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

Visit L9a38 at Knotilus!

L9a38 is 9^2_{5} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a38's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,6,15,5 X18,14,9,13 X16,8,17,7 X6,16,7,15 X8,18,1,17
Gauss code {1, -2, 3, -4, 5, -8, 7, -9}, {4, -1, 2, -3, 6, -5, 8, -7, 9, -6}
A Braid Representative
A Morse Link Presentation L9a38 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) (t(1) t(2)+1)^2}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -2 q^{9/2}+3 q^{7/2}-\frac{1}{q^{7/2}}-5 q^{5/2}+\frac{2}{q^{5/2}}+5 q^{3/2}-\frac{3}{q^{3/2}}+q^{11/2}-6 \sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +a z^5-6 z^5 a^{-1} +z^5 a^{-3} +4 a z^3-12 z^3 a^{-1} +4 z^3 a^{-3} +4 a z-8 z a^{-1} +4 z a^{-3} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} -2 z^2 a^{-6} +2 z^5 a^{-5} -4 z^3 a^{-5} +z a^{-5} +2 z^6 a^{-4} -3 z^4 a^{-4} +z^2 a^{-4} +2 z^7 a^{-3} +a^3 z^5-5 z^5 a^{-3} -3 a^3 z^3+8 z^3 a^{-3} +a^3 z-4 z a^{-3} +z^8 a^{-2} +2 a^2 z^6-z^6 a^{-2} -6 a^2 z^4+3 a^2 z^2+3 z^2 a^{-2} +2 a z^7+4 z^7 a^{-1} -6 a z^5-14 z^5 a^{-1} +6 a z^3+21 z^3 a^{-1} -4 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^8-z^6-2 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 \
12         1-1
10        1 1
8       21 -1
6      31  2
4     22   0
2    43    1
0   24     2
-2  12      -1
-4 12       1
-6 1        -1
-81         1
Integral Khovanov Homology

(db, data source)

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