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

Visit L9a22 at Knotilus!

L9a22 is 9^2_{35} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a22's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X16,12,17,11 X12,6,13,5 X4,17,5,18 X14,7,15,8 X18,13,7,14 X6,16,1,15
Gauss code {1, -2, 3, -6, 5, -9}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -4, 6, -8}
A Braid Representative
A Morse Link Presentation L9a22 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-2 u^2 v^3+2 u^2 v^2-u^2 v-u v^4+3 u v^3-3 u v^2+3 u v-u-v^3+2 v^2-2 v+1}{u v^2} (db)
Jones polynomial q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^3 z^7+a^5 z^5-5 a^3 z^5+a z^5+3 a^5 z^3-8 a^3 z^3+3 a z^3+2 a^5 z-4 a^3 z+a z+a^3 z^{-1} -a z^{-1} (db)
Kauffman polynomial -z^3 a^9-3 z^4 a^8+z^2 a^8-5 z^5 a^7+4 z^3 a^7-z a^7-6 z^6 a^6+8 z^4 a^6-3 z^2 a^6-5 z^7 a^5+8 z^5 a^5-3 z^3 a^5+z a^5-2 z^8 a^4-2 z^6 a^4+13 z^4 a^4-7 z^2 a^4-8 z^7 a^3+24 z^5 a^3-19 z^3 a^3+4 z a^3+a^3 z^{-1} -2 z^8 a^2+3 z^6 a^2+5 z^4 a^2-5 z^2 a^2-a^2-3 z^7 a+11 z^5 a-11 z^3 a+2 z a+a z^{-1} -z^6+3 z^4-2 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 \
4         1-1
2        2 2
0       21 -1
-2      52  3
-4     33   0
-6    54    1
-8   34     1
-10  24      -2
-12 13       2
-14 2        -2
-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}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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