L10n62

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L10n61.gif

L10n61

L10n63.gif

L10n63

Contents

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

Visit L10n62 at Knotilus!


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
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n62 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2+t(1)^2-t(2) t(1)+t(2)^2+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -4 q^{9/2}+3 q^{7/2}-4 q^{5/2}+q^{3/2}-q^{17/2}+3 q^{15/2}-3 q^{13/2}+4 q^{11/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial -z a^{-9} - a^{-9} z^{-1} +z^5 a^{-7} +5 z^3 a^{-7} +8 z a^{-7} +5 a^{-7} z^{-1} -z^7 a^{-5} -6 z^5 a^{-5} -13 z^3 a^{-5} -15 z a^{-5} -8 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +8 z a^{-3} +4 a^{-3} z^{-1} (db)
Kauffman polynomial -z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -5 z^7 a^{-5} -4 z^7 a^{-7} +3 z^6 a^{-4} -2 z^6 a^{-6} -5 z^6 a^{-8} +6 z^5 a^{-3} +24 z^5 a^{-5} +16 z^5 a^{-7} -2 z^5 a^{-9} +3 z^4 a^{-4} +23 z^4 a^{-6} +20 z^4 a^{-8} -13 z^3 a^{-3} -35 z^3 a^{-5} -18 z^3 a^{-7} +4 z^3 a^{-9} -13 z^2 a^{-4} -33 z^2 a^{-6} -23 z^2 a^{-8} -3 z^2 a^{-10} +12 z a^{-3} +23 z a^{-5} +11 z a^{-7} -z a^{-9} -z a^{-11} +8 a^{-4} +14 a^{-6} +9 a^{-8} +2 a^{-10} -4 a^{-3} z^{-1} -8 a^{-5} z^{-1} -5 a^{-7} z^{-1} - a^{-9} 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 \
-2-10123456χ
18        11
16       2 -2
14      22 0
12     32  -1
10    121  0
8   23    1
6  21     1
4 14      3
2         0
01        1
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}

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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