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

Visit L9a31 at Knotilus!

L9a31 is 9^2_{39} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a31's Link Presentations]

Planar diagram presentation X8192 X16,9,17,10 X6718 X18,13,7,14 X10,4,11,3 X14,6,15,5 X4,12,5,11 X12,17,13,18 X2,16,3,15
Gauss code {1, -9, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 4, -6, 9, -2, 8, -4}
A Braid Representative
A Morse Link Presentation L9a31 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(v^2-v+1\right) (u v-u+1) (u v-v+1)}{u v^2} (db)
Jones polynomial -q^{7/2}+3 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{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^3-3 a^3 z-a^3 z^{-1} +a z^7+5 a z^5-z^5 a^{-1} +9 a z^3-3 z^3 a^{-1} +7 a z-3 z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial a^6 z^4-a^6 z^2+3 a^5 z^5-3 a^5 z^3+5 a^4 z^6-7 a^4 z^4+4 a^4 z^2-a^4+5 a^3 z^7-8 a^3 z^5+z^5 a^{-3} +8 a^3 z^3-2 z^3 a^{-3} -4 a^3 z+z a^{-3} +a^3 z^{-1} +2 a^2 z^8+4 a^2 z^6+3 z^6 a^{-2} -12 a^2 z^4-6 z^4 a^{-2} +10 a^2 z^2+2 z^2 a^{-2} -3 a^2+9 a z^7+4 z^7 a^{-1} -20 a z^5-8 z^5 a^{-1} +18 a z^3+5 z^3 a^{-1} -9 a z-4 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +2 z^8+2 z^6-10 z^4+7 z^2-3 (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 \
8         11
6        2 -2
4       41 3
2      43  -1
0     63   3
-2    45    1
-4   45     -1
-6  24      2
-8 14       -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
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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