# L8a18

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a18 at Knotilus! L8a18 is $8^3_{1}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^2 w^2-u v w^2+u v w-u w+u-v^2 w^2+v^2 w-v w+v-1}{\sqrt{u} v w}$ (db) Jones polynomial $q^8-2 q^7+3 q^6-3 q^5+4 q^4-2 q^3+3 q^2-q+1$ (db) Signature 4 (db) HOMFLY-PT polynomial $z^4 a^{-6} +3 z^2 a^{-6} + a^{-6} z^{-2} +2 a^{-6} -z^6 a^{-4} -5 z^4 a^{-4} -8 z^2 a^{-4} -2 a^{-4} z^{-2} -6 a^{-4} +z^4 a^{-2} +4 z^2 a^{-2} + a^{-2} z^{-2} +4 a^{-2}$ (db) Kauffman polynomial $z^2 a^{-10} +2 z^3 a^{-9} +3 z^4 a^{-8} -3 z^2 a^{-8} + a^{-8} +3 z^5 a^{-7} -4 z^3 a^{-7} +3 z^6 a^{-6} -8 z^4 a^{-6} +6 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +z^7 a^{-5} -7 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +4 z^6 a^{-4} -16 z^4 a^{-4} +18 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +z^7 a^{-3} -3 z^5 a^{-3} -z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +z^6 a^{-2} -5 z^4 a^{-2} +8 z^2 a^{-2} + a^{-2} z^{-2} -5 a^{-2}$ (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 \
-2-10123456χ
17        11
15       21-1
13      1  1
11     22  0
9    21   1
7   13    2
5  21     1
3 13      2
1         0
-11        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}$ ${\mathbb Z}$

### 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.