# L8a16

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a16 at Knotilus! L8a16 is $8^3_{5}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) (t(3)-1) (t(2) t(3)+1)}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $-q^6+3 q^5-4 q^4+6 q^3-5 q^2- q^{-2} +6 q+3 q^{-1} -3$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+4 z^2 a^{-2} -2 z^2 a^{-4} -2 z^2- a^{-2} +1-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2}$ (db) Kauffman polynomial $2 z^7 a^{-1} +2 z^7 a^{-3} +7 z^6 a^{-2} +4 z^6 a^{-4} +3 z^6+a z^5-3 z^5 a^{-1} +4 z^5 a^{-5} -17 z^4 a^{-2} -5 z^4 a^{-4} +3 z^4 a^{-6} -9 z^4-2 a z^3-z^3 a^{-1} -3 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +10 z^2 a^{-2} +2 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2-z a^{-1} -z a^{-3} + a^{-2} + a^{-4} +1+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
-3-2-1012345χ
13        1-1
11       2 2
9      21 -1
7     42  2
5    34   1
3   32    1
1  25     3
-1 11      0
-3 2       2
-51        -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.