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

Visit L9a48 at Knotilus!

L9a48 is 9^3_{5} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a48's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X18,14,11,13 X8,16,9,15 X14,8,15,7 X16,10,17,9 X10,18,5,17 X2536 X4,11,1,12
Gauss code {1, -8, 2, -9}, {8, -1, 5, -4, 6, -7}, {9, -2, 3, -5, 4, -6, 7, -3}
A Braid Representative
A Morse Link Presentation L9a48 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^2 w^2-u v^2 w-u v w^2+2 u v w-u v-u w+2 u-2 v^2 w^2+v^2 w+v w^2-2 v w+v+w-1}{\sqrt{u} v w} (db)
Jones polynomial -q^6+3 q^5-4 q^4+6 q^3-6 q^2+6 q-4+4 q^{-1} - q^{-2} + q^{-3} (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^4 a^{-4} -2 z^2 a^{-4} +z^6 a^{-2} +4 z^4 a^{-2} +a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+3 a^{-2} -2 z^4-7 z^2-2 z^{-2} -6 (db)
Kauffman polynomial z^8 a^{-2} +z^8+a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+2 z^6 a^{-2} +4 z^6 a^{-4} -z^6-2 a z^5-10 z^5 a^{-1} -4 z^5 a^{-3} +4 z^5 a^{-5} -5 a^2 z^4-8 z^4 a^{-2} -4 z^4 a^{-4} +3 z^4 a^{-6} -6 z^4-3 a z^3+2 z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +8 a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-6} +11 z^2+6 a z+6 z a^{-1} -5 a^2-3 a^{-2} + a^{-4} -8-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +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 \
13         1-1
11        2 2
9       21 -1
7      42  2
5     44   0
3    22    0
1   35     2
-1  11      0
-3  3       3
-511        0
-71         1
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

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