L10a24

From Knot Atlas
Jump to: navigation, search

L10a23.gif

L10a23

L10a25.gif

L10a25

Contents

L10a24.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a24 at Knotilus!


Link Presentations

[edit Notes on L10a24's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X14,12,15,11 X10,4,11,3 X12,5,13,6 X20,13,5,14 X16,9,17,10 X2,16,3,15 X8,17,9,18
Gauss code {1, -9, 5, -3}, {6, -1, 2, -10, 8, -5, 4, -6, 7, -4, 9, -8, 10, -2, 3, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a24 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{12}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)+2 a^5 z^3+a^5 z-a^5 z^{-1} -a^3 z^5+2 a^3 z+3 a^3 z^{-1} -a z^5-a z^3+z^3 a^{-1} -2 a z-2 a z^{-1} (db)
Kauffman polynomial a^8 z^6-3 a^8 z^4+2 a^8 z^2+3 a^7 z^7-9 a^7 z^5+7 a^7 z^3-2 a^7 z+4 a^6 z^8-11 a^6 z^6+9 a^6 z^4-4 a^6 z^2+a^6+2 a^5 z^9+2 a^5 z^7-16 a^5 z^5+14 a^5 z^3-2 a^5 z-a^5 z^{-1} +10 a^4 z^8-26 a^4 z^6+24 a^4 z^4-11 a^4 z^2+3 a^4+2 a^3 z^9+7 a^3 z^7-21 a^3 z^5+12 a^3 z^3+3 a^3 z-3 a^3 z^{-1} +6 a^2 z^8-7 a^2 z^6+4 a^2 z^4+z^4 a^{-2} -5 a^2 z^2+3 a^2+8 a z^7-10 a z^5+4 z^5 a^{-1} +2 a z^3-3 z^3 a^{-1} +3 a z-2 a z^{-1} +7 z^6-7 z^4 (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 \
-7-6-5-4-3-2-10123χ
6          1-1
4         3 3
2        41 -3
0       73  4
-2      76   -1
-4     75    2
-6    57     2
-8   47      -3
-10  25       3
-12 14        -3
-14 2         2
-161          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
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.

L10a23.gif

L10a23

L10a25.gif

L10a25