L11n61

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L11n60.gif

L11n60

L11n62.gif

L11n62

Contents

L11n61.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n61's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,8,15,7 X18,11,19,12 X19,5,20,22 X15,21,16,20 X21,17,22,16 X12,17,13,18 X8,14,9,13 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -8, 9, -3, -6, 7, 8, -4, -5, 6, -7, 5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11n61 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^5+t(1) t(2)^4-3 t(2)^4-2 t(1) t(2)^3+2 t(2)^3+2 t(1) t(2)^2-2 t(2)^2-3 t(1) t(2)+t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -2 q^{5/2}+4 q^{3/2}-5 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+2 a^5 z^{-1} -3 a^3 z^3-8 a^3 z-4 a^3 z^{-1} +2 a z^5+8 a z^3-2 z^3 a^{-1} +9 a z+3 a z^{-1} -4 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^5 z^7-5 a^5 z^5+9 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-8 a^4 z^6+7 a^4 z^4+a^4 z^2-2 a^4+a^3 z^9+a^3 z^7-19 a^3 z^5+29 a^3 z^3-16 a^3 z+z a^{-3} +4 a^3 z^{-1} +5 a^2 z^8-19 a^2 z^6+15 a^2 z^4+z^4 a^{-2} +a^2 z^2+z^2 a^{-2} -3 a^2- a^{-2} +a z^9+2 a z^7+2 z^7 a^{-1} -21 a z^5-7 z^5 a^{-1} +28 a z^3+8 z^3 a^{-1} -13 a z-3 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +3 z^8-11 z^6+9 z^4+z^2-3 (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 \
-6-5-4-3-2-1012χ
6        22
4       2 -2
2      32 1
0     43  -1
-2    231  0
-4   34    1
-6  22     0
-8 14      3
-10 1       -1
-121        1
Integral Khovanov Homology

(db, data source)

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

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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