L11a297

From Knot Atlas
Jump to: navigation, search

L11a296.gif

L11a296

L11a298.gif

L11a298

Contents

L11a297.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a297 at Knotilus!


Link Presentations

[edit Notes on L11a297's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,3,13,4 X14,21,15,22 X16,5,17,6 X18,8,19,7 X20,17,21,18 X4,15,5,16 X6,20,7,19 X22,13,9,14 X2,9,3,10 X8,11,1,12
Gauss code {1, -10, 2, -7, 4, -8, 5, -11}, {10, -1, 11, -2, 9, -3, 7, -4, 6, -5, 8, -6, 3, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a297 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^3 v^2-3 u^3 v+u^3+2 u^2 v^3-7 u^2 v^2+7 u^2 v-3 u^2-3 u v^3+7 u v^2-7 u v+2 u+v^3-3 v^2+2 v}{u^{3/2} v^{3/2}} (db)
Jones polynomial \frac{16}{q^{9/2}}-\frac{14}{q^{7/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{6}{q^{17/2}}-\frac{10}{q^{15/2}}+\frac{13}{q^{13/2}}-\frac{16}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^9-z a^9+z^5 a^7+z^3 a^7+2 z^5 a^5+4 z^3 a^5+3 z a^5+a^5 z^{-1} +z^5 a^3-3 z a^3-a^3 z^{-1} -z^3 a-z a (db)
Kauffman polynomial a^{12} z^6-3 a^{12} z^4+2 a^{12} z^2+3 a^{11} z^7-9 a^{11} z^5+7 a^{11} z^3-2 a^{11} z+4 a^{10} z^8-10 a^{10} z^6+6 a^{10} z^4-2 a^{10} z^2+3 a^9 z^9-3 a^9 z^7-6 a^9 z^5+7 a^9 z^3-3 a^9 z+a^8 z^{10}+6 a^8 z^8-19 a^8 z^6+17 a^8 z^4-5 a^8 z^2+6 a^7 z^9-9 a^7 z^7-a^7 z^5+10 a^7 z^3-3 a^7 z+a^6 z^{10}+7 a^6 z^8-18 a^6 z^6+16 a^6 z^4-2 a^6 z^2+3 a^5 z^9+2 a^5 z^7-14 a^5 z^5+18 a^5 z^3-8 a^5 z+a^5 z^{-1} +5 a^4 z^8-7 a^4 z^6+3 a^4 z^4-a^4+5 a^3 z^7-9 a^3 z^5+6 a^3 z^3-5 a^3 z+a^3 z^{-1} +3 a^2 z^6-5 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z (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 \
-9-8-7-6-5-4-3-2-1012χ
2           11
0          2 -2
-2         51 4
-4        63  -3
-6       84   4
-8      86    -2
-10     88     0
-12    69      3
-14   47       -3
-16  26        4
-18 14         -3
-20 2          2
-221           -1
Integral Khovanov Homology

(db, data source)

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

L11a296.gif

L11a296

L11a298.gif

L11a298