# L9a32

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a32 at Knotilus! L9a32 is $9^2_{40}$ in the Rolfsen table of links.  A traditional symbol of the Christian Trinity (a triquetra interlaced with a circle, or "Trinity knot")  Symmetric version, with three lines touching at center  Symmetric version, with three lines touching at cicrumference

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(2)^4-3 t(1) t(2)^3+3 t(2)^3-3 t(1)^2 t(2)^2+5 t(1) t(2)^2-3 t(2)^2+3 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2}{t(1) t(2)^2}$ (db) Jones polynomial $\frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{9}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -3 a^7 z^3-4 a^7 z-2 a^7 z^{-1} -3 a^5 z^3-3 a^5 z-a^3 z^3$ (db) Kauffman polynomial $a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-10 a^{11} z^5+9 a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +2 a^{10} z^8-a^{10} z^6-12 a^{10} z^4+14 a^{10} z^2-3 a^{10}+9 a^9 z^7-26 a^9 z^5+22 a^9 z^3-11 a^9 z+3 a^9 z^{-1} +2 a^8 z^8+5 a^8 z^6-21 a^8 z^4+14 a^8 z^2-3 a^8+6 a^7 z^7-10 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +7 a^6 z^6-9 a^6 z^4+3 a^6 z^2+6 a^5 z^5-6 a^5 z^3+3 a^5 z+3 a^4 z^4+a^3 z^3$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         11
-4        31-2
-6       3  3
-8      43  -1
-10     53   2
-12    34    1
-14   45     -1
-16  24      2
-18 13       -2
-20 2        2
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### 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.