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

Visit L9a20 at Knotilus!

L9a20 is 9^2_{42} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a20's Link Presentations]

Planar diagram presentation X8192 X16,11,17,12 X10,4,11,3 X2,15,3,16 X12,5,13,6 X6718 X14,10,15,9 X18,14,7,13 X4,18,5,17
Gauss code {1, -4, 3, -9, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -7, 4, -2, 9, -8}
A Braid Representative
A Morse Link Presentation L9a20 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-7 q^{3/2}+9 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-2 a^3 z^3+a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +4 a z^3-2 z^3 a^{-1} -a z-a z^{-1} (db)
Kauffman polynomial -3 a^2 z^8-3 z^8-7 a^3 z^7-13 a z^7-6 z^7 a^{-1} -7 a^4 z^6-6 a^2 z^6-4 z^6 a^{-2} -3 z^6-4 a^5 z^5+8 a^3 z^5+25 a z^5+12 z^5 a^{-1} -z^5 a^{-3} -a^6 z^4+8 a^4 z^4+16 a^2 z^4+7 z^4 a^{-2} +14 z^4+3 a^5 z^3-2 a^3 z^3-12 a z^3-6 z^3 a^{-1} +z^3 a^{-3} -2 a^4 z^2-6 a^2 z^2-2 z^2 a^{-2} -6 z^2-a^3 z-a z-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        3 -3
4       41 3
2      53  -2
0     74   3
-2    56    1
-4   56     -1
-6  36      3
-8 14       -3
-10 3        3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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