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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a9 at Knotilus!

L9a9 is 9^2_{37} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a9's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X2,14,3,13
Gauss code {1, -9, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -6, 9, -2, 4, -3, 5, -8}
A Braid Representative
A Morse Link Presentation L9a9 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{3}{q^{9/2}}-q^{7/2}+\frac{5}{q^{7/2}}+3 q^{5/2}-\frac{7}{q^{5/2}}-5 q^{3/2}+\frac{8}{q^{3/2}}+\frac{1}{q^{11/2}}+6 \sqrt{q}-\frac{9}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-3 a^3 z^3-2 a^3 z+a z^7+5 a z^5-z^5 a^{-1} +8 a z^3-3 z^3 a^{-1} +4 a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 a^2 z^8-2 z^8-4 a^3 z^7-8 a z^7-4 z^7 a^{-1} -4 a^4 z^6-3 z^6 a^{-2} +z^6-3 a^5 z^5+7 a^3 z^5+22 a z^5+11 z^5 a^{-1} -z^5 a^{-3} -a^6 z^4+5 a^4 z^4+3 a^2 z^4+7 z^4 a^{-2} +4 z^4+4 a^5 z^3-8 a^3 z^3-24 a z^3-10 z^3 a^{-1} +2 z^3 a^{-3} +a^6 z^2-a^4 z^2-3 a^2 z^2-2 z^2 a^{-2} -3 z^2+4 a^3 z+8 a z+4 z a^{-1} +1-a z^{-1} - a^{-1} 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 \
8         11
6        2 -2
4       31 2
2      32  -1
0     63   3
-2    45    1
-4   34     -1
-6  24      2
-8 13       -2
-10 2        2
-121         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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).

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