L11n252

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L11n251.gif

L11n251

L11n253.gif

L11n253

Contents

L11n252.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n252's Link Presentations]

Planar diagram presentation X12,1,13,2 X14,3,15,4 X9,18,10,19 X5,16,6,17 X22,7,11,8 X6,21,7,22 X15,21,16,20 X17,8,18,9 X19,4,20,5 X2,11,3,12 X10,13,1,14
Gauss code {1, -10, 2, 9, -4, -6, 5, 8, -3, -11}, {10, -1, 11, -2, -7, 4, -8, 3, -9, 7, 6, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n252 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(2)^2 t(1)^4-t(2)^3 t(1)^3+t(2)^2 t(1)^3-t(1)^3-t(2)^2 t(1)^2-t(2)^4 t(1)+t(2)^2 t(1)-t(2) t(1)-t(2)^2}{t(1)^2 t(2)^2} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}-\frac{1}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^{11} z+a^{11} z^{-1} -a^9 z^{-1} -a^7 z^5-4 a^7 z^3-3 a^7 z-a^5 z^5-4 a^5 z^3-3 a^5 z (db)
Kauffman polynomial -z^7 a^{13}+7 z^5 a^{13}-14 z^3 a^{13}+8 z a^{13}+z^4 a^{12}-2 z^2 a^{12}+z^5 a^{11}-3 z^3 a^{11}+a^{11} z^{-1} -z^2 a^{10}-a^{10}-z^5 a^9+2 z^3 a^9-2 z a^9+a^9 z^{-1} -z^6 a^8+2 z^4 a^8-z^7 a^7+4 z^5 a^7-5 z^3 a^7+3 z a^7-z^6 a^6+3 z^4 a^6-z^2 a^6-z^5 a^5+4 z^3 a^5-3 z a^5 (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 \
-10-9-8-7-6-5-4-3-2-10χ
-4          11
-6         110
-8        1  1
-10       11  0
-12     121   0
-14     11    0
-16   122     -1
-18    1      1
-20  11       0
-221          1
-241          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4 i=-2
r=-10 {\mathbb Z} {\mathbb Z}
r=-9
r=-8 {\mathbb Z}
r=-7 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{2} {\mathbb Z} {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z} {\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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L11n251.gif

L11n251

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L11n253