z^2 a^4+a^4-z^7 a^3-4 z^5 a^3+12 z^3 a^3-9 z a^3+2 a^3 z^{-1} -3 z^6 a^2+4 z^4 a^2-3 z^2 a^2-3 z^5 a+3 z^3 a-3 z a+a z^{-1} -2 z^4+z^2-z^3 a^{-1} +z a^{-1}$ (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 \
-6-5-4-3-2-1012χ
4        11
2       1 -1
0      31 2
-2     32  -1
-4    22   0
-6   23    1
-8  22     0
-10 13      2
-12 1       -1
-141        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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

### Modifying This Page

 Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.