L11a86

From Knot Atlas
Jump to: navigation, search

L11a85.gif

L11a85

L11a87.gif

L11a87

Contents

L11a86.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a86 at Knotilus!


Link Presentations

[edit Notes on L11a86's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X14,10,15,9 X10,14,11,13 X20,17,21,18 X18,7,19,8 X8,19,9,20 X22,15,5,16 X16,21,17,22 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 6, -7, 3, -4, 11, -2, 4, -3, 8, -9, 5, -6, 7, -5, 9, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a86 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(1) t(2)^3-4 t(2)^3-8 t(1) t(2)^2+9 t(2)^2+9 t(1) t(2)-8 t(2)-4 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{14}{q^{9/2}}-\frac{15}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +3 z a^9+2 a^9 z^{-1} -3 z^3 a^7-2 z a^7+z^5 a^5-z^3 a^5-3 z a^5-a^5 z^{-1} +z^5 a^3+z^3 a^3+z a^3-z^3 a-z a (db)
Kauffman polynomial -z^6 a^{12}+4 z^4 a^{12}-5 z^2 a^{12}+2 a^{12}-2 z^7 a^{11}+6 z^5 a^{11}-5 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -2 z^8 a^{10}+z^6 a^{10}+10 z^4 a^{10}-13 z^2 a^{10}+5 a^{10}-2 z^9 a^9+z^7 a^9+5 z^5 a^9-6 z^3 a^9+5 z a^9-2 a^9 z^{-1} -z^{10} a^8-2 z^8 a^8+3 z^6 a^8+4 z^4 a^8-6 z^2 a^8+3 a^8-5 z^9 a^7+9 z^7 a^7-9 z^5 a^7+4 z^3 a^7-z^{10} a^6-4 z^8 a^6+7 z^6 a^6-5 z^4 a^6+2 z^2 a^6-a^6-3 z^9 a^5+2 z^7 a^5-2 z^5 a^5+3 z^3 a^5-2 z a^5+a^5 z^{-1} -4 z^8 a^4+3 z^6 a^4+3 z^4 a^4-3 z^2 a^4-4 z^7 a^3+5 z^5 a^3-3 z^6 a^2+6 z^4 a^2-3 z^2 a^2-z^5 a+2 z^3 a-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 \
-9-8-7-6-5-4-3-2-1012χ
2           11
0          2 -2
-2         41 3
-4        73  -4
-6       73   4
-8      77    0
-10     87     1
-12    58      3
-14   47       -3
-16  15        4
-18 14         -3
-20 1          1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
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.

L11a85.gif

L11a85

L11a87.gif

L11a87