L11n27

From Knot Atlas
Jump to: navigation, search

L11n26.gif

L11n26

L11n28.gif

L11n28

Contents

L11n27.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n27 at Knotilus!


Link Presentations

[edit Notes on L11n27's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X17,1,18,4 X5,12,6,13 X3849 X13,22,14,5 X21,14,22,15 X11,18,12,19 X9,20,10,21 X19,10,20,11 X2,16,3,15
Gauss code {1, -11, -5, 3}, {-4, -1, 2, 5, -9, 10, -8, 4, -6, 7, 11, -2, -3, 8, -10, 9, -7, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n27 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{3 (u-1) (v-1)}{\sqrt{u} \sqrt{v}} (db)
Jones polynomial -\frac{4}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^9-a^9 z^{-1} +z^3 a^7+z a^7+a^7 z^{-1} +2 z^3 a^5+4 z a^5+2 a^5 z^{-1} -3 z a^3-2 a^3 z^{-1} -z a (db)
Kauffman polynomial -z^8 a^{10}+7 z^6 a^{10}-17 z^4 a^{10}+16 z^2 a^{10}-4 a^{10}-z^9 a^9+5 z^7 a^9-6 z^5 a^9+a^9 z^{-1} -4 z^8 a^8+23 z^6 a^8-42 z^4 a^8+31 z^2 a^8-9 a^8-z^9 a^7+z^7 a^7+13 z^5 a^7-22 z^3 a^7+5 z a^7+a^7 z^{-1} -3 z^8 a^6+14 z^6 a^6-18 z^4 a^6+11 z^2 a^6-4 a^6-4 z^7 a^5+19 z^5 a^5-25 z^3 a^5+13 z a^5-2 a^5 z^{-1} -2 z^6 a^4+7 z^4 a^4-5 z^2 a^4+2 a^4-3 z^3 a^3+7 z a^3-2 a^3 z^{-1} -z^2 a^2-z a (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 \
-9-8-7-6-5-4-3-2-10χ
0         11
-2        220
-4       1 12
-6      33  0
-8     21   1
-10    131   1
-12   32     1
-14   1      1
-16 13       -2
-18          0
-201         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2 i=0
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}_2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\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.

L11n26.gif

L11n26

L11n28.gif

L11n28