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

Visit L10n106 at Knotilus!

Link Presentations

[edit Notes on L10n106's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X9,11,10,14 X13,15,14,20 X19,5,20,10 X11,18,12,19 X4,17,1,18
Gauss code {1, 4, -3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -6, 8}, {-4, -5, 10, 9, -8, 7}
A Braid Representative
A Morse Link Presentation L10n106 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(2)-1) (t(3)-1) (t(4)-1) (t(1) t(3) t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} (db)
Jones polynomial q^{3/2}-3 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{1}{q^{13/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z a^7+a^7 z^{-3} +z^5 a^5+4 z^3 a^5+3 z a^5-2 a^5 z^{-1} -3 a^5 z^{-3} -z^7 a^3-5 z^5 a^3-7 z^3 a^3-2 z a^3+4 a^3 z^{-1} +3 a^3 z^{-3} +z^5 a+3 z^3 a-2 a z^{-1} -a z^{-3} (db)
Kauffman polynomial -z^2 a^8-4 z^3 a^7+3 z a^7+2 a^7 z^{-1} -a^7 z^{-3} -2 z^6 a^6+4 z^4 a^6-3 z^2 a^6+3 a^6 z^{-2} -4 a^6-4 z^7 a^5+16 z^5 a^5-24 z^3 a^5+11 z a^5+3 a^5 z^{-1} -3 a^5 z^{-3} -2 z^8 a^4+4 z^6 a^4+3 z^4 a^4-4 z^2 a^4+6 a^4 z^{-2} -7 a^4-7 z^7 a^3+28 z^5 a^3-32 z^3 a^3+11 z a^3+3 a^3 z^{-1} -3 a^3 z^{-3} -2 z^8 a^2+5 z^6 a^2+2 z^4 a^2-3 z^2 a^2+3 a^2 z^{-2} -4 a^2-3 z^7 a+12 z^5 a-12 z^3 a+3 z a+2 a z^{-1} -a z^{-3} -z^6+3 z^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       2 2
0      11 0
-2     52  3
-4   123   2
-6   63    3
-8 125     4
-10 32      1
-12 3       3
-141        -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2 i=0
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{3} {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{6} {\mathbb Z}
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}^{5}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
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

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

See/edit the Link_Splice_Base (expert).

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