L10n41

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L10n40.gif

L10n40

L10n42.gif

L10n42

Contents

L10n41.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n41 at Knotilus!


Link Presentations

[edit Notes on L10n41's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X20,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X11,16,12,17 X17,6,18,1 X14,19,15,20 X18,13,19,14
Gauss code {1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -10, 9, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n41 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^2-2 u^2 v+u^2-u v^2+3 u v-u+v^2-2 v+1}{u v} (db)
Jones polynomial -\frac{4}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{3}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7-a^7 z^{-1} +2 z^3 a^5+5 z a^5+3 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-6 z a^3-2 a^3 z^{-1} +z^3 a+z a (db)
Kauffman polynomial a^8 z^6-4 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^7-8 a^7 z^5+8 a^7 z^3-3 a^7 z+a^7 z^{-1} +a^6 z^8-a^6 z^6-7 a^6 z^4+9 a^6 z^2-3 a^6+4 a^5 z^7-15 a^5 z^5+17 a^5 z^3-10 a^5 z+3 a^5 z^{-1} +a^4 z^8-a^4 z^6-4 a^4 z^4+6 a^4 z^2-3 a^4+2 a^3 z^7-7 a^3 z^5+12 a^3 z^3-9 a^3 z+2 a^3 z^{-1} +a^2 z^6-a^2 z^4+2 a^2 z^2+3 a z^3-2 a z+z^2 (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 \
-7-6-5-4-3-2-101χ
2        1-1
0       2 2
-2      22 0
-4     31  2
-6    23   1
-8   22    0
-10  12     1
-12 12      -1
-14 1       1
-161        -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\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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L10n40.gif

L10n40

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L10n42