# L8a5

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a5 at Knotilus! L8a5 is $8^2_{11}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-2 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-4 q^{5/2}+5 q^{3/2}-5 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^5 a^{-1} -2 a z^3+3 z^3 a^{-1} -z^3 a^{-3} +a^3 z-5 a z+3 z a^{-1} -z a^{-3} +2 a^3 z^{-1} -3 a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-a z^7-z^7 a^{-1} -a^2 z^6-3 z^6 a^{-2} -4 z^6-a^3 z^5-3 z^5 a^{-1} -4 z^5 a^{-3} +a^2 z^4+2 z^4 a^{-2} -3 z^4 a^{-4} +6 z^4+4 a^3 z^3+5 a z^3+6 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +3 a^2 z^2+z^2 a^{-2} +2 z^2 a^{-4} +2 z^2-5 a^3 z-6 a z-2 z a^{-1} -z a^{-3} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1}$ (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 \
-4-3-2-101234χ
10        11
8       2 -2
6      21 1
4     32  -1
2    22   0
0   34    1
-2  11     0
-4  3      3
-611       0
-81        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.