L11n421

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L11n420.gif

L11n420

L11n422.gif

L11n422

Contents

L11n421.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n421 at Knotilus!


Link Presentations

[edit Notes on L11n421's Link Presentations]

Planar diagram presentation X8192 X16,8,17,7 X3,10,4,11 X17,2,18,3 X9,19,10,18 X20,12,21,11 X14,6,15,5 X22,15,13,16 X6,14,1,13 X4,19,5,20 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
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n421 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^2 t(3)^3-t(2)^2 t(3)^3-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2+t(1) t(2) t(3)^2-t(1)^2 t(3)+2 t(1) t(3)-t(1) t(2) t(3)+t(1)^2-t(1)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial q^5-q^4+3 q^3-3 q^2+4 q-3+4 q^{-1} -2 q^{-2} +2 q^{-3} - q^{-4} (db)
Signature 2 (db)
HOMFLY-PT polynomial z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -a^2 z^4-2 z^4 a^{-2} -3 a^2 z^2-8 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2-8 a^{-2} +z^6+5 z^4+8 z^2+ z^{-2} +6 (db)
Kauffman polynomial z^2 a^{-6} - a^{-6} +z^3 a^{-5} +2 z^4 a^{-4} -4 z^2 a^{-4} - a^{-4} z^{-2} +4 a^{-4} +a^3 z^7+z^7 a^{-3} -5 a^3 z^5-4 z^5 a^{-3} +6 a^3 z^3+7 z^3 a^{-3} -a^3 z-6 z a^{-3} +2 a^{-3} z^{-1} +2 a^2 z^8+2 z^8 a^{-2} -11 a^2 z^6-11 z^6 a^{-2} +18 a^2 z^4+23 z^4 a^{-2} -11 a^2 z^2-25 z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2+12 a^{-2} +a z^9+z^9 a^{-1} -3 a z^7-3 z^7 a^{-1} -3 a z^5-2 z^5 a^{-1} +9 a z^3+9 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-22 z^6+39 z^4-31 z^2- z^{-2} +10 (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 \
-5-4-3-2-101234χ
11         11
9        110
7       2  2
5      22  0
3     21   1
1    23    1
-1   21     1
-3  13      2
-5 11       0
-7 1        1
-91         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\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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L11n420.gif

L11n420

L11n422.gif

L11n422