L11n5

From Knot Atlas
Jump to: navigation, search

L11n4.gif

L11n4

L11n6.gif

L11n6

Contents

L11n5.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n5 at Knotilus!


Link Presentations

[edit Notes on L11n5's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X17,1,18,4 X9,14,10,15 X3849 X5,11,6,10 X11,19,12,18 X13,20,14,21 X19,5,20,22 X21,12,22,13 X2,16,3,15
Gauss code {1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -8, 4, 11, -2, -3, 7, -9, 8, -10, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11n5 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^{7/2}-3 q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} -2 a^3 z^3-5 a^3 z-3 a^3 z^{-1} +z a^{-3} +a z^5+4 a z^3-2 z^3 a^{-1} +7 a z+4 a z^{-1} -4 z a^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-2 a^4 z^8-5 a^2 z^8-3 z^8-a^5 z^7+a^3 z^7-a z^7-3 z^7 a^{-1} +10 a^4 z^6+24 a^2 z^6-z^6 a^{-2} +13 z^6+5 a^5 z^5+13 a^3 z^5+21 a z^5+13 z^5 a^{-1} -14 a^4 z^4-33 a^2 z^4+z^4 a^{-2} -18 z^4-7 a^5 z^3-26 a^3 z^3-37 a z^3-21 z^3 a^{-1} -3 z^3 a^{-3} +7 a^4 z^2+17 a^2 z^2-z^2 a^{-4} +11 z^2+4 a^5 z+16 a^3 z+22 a z+13 z a^{-1} +3 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 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 \
-6-5-4-3-2-10123χ
8         1-1
6        2 2
4       22 0
2      42  2
0     341  0
-2    341   0
-4   23     1
-6  131     -1
-8 12       1
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z} {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z} {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\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).

Back to the top.

L11n4.gif

L11n4

L11n6.gif

L11n6