L11n254

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L11n253.gif

L11n253

L11n255.gif

L11n255

Contents

L11n254.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n254 at Knotilus!


Link Presentations

[edit Notes on L11n254's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X22,7,11,8 X9,18,10,19 X17,20,18,21 X19,10,20,1 X8,11,9,12 X4,15,5,16 X6,21,7,22
Gauss code {1, -2, 3, -10, 4, -11, 5, -9, -6, 8}, {9, -1, 2, -3, 10, -4, -7, 6, -8, 7, 11, -5}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11n254 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^4 v^4+u^3 v^2+u^2 v^3-u^2 v^2+u^2 v+u v^2+1}{u^2 v^2} (db)
Jones polynomial -\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{27/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial -z^3 a^{13}-2 z a^{13}+z^7 a^{11}+8 z^5 a^{11}+19 z^3 a^{11}+14 z a^{11}+a^{11} z^{-1} -z^9 a^9-9 z^7 a^9-28 z^5 a^9-36 z^3 a^9-17 z a^9-a^9 z^{-1} (db)
Kauffman polynomial a^{17} z^3-2 a^{17} z+a^{16} z^4-a^{16} z^2+a^{15} z^5-2 a^{15} z^3+2 a^{15} z+a^{14} z^4+2 a^{13} z+a^{12} z^8-8 a^{12} z^6+19 a^{12} z^4-13 a^{12} z^2+a^{11} z^9-9 a^{11} z^7+27 a^{11} z^5-33 a^{11} z^3+15 a^{11} z-a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+19 a^{10} z^4-14 a^{10} z^2+a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-36 a^9 z^3+17 a^9 z-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 \
-10-9-8-7-6-5-4-3-2-10χ
-8          11
-10          11
-12        1  1
-14      1    1
-16     111   -1
-18    11     0
-20   111     -1
-22  11       0
-24  1        1
-2611         0
-281          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-10 i=-8 i=-6
r=-10 {\mathbb Z} {\mathbb Z}
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}_2 {\mathbb Z}
r=-1
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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L11n253.gif

L11n253

L11n255.gif

L11n255