L11n279

From Knot Atlas
Jump to: navigation, search

L11n278.gif

L11n278

L11n280.gif

L11n280

Contents

L11n279.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n279 at Knotilus!


Link Presentations

[edit Notes on L11n279's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X15,1,16,4 X10,20,5,19
Gauss code {1, -4, -3, 10}, {-2, -1, 5, 3, 6, -11}, {-8, 2, 4, -5, -10, 9, -7, -6, 11, 8, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11n279 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v w^2-2 u v w+u v-u w^2+u w+v^2 (-w)+v^2-v w^2+2 v w-2 v}{\sqrt{u} v w} (db)
Jones polynomial -2 q^6+3 q^5-4 q^4+6 q^3-4 q^2+5 q-2+2 q^{-1} (db)
Signature 2 (db)
HOMFLY-PT polynomial -2 z^4 a^{-2} -7 z^2 a^{-2} +4 z^2 a^{-4} +2 z^2-10 a^{-2} +8 a^{-4} -2 a^{-6} +4-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} (db)
Kauffman polynomial 3 z^3 a^{-7} -5 z a^{-7} + a^{-7} z^{-1} +z^6 a^{-6} -2 z^2 a^{-6} - a^{-6} z^{-2} +3 a^{-6} +2 z^7 a^{-5} -8 z^5 a^{-5} +20 z^3 a^{-5} -18 z a^{-5} +5 a^{-5} z^{-1} +z^8 a^{-4} -3 z^6 a^{-4} +10 z^4 a^{-4} -14 z^2 a^{-4} -4 a^{-4} z^{-2} +12 a^{-4} +3 z^7 a^{-3} -11 z^5 a^{-3} +24 z^3 a^{-3} -24 z a^{-3} +9 a^{-3} z^{-1} +z^8 a^{-2} -4 z^6 a^{-2} +13 z^4 a^{-2} -21 z^2 a^{-2} -5 a^{-2} z^{-2} +15 a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +7 z^3 a^{-1} -11 z a^{-1} +5 a^{-1} z^{-1} +3 z^4-9 z^2-2 z^{-2} +7 (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       2-2
11      1 1
9     32 -1
7    31  2
5   24   2
3  32    1
1 14     3
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}_2^{2} {\mathbb Z}^{2}

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.

L11n278.gif

L11n278

L11n280.gif

L11n280