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

Visit L9a50 at Knotilus!

L9a50 is 9^3_{1} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a50's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X18,16,11,15 X14,8,15,7 X10,12,5,11 X8,17,9,18 X16,9,17,10 X2536 X4,14,1,13
Gauss code {1, -8, 2, -9}, {8, -1, 4, -6, 7, -5}, {5, -2, 9, -4, 3, -7, 6, -3}
A Braid Representative
A Morse Link Presentation L9a50 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial -q^6+3 q^5-5 q^4+7 q^3+ q^{-3} -7 q^2-2 q^{-2} +8 q+5 q^{-1} -5 (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} +z^6 a^{-2} +4 z^4 a^{-2} +a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +2 a^2+4 a^{-2} -2 z^4-6 z^2-2 z^{-2} -5 (db)
Kauffman polynomial z^8 a^{-2} +z^8+2 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} +a^2 z^6+8 z^6 a^{-2} +6 z^6 a^{-4} +3 z^6-6 a z^5-14 z^5 a^{-1} -3 z^5 a^{-3} +5 z^5 a^{-5} -4 a^2 z^4-27 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} -19 z^4+3 a z^3+3 z^3 a^{-1} -5 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +6 a^2 z^2+23 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +22 z^2+3 a z+5 z a^{-1} +3 z a^{-3} +z a^{-5} -4 a^2-8 a^{-2} -2 a^{-4} -9-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 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 \
13         1-1
11        2 2
9       31 -2
7      42  2
5     44   0
3    43    1
1   36     3
-1  22      0
-3 14       3
-5 1        -1
-71         1
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

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