L10n24

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L10n23

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L10n25

Contents

L10n24.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n24 at Knotilus!


Link Presentations

[edit Notes on L10n24's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X9,17,10,16 X17,9,18,8 X15,20,16,5 X19,14,20,15 X2536 X4,12,1,11
Gauss code {1, -9, 2, -10}, {9, -1, 3, 6, -5, -2, 10, -3, -4, 8, -7, 5, -6, 4, -8, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10n24 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^2+1\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial q^{9/2}-2 q^{7/2}+2 q^{5/2}-3 q^{3/2}+2 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z^3 a^{-3} +2 z a^{-3} -z^5 a^{-1} +a z^3-4 z^3 a^{-1} +2 a z-4 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^6 a^{-4} -4 z^4 a^{-4} +3 z^2 a^{-4} +2 z^7 a^{-3} -9 z^5 a^{-3} +10 z^3 a^{-3} -4 z a^{-3} +z^8 a^{-2} -3 z^6 a^{-2} -z^4 a^{-2} +a^2 z^2+3 z^2 a^{-2} +a z^7+3 z^7 a^{-1} -5 a z^5-14 z^5 a^{-1} +8 a z^3+18 z^3 a^{-1} -4 a z-8 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^8-4 z^6+3 z^4+z^2-1 (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
10       1-1
8      1 1
6     11 0
4    21  1
2  111   1
0  32    1
-2 12     1
-4 1      -1
-61       1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{3} {\mathbb Z}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\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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L10n23

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L10n25