L11n419

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L11n418.gif

L11n418

L11n420.gif

L11n420

Contents

L11n419.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n419 at Knotilus!


Link Presentations

[edit Notes on L11n419's Link Presentations]

Planar diagram presentation X8192 X16,8,17,7 X3,10,4,11 X2,18,3,17 X9,19,10,18 X20,12,21,11 X5,14,6,15 X15,13,16,22 X13,6,14,1 X19,5,20,4 X12,22,7,21
Gauss code {1, -4, -3, 10, -7, 9}, {2, -1, -5, 3, 6, -11}, {-9, 7, -8, -2, 4, 5, -10, -6, 11, 8}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n419 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2) t(3)^3-t(2) t(3)^3+t(1) t(2)^2 t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2-t(1) t(3)-t(1)^2 t(2) t(3)+2 t(1) t(2) t(3)+t(1)^2 t(2)-t(1) t(2)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial -q^6+3 q^5-3 q^4+5 q^3-4 q^2+4 q+2 q^{-1} -2 (db)
Signature 2 (db)
HOMFLY-PT polynomial -2 z^4 a^{-2} -7 z^2 a^{-2} +3 z^2 a^{-4} +2 z^2-8 a^{-2} +5 a^{-4} - a^{-6} +4-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} (db)
Kauffman polynomial z^3 a^{-7} -z a^{-7} +3 z^4 a^{-6} -4 z^2 a^{-6} + a^{-6} +z^7 a^{-5} -3 z^5 a^{-5} +7 z^3 a^{-5} -3 z a^{-5} +z^8 a^{-4} -5 z^6 a^{-4} +16 z^4 a^{-4} -18 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4} +2 z^7 a^{-3} -6 z^5 a^{-3} +11 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +z^8 a^{-2} -5 z^6 a^{-2} +16 z^4 a^{-2} -23 z^2 a^{-2} -2 a^{-2} z^{-2} +12 a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +5 z^3 a^{-1} -6 z a^{-1} +2 a^{-1} z^{-1} +3 z^4-9 z^2- z^{-2} +6 (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 \
-2-1012345χ
13       1-1
11      2 2
9     22 0
7    31  2
5   23   1
3  22    0
1 13     2
-111      0
-32       2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-2 {\mathbb Z}^{2} {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_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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L11n418.gif

L11n418

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L11n420