L11a469

From Knot Atlas
Jump to: navigation, search

L11a468.gif

L11a468

L11a470.gif

L11a470

Contents

L11a469.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a469 at Knotilus!


Link Presentations

[edit Notes on L11a469's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,11,19,12 X16,8,17,7 X8,16,9,15 X22,17,15,18 X12,21,13,22 X20,13,21,14 X14,19,5,20 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {5, -4, 6, -3, 9, -8, 7, -6}, {10, -1, 4, -5, 11, -2, 3, -7, 8, -9}
A Braid Representative
BraidPart1.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.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11a469 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(w-1) \left(2 u v^2 w-u v^2+2 u v w^2-4 u v w+u v-2 u w^2+u w+v^2 w^2-2 v^2 w+v w^3-4 v w^2+2 v w-w^3+2 w^2\right)}{\sqrt{u} v w^2} (db)
Jones polynomial - q^{-10} +3 q^{-9} -6 q^{-8} +11 q^{-7} -13 q^{-6} +17 q^{-5} -16 q^{-4} +15 q^{-3} -11 q^{-2} +q+7 q^{-1} -3 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^{10}+3 a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^4-a^6 z^2-2 a^6 z^{-2} -2 a^6-3 a^4 z^4-3 a^4 z^2+a^4 z^{-2} -a^4-a^2 z^4+2 a^2 z^2+2 a^2+z^2 (db)
Kauffman polynomial a^{11} z^7-4 a^{11} z^5+5 a^{11} z^3-2 a^{11} z+3 a^{10} z^8-12 a^{10} z^6+16 a^{10} z^4-9 a^{10} z^2+3 a^{10}+3 a^9 z^9-6 a^9 z^7-7 a^9 z^5+18 a^9 z^3-7 a^9 z+a^8 z^{10}+8 a^8 z^8-38 a^8 z^6+51 a^8 z^4-34 a^8 z^2-a^8 z^{-2} +12 a^8+7 a^7 z^9-11 a^7 z^7-13 a^7 z^5+25 a^7 z^3-14 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+12 a^6 z^8-38 a^6 z^6+39 a^6 z^4-28 a^6 z^2-2 a^6 z^{-2} +12 a^6+4 a^5 z^9+4 a^5 z^7-24 a^5 z^5+22 a^5 z^3-10 a^5 z+2 a^5 z^{-1} +7 a^4 z^8-6 a^4 z^6-4 a^4 z^4+4 a^4 z^2-a^4 z^{-2} +2 a^4+8 a^3 z^7-11 a^3 z^5+8 a^3 z^3-a^3 z+6 a^2 z^6-7 a^2 z^4+6 a^2 z^2-2 a^2+3 a z^5-2 a z^3+z^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 \
-9-8-7-6-5-4-3-2-1012χ
3           11
1          2 -2
-1         51 4
-3        73  -4
-5       84   4
-7      98    -1
-9     87     1
-11    610      4
-13   57       -2
-15  27        5
-17 14         -3
-19 2          2
-211           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
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}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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.

L11a468.gif

L11a468

L11a470.gif

L11a470