# L9a14

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a14 at Knotilus! L9a14 is $9^2_{13}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(v^2-v+1\right) \left(v^2+v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-4 q^{9/2}+2 q^{7/2}-3 q^{5/2}+q^{3/2}+q^{19/2}-2 q^{17/2}+3 q^{15/2}-3 q^{13/2}+4 q^{11/2}-\sqrt{q}$ (db) Signature 5 (db) HOMFLY-PT polynomial $z^5 a^{-7} +4 z^3 a^{-7} +4 z a^{-7} +2 a^{-7} z^{-1} -z^7 a^{-5} -6 z^5 a^{-5} -12 z^3 a^{-5} -11 z a^{-5} -5 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +7 z a^{-3} +3 a^{-3} z^{-1}$ (db) Kauffman polynomial $-z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -4 z^7 a^{-5} -3 z^7 a^{-7} +4 z^6 a^{-4} +z^6 a^{-6} -3 z^6 a^{-8} +6 z^5 a^{-3} +20 z^5 a^{-5} +11 z^5 a^{-7} -3 z^5 a^{-9} -z^4 a^{-4} +8 z^4 a^{-6} +6 z^4 a^{-8} -3 z^4 a^{-10} -12 z^3 a^{-3} -30 z^3 a^{-5} -12 z^3 a^{-7} +4 z^3 a^{-9} -2 z^3 a^{-11} -8 z^2 a^{-4} -12 z^2 a^{-6} +3 z^2 a^{-10} -z^2 a^{-12} +10 z a^{-3} +17 z a^{-5} +7 z a^{-7} +5 a^{-4} +5 a^{-6} - a^{-10} -3 a^{-3} z^{-1} -5 a^{-5} z^{-1} -2 a^{-7} z^{-1}$ (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 \
-2-101234567χ
20         1-1
18        1 1
16       21 -1
14      11  0
12     32   -1
10    11    0
8   13     2
6  21      1
4 13       2
2          0
01         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=4$ $i=6$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.