# L9a29

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a29 at Knotilus! L9a29 is $9^2_{19}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^4-u^2 v^3+u^2 v^2-u^2 v-u v^4+u v^3-u v^2+u v-u-v^3+v^2-v+1}{u v^2}$ (db) Jones polynomial $-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^7 \left(-z^5\right)-4 a^7 z^3-4 a^7 z-a^7 z^{-1} +a^5 z^7+6 a^5 z^5+12 a^5 z^3+10 a^5 z+3 a^5 z^{-1} -a^3 z^5-5 a^3 z^3-7 a^3 z-2 a^3 z^{-1}$ (db) Kauffman polynomial $a^{12} z^2+2 a^{11} z^3+3 a^{10} z^4-2 a^{10} z^2+4 a^9 z^5-7 a^9 z^3+2 a^9 z+4 a^8 z^6-10 a^8 z^4+5 a^8 z^2-a^8+3 a^7 z^7-9 a^7 z^5+6 a^7 z^3-3 a^7 z+a^7 z^{-1} +a^6 z^8-11 a^6 z^4+13 a^6 z^2-3 a^6+4 a^5 z^7-19 a^5 z^5+27 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +a^4 z^8-4 a^4 z^6+2 a^4 z^4+5 a^4 z^2-3 a^4+a^3 z^7-6 a^3 z^5+12 a^3 z^3-9 a^3 z+2 a^3 z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
0         11
-2          0
-4       31 2
-6      11  0
-8     32   1
-10    22    0
-12   22     0
-14  12      1
-16 12       -1
-18 1        1
-201         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.