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

Visit L10a28 at Knotilus!

Link Presentations

[edit Notes on L10a28's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,8,15,7 X20,16,5,15 X16,9,17,10 X8,19,9,20 X18,11,19,12 X10,17,11,18 X2536 X4,14,1,13
Gauss code {1, -9, 2, -10}, {9, -1, 3, -6, 5, -8, 7, -2, 10, -3, 4, -5, 8, -7, 6, -4}
A Braid Representative
A Morse Link Presentation L10a28 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)^2}{\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{12}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+2 a^5 z+a^5 z^{-1} -2 a^3 z^5-7 a^3 z^3-8 a^3 z-3 a^3 z^{-1} +a z^7+5 a z^5-z^5 a^{-1} +10 a z^3-3 z^3 a^{-1} +9 a z-3 z a^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-3 a^4 z^8-7 a^2 z^8-4 z^8-4 a^5 z^7-8 a^3 z^7-9 a z^7-5 z^7 a^{-1} -3 a^6 z^6+13 a^2 z^6-3 z^6 a^{-2} +7 z^6-a^7 z^5+6 a^5 z^5+24 a^3 z^5+31 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +6 a^6 z^4+9 a^4 z^4-7 a^2 z^4+6 z^4 a^{-2} -4 z^4+2 a^7 z^3-a^5 z^3-28 a^3 z^3-40 a z^3-13 z^3 a^{-1} +2 z^3 a^{-3} -3 a^6 z^2-8 a^4 z^2-5 a^2 z^2-z^2 a^{-2} -z^2-a^7 z+2 a^5 z+15 a^3 z+19 a z+7 z a^{-1} +a^6+3 a^4+3 a^2+2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 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      84   4
-2     66    0
-4    56     -1
-6   46      2
-8  25       -3
-10 14        3
-12 2         -2
-141          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
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

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