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

Visit L9a41 at Knotilus!

L9a41 is 9^2_{23} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a41's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,5,9,6 X14,8,15,7 X16,14,17,13 X8,16,1,15 X6,9,7,10 X4,17,5,18
Gauss code {1, -2, 3, -9, 4, -8, 5, -7}, {8, -1, 2, -3, 6, -5, 7, -6, 9, -4}
A Braid Representative
A Morse Link Presentation L9a41 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1) t(2)+1) (t(2) t(1)-t(1)+1) (t(1) t(2)-t(2)+1)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial q^{3/2}-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{1}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^5 z^5+4 a^5 z^3+5 a^5 z+a^5 z^{-1} -a^3 z^7-6 a^3 z^5-13 a^3 z^3-11 a^3 z-a^3 z^{-1} +a z^5+4 a z^3+4 a z (db)
Kauffman polynomial -z^3 a^9+z a^9-2 z^4 a^8+z^2 a^8-3 z^5 a^7+2 z^3 a^7-z a^7-4 z^6 a^6+7 z^4 a^6-6 z^2 a^6-3 z^7 a^5+5 z^5 a^5-4 z^3 a^5+4 z a^5-a^5 z^{-1} -z^8 a^4-3 z^6 a^4+13 z^4 a^4-10 z^2 a^4+a^4-5 z^7 a^3+16 z^5 a^3-17 z^3 a^3+11 z a^3-a^3 z^{-1} -z^8 a^2+8 z^4 a^2-7 z^2 a^2-2 z^7 a+8 z^5 a-10 z^3 a+5 z a-z^6+4 z^4-4 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        1 1
0       21 -1
-2      41  3
-4     23   1
-6    43    1
-8   22     0
-10  24      -2
-12 13       2
-14 1        -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}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
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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