L10n110

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L10n109.gif

L10n109

L10n111.gif

L10n111

Contents

L10n110.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n110's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X13,17,14,20 X19,11,20,16 X7,19,8,18 X15,8,16,9 X9,14,10,15 X17,5,18,10 X2536 X11,1,12,4
Gauss code {1, -9, -2, 10}, {-8, 5, -4, 3}, {9, -1, -5, 6, -7, 8}, {-10, 2, -3, 7, -6, 4}
A Braid Representative
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A Morse Link Presentation L10n110 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^2-t(1) t(2) t(3)^2-t(1) t(4) t(3)^2+t(1) t(2) t(4) t(3)^2+t(4) t(3)^2-t(1) t(4)^2 t(3)-t(2) t(4)^2 t(3)+t(4)^2 t(3)-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+2 t(1) t(4) t(3)-2 t(1) t(2) t(4) t(3)+2 t(2) t(4) t(3)-2 t(4) t(3)+t(2) t(4)^2-t(4)^2+t(1) t(2) t(4)-t(2) t(4)+t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} (db)
Jones polynomial -6 q^{9/2}+6 q^{7/2}-10 q^{5/2}+7 q^{3/2}-\frac{3}{q^{3/2}}-q^{13/2}+3 q^{11/2}-8 \sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-5} - a^{-5} z^{-3} -z a^{-5} -2 a^{-5} z^{-1} +z^5 a^{-3} +3 z^3 a^{-3} +3 a^{-3} z^{-3} +7 z a^{-3} +7 a^{-3} z^{-1} -4 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a z-9 z a^{-1} +3 a z^{-1} -8 a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^8 a^{-2} -2 z^8 a^{-4} -5 z^7 a^{-1} -9 z^7 a^{-3} -4 z^7 a^{-5} -z^6 a^{-2} -z^6 a^{-4} -3 z^6 a^{-6} -3 z^6+17 z^5 a^{-1} +27 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} +11 z^4 a^{-2} +11 z^4 a^{-4} +6 z^4 a^{-6} +6 z^4-6 a z^3-33 z^3 a^{-1} -36 z^3 a^{-3} -7 z^3 a^{-5} +2 z^3 a^{-7} -23 z^2 a^{-2} -13 z^2 a^{-4} -z^2 a^{-6} -11 z^2+11 a z+27 z a^{-1} +25 z a^{-3} +8 z a^{-5} -z a^{-7} +19 a^{-2} +10 a^{-4} +10-5 a z^{-1} -12 a^{-1} z^{-1} -12 a^{-3} z^{-1} -5 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3} (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 \
-2-10123456χ
14        11
12       2 -2
10      41 3
8     44  0
6    62   4
4   36    3
2  54     1
0 15      4
-223       -1
-43        3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-2 {\mathbb Z}^{3} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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L10n109.gif

L10n109

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L10n111