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

Visit L9a28 at Knotilus!

L9a28 is 9^2_{20} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a28's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,16,7,15 X16,12,17,11 X12,18,13,17 X6718 X4,13,5,14
Gauss code {1, -2, 3, -9, 4, -8}, {8, -1, 2, -3, 6, -7, 9, -4, 5, -6, 7, -5}
A Braid Representative
A Morse Link Presentation L9a28 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-2 t(1)^2 t(2)^3+2 t(1) t(2)^3+2 t(1)^2 t(2)^2-3 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-2 t(2)+1}{t(1) t(2)^2} (db)
Jones polynomial \frac{5}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{5}{q^{5/2}}+q^{3/2}-\frac{5}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{4}{q^{11/2}}-2 \sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^5 z^5+4 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-6 a^3 z^5-13 a^3 z^3-12 a^3 z-3 a^3 z^{-1} +a z^5+4 a z^3+4 a z+a z^{-1} (db)
Kauffman polynomial a^9 z^3-a^9 z+2 a^8 z^4-a^8 z^2+3 a^7 z^5-3 a^7 z^3+2 a^7 z+3 a^6 z^6-3 a^6 z^4+a^6 z^2+3 a^5 z^7-7 a^5 z^5+9 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^8+2 a^4 z^6-11 a^4 z^4+10 a^4 z^2-3 a^4+5 a^3 z^7-18 a^3 z^5+22 a^3 z^3-14 a^3 z+3 a^3 z^{-1} +a^2 z^8-10 a^2 z^4+12 a^2 z^2-3 a^2+2 a z^7-8 a z^5+9 a z^3-4 a z+a z^{-1} +z^6-4 z^4+4 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 \
4         1-1
2        1 1
0       21 -1
-2      31  2
-4     33   0
-6    32    1
-8   23     1
-10  23      -1
-12  2       2
-1412        -1
-161         1
Integral Khovanov Homology

(db, data source)

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

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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