L11n246

From Knot Atlas
Jump to: navigation, search

L11n245.gif

L11n245

L11n247.gif

L11n247

Contents

L11n246.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n246 at Knotilus!


Link Presentations

[edit Notes on L11n246's Link Presentations]

Planar diagram presentation X12,1,13,2 X3849 X5,14,6,15 X7,18,8,19 X9,21,10,20 X10,11,1,12 X13,6,14,7 X17,4,18,5 X15,11,16,22 X19,3,20,2 X21,17,22,16
Gauss code {1, 10, -2, 8, -3, 7, -4, 2, -5, -6}, {6, -1, -7, 3, -9, 11, -8, 4, -10, 5, -11, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n246 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u+v) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+5 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -q^{3/2}+4 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)+3 a^5 z^3+4 a^5 z+a^5 z^{-1} -2 a^3 z^5-6 a^3 z^3-7 a^3 z-a^3 z^{-1} +3 a z^3+3 a z-z a^{-1} (db)
Kauffman polynomial -2 a^5 z^9-2 a^3 z^9-5 a^6 z^8-9 a^4 z^8-4 a^2 z^8-4 a^7 z^7-4 a^5 z^7-2 a^3 z^7-2 a z^7-a^8 z^6+12 a^6 z^6+23 a^4 z^6+10 a^2 z^6+11 a^7 z^5+24 a^5 z^5+15 a^3 z^5+2 a z^5+2 a^8 z^4-3 a^6 z^4-15 a^4 z^4-14 a^2 z^4-4 z^4-7 a^7 z^3-20 a^5 z^3-19 a^3 z^3-7 a z^3-z^3 a^{-1} -a^8 z^2-a^6 z^2+2 a^4 z^2+4 a^2 z^2+2 z^2+a^7 z+7 a^5 z+8 a^3 z+3 a z+z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-1} (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 \
-7-6-5-4-3-2-1012χ
4         11
2        3 -3
0       51 4
-2      64  -2
-4     64   2
-6    56    1
-8   56     -1
-10  36      3
-12 14       -3
-14 3        3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\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.

L11n245.gif

L11n245

L11n247.gif

L11n247