# L9a51

Jump to: navigation, search

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a51 at Knotilus! L9a51 is $9^3_{11}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^2 w^2-2 u v^2 w-2 u v w^2+4 u v w-2 u v+u w^2-2 u w+u-v^2 w^2+2 v^2 w-v^2+2 v w^2-4 v w+2 v+2 w-1}{\sqrt{u} v w}$ (db) Jones polynomial $q^7-3 q^6+7 q^5-8 q^4+11 q^3-10 q^2- q^{-2} +9 q+4 q^{-1} -6$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^2 a^{-6} + a^{-6} z^{-2} + a^{-6} -2 z^4 a^{-4} -4 z^2 a^{-4} -2 a^{-4} z^{-2} -3 a^{-4} +z^6 a^{-2} +3 z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} z^{-2} + a^{-2} -z^4-z^2+1$ (db) Kauffman polynomial $z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -10 z^4 a^{-6} +10 z^2 a^{-6} + a^{-6} z^{-2} -5 a^{-6} +5 z^7 a^{-5} -3 z^5 a^{-5} -4 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +2 z^8 a^{-4} +8 z^6 a^{-4} -21 z^4 a^{-4} +17 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +10 z^7 a^{-3} -16 z^5 a^{-3} +2 z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +2 z^8 a^{-2} +6 z^6 a^{-2} -18 z^4 a^{-2} +9 z^2 a^{-2} + a^{-2} z^{-2} -3 a^{-2} +5 z^7 a^{-1} +a z^5-9 z^5 a^{-1} -a z^3+3 z^3 a^{-1} +4 z^6-8 z^4+3 z^2+1$ (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 \
-3-2-10123456χ
15         11
13        31-2
11       4  4
9      43  -1
7     74   3
5    45    1
3   56     -1
1  36      3
-1 13       -2
-3 3        3
-51         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}$ ${\mathbb Z}$

### 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.