L11n122

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L11n121.gif

L11n121

L11n123.gif

L11n123

Contents

L11n122.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n122 at Knotilus!


Link Presentations

[edit Notes on L11n122's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X16,10,17,9 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X10,16,11,15 X22,17,5,18 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, -6, 5, 3, -8, -4, 7, 11, -2, 8, -3, 9, 6, -7, 4, -5, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n122 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-2 u v^2+2 u v+u+v^3+2 v^2-2 v}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -q^{11/2}+2 q^{9/2}-2 q^{7/2}+3 q^{5/2}-2 q^{3/2}+\sqrt{q}-\frac{2}{\sqrt{q}}-\frac{1}{q^{7/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z a^3+2 a^3 z^{-1} -z^3 a-5 z a-3 a z^{-1} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} +z^3 a^{-3} +z a^{-3} -z a^{-5} (db)
Kauffman polynomial -z^9 a^{-1} -z^9 a^{-3} -3 z^8 a^{-2} -2 z^8 a^{-4} -z^8+5 z^7 a^{-1} +4 z^7 a^{-3} -z^7 a^{-5} +17 z^6 a^{-2} +11 z^6 a^{-4} +6 z^6-a^3 z^5-5 z^5 a^{-1} -z^5 a^{-3} +5 z^5 a^{-5} -26 z^4 a^{-2} -17 z^4 a^{-4} -9 z^4+5 a^3 z^3+4 a z^3+2 z^3 a^{-1} -3 z^3 a^{-3} -6 z^3 a^{-5} +3 a^2 z^2+13 z^2 a^{-2} +9 z^2 a^{-4} +7 z^2-5 a^3 z-7 a z-3 z a^{-1} +z a^{-5} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} 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 \
-4-3-2-10123456χ
12          11
10         1 -1
8        11 0
6       21  -1
4     111   -1
2     12    1
0   131     1
-2    2      2
-4  11       0
-61          1
-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=-4 {\mathbb Z} {\mathbb Z}
r=-3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\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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L11n123