L11n53

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L11n52.gif

L11n52

L11n54.gif

L11n54

Contents

L11n53.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n53 at Knotilus!


Link Presentations

[edit Notes on L11n53's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X13,20,14,21 X17,5,18,22 X21,19,22,18 X19,14,20,15 X15,8,16,9 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, 9, -4, -2, 11, -3, -5, 8, -9, 4, -6, 7, -8, 5, -7, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11n53 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+\frac{1}{q^{7/2}}+4 q^{5/2}-\frac{3}{q^{5/2}}-5 q^{3/2}+\frac{4}{q^{3/2}}+5 \sqrt{q}-\frac{6}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z^5 a^{-1} +2 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -a^3 z+3 a z-3 z a^{-1} +z a^{-3} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^6 a^{-4} -3 z^4 a^{-4} +z^2 a^{-4} +3 z^7 a^{-3} -11 z^5 a^{-3} +2 a^3 z^3+8 z^3 a^{-3} -2 a^3 z-2 z a^{-3} +a^2 z^8+3 z^8 a^{-2} -5 a^2 z^6-11 z^6 a^{-2} +9 a^2 z^4+8 z^4 a^{-2} -3 a^2 z^2-z^2 a^{-2} +a z^9+z^9 a^{-1} -3 a z^7-a z^5-12 z^5 a^{-1} +10 a z^3+16 z^3 a^{-1} -6 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +4 z^8-17 z^6+20 z^4-5 z^2-1 (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 \
-3-2-1012345χ
10        1-1
8       2 2
6      21 -1
4     32  1
2   132   0
0   43    1
-2  24     2
-4 12      -1
-6 2       2
-81        -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4} {\mathbb Z}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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L11n52.gif

L11n52

L11n54.gif

L11n54