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

Visit L9a55 at Knotilus!

L9a55 is 9^4_{1} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a55's Link Presentations]

Planar diagram presentation X6172 X2536 X16,12,17,11 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X18,16,13,15 X12,18,9,17
Gauss code {1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -9}, {7, -6, 8, -3, 9, -8}
A Braid Representative
A Morse Link Presentation L9a55 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(2) t(1)-t(2) t(3) t(1)+t(3) t(1)-t(2) t(4) t(1)+t(2) t(3) t(4) t(1)-2 t(3) t(4) t(1)+2 t(4) t(1)-2 t(1)-2 t(2)+2 t(2) t(3)-t(3)+t(2) t(4)-2 t(2) t(3) t(4)+2 t(3) t(4)-t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} (db)
Jones polynomial q^{5/2}-4 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^{-1} +a^7 z^{-3} -3 z a^5-5 a^5 z^{-1} -3 a^5 z^{-3} +3 z^3 a^3+6 z a^3+7 a^3 z^{-1} +3 a^3 z^{-3} -z^5 a-2 z^3 a-3 z a-3 a z^{-1} -a z^{-3} +z^3 a^{-1} (db)
Kauffman polynomial -a^4 z^8-a^2 z^8-a^5 z^7-5 a^3 z^7-4 a z^7-a^6 z^6-a^4 z^6-6 a^2 z^6-6 z^6-a^7 z^5-3 a^5 z^5+4 a^3 z^5+2 a z^5-4 z^5 a^{-1} -2 a^4 z^4+7 a^2 z^4-z^4 a^{-2} +8 z^4+4 a^7 z^3+12 a^5 z^3+8 a^3 z^3+4 a z^3+4 z^3 a^{-1} +6 a^6 z^2+12 a^4 z^2+6 a^2 z^2-6 a^7 z-14 a^5 z-14 a^3 z-6 a z-8 a^6-15 a^4-8 a^2+4 a^7 z^{-1} +9 a^5 z^{-1} +9 a^3 z^{-1} +4 a z^{-1} +3 a^6 z^{-2} +6 a^4 z^{-2} +3 a^2 z^{-2} -a^7 z^{-3} -3 a^5 z^{-3} -3 a^3 z^{-3} -a z^{-3} (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 \
6         1-1
4        3 3
2       31 -2
0      53  2
-2     67   1
-4    41    3
-6   16     5
-8  44      0
-10 15       4
-12          0
-141         1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2 {\mathbb Z}^{5}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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

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