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

Visit L10n29 at Knotilus!

Link Presentations

[edit Notes on L10n29's Link Presentations]

Planar diagram presentation X6172 X3,10,4,11 X7,14,8,15 X15,20,16,5 X9,17,10,16 X19,9,20,8 X13,19,14,18 X17,13,18,12 X2536 X11,4,12,1
Gauss code {1, -9, -2, 10}, {9, -1, -3, 6, -5, 2, -10, 8, -7, 3, -4, 5, -8, 7, -6, 4}
A Braid Representative
A Morse Link Presentation L10n29 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^5-3 t(1) t(2)^4+t(2)^4+4 t(1) t(2)^3-4 t(2)^3-4 t(1) t(2)^2+4 t(2)^2+t(1) t(2)-3 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -q^{7/2}+3 q^{5/2}-6 q^{3/2}+8 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-a^3 z^5+5 a z^5-z^5 a^{-1} -4 a^3 z^3+9 a z^3-3 z^3 a^{-1} +a^5 z-7 a^3 z+7 a z-3 z a^{-1} +2 a^5 z^{-1} -4 a^3 z^{-1} +3 a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 a^2 z^8-2 z^8-5 a^3 z^7-9 a z^7-4 z^7 a^{-1} -3 a^4 z^6-3 a^2 z^6-3 z^6 a^{-2} -3 z^6+13 a^3 z^5+21 a z^5+7 z^5 a^{-1} -z^5 a^{-3} +3 a^4 z^4+11 a^2 z^4+6 z^4 a^{-2} +14 z^4-6 a^5 z^3-23 a^3 z^3-21 a z^3-2 z^3 a^{-1} +2 z^3 a^{-3} -4 a^4 z^2-11 a^2 z^2-3 z^2 a^{-2} -10 z^2+8 a^5 z+17 a^3 z+12 a z+2 z a^{-1} -z a^{-3} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} z^{-1} (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     42  -2
0    54   1
-2   55    0
-4  34     -1
-6 25      3
-813       -2
-103        3
Integral Khovanov Homology

(db, data source)

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