L10n62

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L10n61

L10n63

Contents

Image:L10n62.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n62's page at Knotilus.

Visit L10n62's page at the original Knot Atlas.


[edit] Link Presentations

[edit Notes on L10n62's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,8,17,7 X5,1,6,10 X3746 X9,5,10,4 X18,14,19,13 X20,16,11,15 X14,20,15,19 X2,11,3,12 X8,18,9,17
Gauss code {1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, {9, -1, 6, -8, 7, -2, 10, -6, 8, -7}
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart2.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gif
A Morse Link Presentation Image:L10n62_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) v3u3 + v2u3vu3 + u3 + v3u2u2v3u + u + v3v2 + v−1 (db)
Jones polynomial -q^{17/2}+3 q^{15/2}-3 q^{13/2}+4 q^{11/2}-4 q^{9/2}+3 q^{7/2}-4 q^{5/2}+q^{3/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial z7a−5 + z5a−3−6z5a−5 + z5a−7 + 5z3a−3−13z3a−5 + 5z3a−7 + 8za−3−15za−5 + 8za−7za−9 + 4a−3z−1−8a−5z−1 + 5a−7z−1a−9z−1 (db)
Kauffman polynomial z8a−4z8a−6z7a−3−5z7a−5−4z7a−7 + 3z6a−4−2z6a−6−5z6a−8 + 6z5a−3 + 24z5a−5 + 16z5a−7−2z5a−9 + 3z4a−4 + 23z4a−6 + 20z4a−8−13z3a−3−35z3a−5−18z3a−7 + 4z3a−9−13z2a−4−33z2a−6−23z2a−8−3z2a−10 + 12za−3 + 23za−5 + 11za−7za−9za−11 + 8a−4 + 14a−6 + 9a−8 + 2a−10−4a−3z−1−8a−5z−1−5a−7z−1a−9z−1 (db)

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 5 is the signature of L10n62. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n62/KhovanovTable
Integral Khovanov Homology

(db, data source)

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

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[edit] 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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L10n61

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