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

Visit L9a39 at Knotilus!

L9a39 is 9^2_{2} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a39's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X6,9,7,10 X16,8,17,7 X18,16,9,15 X8,18,1,17 X4,13,5,14
Gauss code {1, -2, 3, -9, 4, -5, 6, -8}, {5, -1, 2, -3, 9, -4, 7, -6, 8, -7}
A Braid Representative
A Morse Link Presentation L9a39 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^3 t(1)^3-t(2)^2 t(1)^3-t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+1}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial \frac{4}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{5/2}}+q^{3/2}-\frac{4}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}-2 \sqrt{q}+\frac{2}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^3 z^7+a^5 z^5-6 a^3 z^5+a z^5+4 a^5 z^3-12 a^3 z^3+4 a z^3+4 a^5 z-9 a^3 z+3 a z+a^5 z^{-1} -a^3 z^{-1} (db)
Kauffman polynomial -z^3 a^9+z a^9-2 z^4 a^8+2 z^2 a^8-2 z^5 a^7+z^3 a^7-2 z^6 a^6+2 z^4 a^6-z^2 a^6-2 z^7 a^5+5 z^5 a^5-7 z^3 a^5+5 z a^5-a^5 z^{-1} -z^8 a^4+z^6 a^4+3 z^4 a^4-4 z^2 a^4+a^4-4 z^7 a^3+16 z^5 a^3-20 z^3 a^3+10 z a^3-a^3 z^{-1} -z^8 a^2+2 z^6 a^2+3 z^4 a^2-4 z^2 a^2-2 z^7 a+9 z^5 a-11 z^3 a+4 z a-z^6+4 z^4-3 z^2 (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 \
4         1-1
2        1 1
0       11 0
-2      31  2
-4     22   0
-6    32    1
-8   12     1
-10  23      -1
-12 12       1
-14 1        -1
-161         1
Integral Khovanov Homology

(db, data source)

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