L11n357

From Knot Atlas
Jump to: navigation, search

L11n356.gif

L11n356

L11n358.gif

L11n358

Contents

L11n357.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n357 at Knotilus!


Link Presentations

[edit Notes on L11n357's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X11,19,12,18 X7,16,8,17 X15,8,16,9 X17,15,18,22 X13,21,14,20 X19,13,20,12 X21,5,22,14 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {-5, 4, -6, 3, -8, 7, -9, 6}, {10, -1, -4, 5, 11, -2, -3, 8, -7, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n357 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) \left(-u v^2 w-u v+u+v^2 w^3-v w^3-w^2\right)}{\sqrt{u} v w^2} (db)
Jones polynomial  q^{-5} -q^4- q^{-4} +2 q^3+3 q^{-3} -2 q^2-3 q^{-2} +4 q+4 q^{-1} -3 (db)
Signature 2 (db)
HOMFLY-PT polynomial a^4 z^2+a^4 z^{-2} +3 a^4-2 a^2 z^4-z^4 a^{-2} -8 a^2 z^2-3 z^2 a^{-2} -2 a^2 z^{-2} -8 a^2- a^{-2} +z^6+5 z^4+8 z^2+ z^{-2} +6 (db)
Kauffman polynomial a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8-5 a^3 z^7-2 a z^7+3 z^7 a^{-1} -7 a^4 z^6-24 a^2 z^6+2 z^6 a^{-2} -15 z^6+5 a^3 z^5-8 a z^5-12 z^5 a^{-1} +z^5 a^{-3} +17 a^4 z^4+47 a^2 z^4-6 z^4 a^{-2} +24 z^4+5 a^3 z^3+18 a z^3+11 z^3 a^{-1} -2 z^3 a^{-3} -17 a^4 z^2-38 a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -20 z^2-7 a^3 z-10 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +7 a^4+14 a^2- a^{-2} - a^{-4} +8+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2} (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
9         1-1
7        1 1
5       22 0
3      211 2
1     23   1
-1    221   1
-3   23     1
-5  11      0
-7 13       2
-9          0
-111         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3} {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
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.

L11n356.gif

L11n356

L11n358.gif

L11n358