L10n23

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L10n22.gif

L10n22

L10n24.gif

L10n24

Contents

L10n23.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n23 at Knotilus!


Link Presentations

[edit Notes on L10n23's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X17,5,18,20 X13,19,14,18 X19,15,20,14 X8,16,9,15 X2536 X4,12,1,11
Gauss code {1, -9, 2, -10}, {9, -1, 3, -8, 4, -2, 10, -3, -6, 7, 8, -4, -5, 6, -7, 5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n23 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2)^4+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial 2 q^{15/2}-2 q^{13/2}+2 q^{11/2}-3 q^{9/2}+2 q^{7/2}-3 q^{5/2}+q^{3/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial -z a^{-9} +z^5 a^{-7} +5 z^3 a^{-7} +6 z a^{-7} +2 a^{-7} z^{-1} -z^7 a^{-5} -6 z^5 a^{-5} -12 z^3 a^{-5} -12 z a^{-5} -5 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +7 z a^{-3} +3 a^{-3} z^{-1} (db)
Kauffman polynomial  a^{-10} +z^3 a^{-9} -2 z a^{-9} +2 z^6 a^{-8} -8 z^4 a^{-8} +6 z^2 a^{-8} +3 z^7 a^{-7} -15 z^5 a^{-7} +21 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +z^8 a^{-6} -2 z^6 a^{-6} -7 z^4 a^{-6} +13 z^2 a^{-6} -5 a^{-6} +4 z^7 a^{-5} -21 z^5 a^{-5} +32 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +z^8 a^{-4} -4 z^6 a^{-4} +z^4 a^{-4} +7 z^2 a^{-4} -5 a^{-4} +z^7 a^{-3} -6 z^5 a^{-3} +12 z^3 a^{-3} -10 z a^{-3} +3 a^{-3} 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 \
-2-1012345χ
16       2-2
14      110
12     22 0
10    111 1
8   12   1
6  21    1
4 13     2
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}^{3} {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}

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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L10n22

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L10n24