# L8a17

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a17 at Knotilus! L8a17 is $8^3_{2}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(3)^2 t(2)^2+t(1) t(3) t(2)^2-t(3) t(2)^2+t(1) t(3)^2 t(2)-t(3)^2 t(2)+t(1) t(2)-2 t(1) t(3) t(2)+2 t(3) t(2)-t(2)-t(1)+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $q^{-10} -2 q^{-9} +4 q^{-8} -4 q^{-7} +6 q^{-6} -4 q^{-5} +4 q^{-4} -2 q^{-3} + q^{-2}$ (db) Signature -4 (db) HOMFLY-PT polynomial $a^{10} z^{-2} +a^{10}-3 a^8 z^2-2 a^8 z^{-2} -6 a^8+2 a^6 z^4+6 a^6 z^2+a^6 z^{-2} +5 a^6+a^4 z^4+2 a^4 z^2$ (db) Kauffman polynomial $a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-3 a^{11} z^3+2 a^{10} z^6-2 a^{10} z^4+a^{10} z^2+a^{10} z^{-2} -3 a^{10}+a^9 z^7+2 a^9 z^5-6 a^9 z^3+6 a^9 z-2 a^9 z^{-1} +5 a^8 z^6-12 a^8 z^4+15 a^8 z^2+2 a^8 z^{-2} -8 a^8+a^7 z^7+2 a^7 z^5-6 a^7 z^3+6 a^7 z-2 a^7 z^{-1} +3 a^6 z^6-8 a^6 z^4+10 a^6 z^2+a^6 z^{-2} -5 a^6+2 a^5 z^5-3 a^5 z^3+a^4 z^4-2 a^4 z^2$ (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 \
-8-7-6-5-4-3-2-10χ
-3        11
-5       21-1
-7      2  2
-9     22  0
-11    42   2
-13   13    2
-15  33     0
-17 13      2
-19 1       -1
-211        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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

### 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.