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

Visit L8a19 at Knotilus!

L8a19 is 8^3_{6} in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a19's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,5,15,6 X8,12,9,11 X16,8,11,7 X10,13,5,14 X2,9,3,10 X4,16,1,15
Gauss code {1, -7, 2, -8}, {3, -1, 5, -4, 7, -6}, {4, -2, 6, -3, 8, -5}
A Braid Representative
A Morse Link Presentation L8a19 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(3) t(2)-t(2)+1) (t(2) t(3)-t(3)+1)}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial q^4+ q^{-4} -3 q^3-3 q^{-3} +5 q^2+5 q^{-2} -5 q-5 q^{-1} +8 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^6+a^2 z^4+z^4 a^{-2} -4 z^4+2 a^2 z^2+2 z^2 a^{-2} -5 z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial 2 a z^7+2 z^7 a^{-1} +4 a^2 z^6+4 z^6 a^{-2} +8 z^6+3 a^3 z^5+a z^5+z^5 a^{-1} +3 z^5 a^{-3} +a^4 z^4-7 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} -16 z^4-4 a^3 z^3-4 a z^3-4 z^3 a^{-1} -4 z^3 a^{-3} -a^4 z^2+5 a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} +12 z^2+2 a z+2 z a^{-1} -2 a^2-2 a^{-2} -3-2 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 \
9        11
7       2 -2
5      31 2
3     33  0
1    52   3
-1   25    3
-3  33     0
-5 13      2
-7 2       -2
-91        1
Integral Khovanov Homology

(db, data source)

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