L11n263

From Knot Atlas
Jump to: navigation, search

L11n262.gif

L11n262

L11n264.gif

L11n264

Contents

L11n263.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n263 at Knotilus!


Link Presentations

[edit Notes on L11n263's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X22,20,9,19 X20,16,21,15 X16,22,17,21 X17,12,18,13 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, 7, -8, -9, 5, 6, -7, 8, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n263 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(3)^3+t(1) t(2) t(3)^3-t(2) t(3)^3+t(3)^3+3 t(1) t(3)^2-2 t(1) t(2) t(3)^2+3 t(2) t(3)^2-3 t(3)^2-3 t(1) t(3)+3 t(1) t(2) t(3)-3 t(2) t(3)+2 t(3)+t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial  q^{-7} -2 q^{-6} +7 q^{-5} -8 q^{-4} +11 q^{-3} -q^2-10 q^{-2} +4 q+9 q^{-1} -7 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^8 z^{-2} -2 a^6 z^{-2} -3 a^6-a^4 z^4+a^4 z^2+a^4 z^{-2} +3 a^4+a^2 z^6+3 a^2 z^4+3 a^2 z^2-z^4-z^2 (db)
Kauffman polynomial a^5 z^9+a^3 z^9+a^6 z^8+5 a^4 z^8+4 a^2 z^8-a^5 z^7+5 a^3 z^7+6 a z^7-3 a^6 z^6-12 a^4 z^6-5 a^2 z^6+4 z^6+2 a^7 z^5+3 a^5 z^5-12 a^3 z^5-12 a z^5+z^5 a^{-1} +a^8 z^4+12 a^6 z^4+17 a^4 z^4-a^2 z^4-7 z^4-a^7 z^3+2 a^5 z^3+7 a^3 z^3+3 a z^3-z^3 a^{-1} -3 a^8 z^2-12 a^6 z^2-10 a^4 z^2+z^2-3 a^7 z-3 a^5 z+3 a^8+5 a^6+3 a^4+2 a^7 z^{-1} +2 a^5 z^{-1} -a^8 z^{-2} -2 a^6 z^{-2} -a^4 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
5         1-1
3        3 3
1       41 -3
-1      53  2
-3     65   -1
-5    54    1
-7   36     3
-9  45      -1
-11 16       5
-13 1        -1
-151         1
Integral Khovanov Homology

(db, data source)

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

L11n262.gif

L11n262

L11n264.gif

L11n264