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

Visit L9a42 at Knotilus!

L9a42 is 9^2_{41} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a42's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X18,5,9,6 X6,9,7,10 X16,12,17,11 X14,8,15,7 X4,14,5,13 X8,16,1,15 X2,17,3,18
Gauss code {1, -9, 2, -7, 3, -4, 6, -8}, {4, -1, 5, -2, 7, -6, 8, -5, 9, -3}
A Braid Representative
A Morse Link Presentation L9a42 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(u^2 v^2-u v^2+3 u v-u+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -3 q^{9/2}+6 q^{7/2}-\frac{1}{q^{7/2}}-9 q^{5/2}+\frac{3}{q^{5/2}}+9 q^{3/2}-\frac{6}{q^{3/2}}+q^{11/2}-10 \sqrt{q}+\frac{8}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +a z^5-5 z^5 a^{-1} +z^5 a^{-3} +3 a z^3-9 z^3 a^{-1} +3 z^3 a^{-3} +3 a z-6 z a^{-1} +3 z a^{-3} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +5 z^6 a^{-4} -6 z^4 a^{-4} +3 z^2 a^{-4} +5 z^7 a^{-3} +a^3 z^5-6 z^5 a^{-3} -2 a^3 z^3+4 z^3 a^{-3} +a^3 z-2 z a^{-3} +2 z^8 a^{-2} +3 a^2 z^6+5 z^6 a^{-2} -6 a^2 z^4-12 z^4 a^{-2} +3 a^2 z^2+6 z^2 a^{-2} +4 a z^7+9 z^7 a^{-1} -7 a z^5-17 z^5 a^{-1} +3 a z^3+12 z^3 a^{-1} -2 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^8+3 z^6-11 z^4+5 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        2 2
8       41 -3
6      52  3
4     44   0
2    65    1
0   46     2
-2  24      -2
-4 14       3
-6 2        -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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