# L9a50

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a50 at Knotilus! L9a50 is $9^3_{1}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $-q^6+3 q^5-5 q^4+7 q^3+ q^{-3} -7 q^2-2 q^{-2} +8 q+5 q^{-1} -5$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} +z^6 a^{-2} +4 z^4 a^{-2} +a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +2 a^2+4 a^{-2} -2 z^4-6 z^2-2 z^{-2} -5$ (db) Kauffman polynomial $z^8 a^{-2} +z^8+2 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} +a^2 z^6+8 z^6 a^{-2} +6 z^6 a^{-4} +3 z^6-6 a z^5-14 z^5 a^{-1} -3 z^5 a^{-3} +5 z^5 a^{-5} -4 a^2 z^4-27 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} -19 z^4+3 a z^3+3 z^3 a^{-1} -5 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +6 a^2 z^2+23 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +22 z^2+3 a z+5 z a^{-1} +3 z a^{-3} +z a^{-5} -4 a^2-8 a^{-2} -2 a^{-4} -9-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-4-3-2-1012345χ
13         1-1
11        2 2
9       31 -2
7      42  2
5     44   0
3    43    1
1   36     3
-1  22      0
-3 14       3
-5 1        -1
-71         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.