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

Visit L9a21 at Knotilus!

L9a21 is 9^2_{34} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a21's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X18,14,7,13 X6,17,1,18 X16,11,17,12 X12,6,13,5 X4,16,5,15
Gauss code {1, -2, 3, -9, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -7, 6, -5}
A Braid Representative
A Morse Link Presentation L9a21 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(t(1) t(2)^2-t(1) t(2)+t(2)+t(1)-1\right) \left(t(1) t(2)^2-t(2)^2-t(1) t(2)+t(2)-1\right)}{t(1) t(2)^2} (db)
Jones polynomial -q^{7/2}+3 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-3 a^3 z^3-2 a^3 z+a^3 z^{-1} +a z^7+5 a z^5-z^5 a^{-1} +8 a z^3-3 z^3 a^{-1} +3 a z-2 z a^{-1} -a z^{-1} (db)
Kauffman polynomial -2 a^2 z^8-2 z^8-4 a^3 z^7-8 a z^7-4 z^7 a^{-1} -4 a^4 z^6-a^2 z^6-3 z^6 a^{-2} -3 a^5 z^5+5 a^3 z^5+19 a z^5+10 z^5 a^{-1} -z^5 a^{-3} -a^6 z^4+4 a^4 z^4+4 a^2 z^4+7 z^4 a^{-2} +6 z^4+4 a^5 z^3-3 a^3 z^3-17 a z^3-8 z^3 a^{-1} +2 z^3 a^{-3} +a^6 z^2-2 a^2 z^2-3 z^2 a^{-2} -4 z^2-a^5 z+4 a z+3 z a^{-1} -a^2+a^3 z^{-1} +a z^{-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 \
8         11
6        2 -2
4       31 2
2      42  -2
0     53   2
-2    45    1
-4   44     0
-6  25      3
-8 13       -2
-10 2        2
-121         -1
Integral Khovanov Homology

(db, data source)

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