L11n62

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L11n61.gif

L11n61

L11n63.gif

L11n63

Contents

L11n62.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n62 at Knotilus!


Link Presentations

[edit Notes on L11n62's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,14,8,15 X15,20,16,21 X11,18,12,19 X19,12,20,13 X17,22,18,5 X21,16,22,17 X13,8,14,9 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -4, 8, -7, 5, -6, 4, -8, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n62 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^5+t(2)^5-2 t(2)^4-2 t(1) t(2)^3+3 t(2)^3+3 t(1) t(2)^2-2 t(2)^2-2 t(1) t(2)+t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{1}{q^{7/2}}+\frac{1}{q^{27/2}}-\frac{2}{q^{25/2}}+\frac{4}{q^{23/2}}-\frac{5}{q^{21/2}}+\frac{5}{q^{19/2}}-\frac{5}{q^{17/2}}+\frac{4}{q^{15/2}}-\frac{3}{q^{13/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z a^{13}-a^{13} z^{-1} +2 z^3 a^{11}+3 z a^{11}+a^{11} z^{-1} +3 z^3 a^9+6 z a^9+2 a^9 z^{-1} -z^7 a^7-7 z^5 a^7-14 z^3 a^7-10 z a^7-2 a^7 z^{-1} (db)
Kauffman polynomial a^{16} z^6-4 a^{16} z^4+4 a^{16} z^2-a^{16}+2 a^{15} z^7-7 a^{15} z^5+5 a^{15} z^3-a^{15} z+2 a^{14} z^8-6 a^{14} z^6+2 a^{14} z^4+a^{13} z^9-2 a^{13} z^7-a^{13} z^5-a^{13} z^3+2 a^{13} z-a^{13} z^{-1} +3 a^{12} z^8-13 a^{12} z^6+20 a^{12} z^4-14 a^{12} z^2+3 a^{12}+a^{11} z^9-5 a^{11} z^7+10 a^{11} z^5-9 a^{11} z^3+5 a^{11} z-a^{11} z^{-1} +a^{10} z^8-6 a^{10} z^6+11 a^{10} z^4-3 a^{10} z^2-3 a^9 z^5+11 a^9 z^3-8 a^9 z+2 a^9 z^{-1} -3 a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^7-7 a^7 z^5+14 a^7 z^3-10 a^7 z+2 a^7 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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-6           11
-8           11
-10        11  0
-12       3    3
-14      331   -1
-16     32     1
-18    231     0
-20   33       0
-22  12        1
-24 13         -2
-26 1          1
-281           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-8 i=-6 i=-4
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=-3 {\mathbb Z} {\mathbb Z}_2 {\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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