# L8a15

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a15 at Knotilus! L8a15 is $8^3_{3}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v w-2 u v-2 u w+2 u-2 v w+2 v+2 w-1}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db) Jones polynomial $q^{-7} - q^{-6} +4 q^{-5} -4 q^{-4} +6 q^{-3} -4 q^{-2} +q+4 q^{-1} -3$ (db) Signature -2 (db) HOMFLY-PT polynomial $a^8 z^{-2} -2 a^6 z^{-2} -3 a^6+3 a^4 z^2+a^4 z^{-2} +3 a^4-a^2 z^4-a^2 z^2+z^2$ (db) Kauffman polynomial $z^4 a^8-3 z^2 a^8-a^8 z^{-2} +3 a^8+z^5 a^7-3 z a^7+2 a^7 z^{-1} +z^6 a^6+2 z^4 a^6-6 z^2 a^6-2 a^6 z^{-2} +5 a^6+z^7 a^5+2 z^3 a^5-3 z a^5+2 a^5 z^{-1} +4 z^6 a^4-5 z^4 a^4-a^4 z^{-2} +3 a^4+z^7 a^3+2 z^5 a^3-3 z^3 a^3+3 z^6 a^2-5 z^4 a^2+2 z^2 a^2+3 z^5 a-5 z^3 a+z^4-z^2$ (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 \
-6-5-4-3-2-1012χ
3        11
1       2 -2
-1      21 1
-3     33  0
-5    31   2
-7   13    2
-9  33     0
-11 14      3
-13         0
-151        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.