L11a133

From Knot Atlas
Jump to: navigation, search

L11a132.gif

L11a132

L11a134.gif

L11a134

Contents

L11a133.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a133 at Knotilus!


Link Presentations

[edit Notes on L11a133's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X18,11,19,12 X22,19,5,20 X20,14,21,13 X12,22,13,21 X2,9,3,10 X8,15,9,16
Gauss code {1, -10, 5, -3}, {4, -1, 2, -11, 10, -4, 6, -9, 8, -5, 11, -2, 3, -6, 7, -8, 9, -7}
A Braid Representative
BraidPart1.gifBraidPart0.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.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.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.gif
A Morse Link Presentation L11a133 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^5+4 t(1) t(2)^4-6 t(2)^4-14 t(1) t(2)^3+14 t(2)^3+14 t(1) t(2)^2-14 t(2)^2-6 t(1) t(2)+4 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial \frac{25}{q^{9/2}}-\frac{22}{q^{7/2}}+\frac{15}{q^{5/2}}-\frac{10}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{5}{q^{19/2}}+\frac{10}{q^{17/2}}-\frac{16}{q^{15/2}}+\frac{22}{q^{13/2}}-\frac{25}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^9+z a^9+a^9 z^{-1} +z^5 a^7-2 z^3 a^7-5 z a^7-3 a^7 z^{-1} +3 z^5 a^5+6 z^3 a^5+7 z a^5+4 a^5 z^{-1} +z^5 a^3-2 z^3 a^3-5 z a^3-2 a^3 z^{-1} -z^3 a (db)
Kauffman polynomial -z^6 a^{12}+z^4 a^{12}-5 z^7 a^{11}+10 z^5 a^{11}-4 z^3 a^{11}-z a^{11}-9 z^8 a^{10}+20 z^6 a^{10}-12 z^4 a^{10}+z^2 a^{10}+a^{10}-7 z^9 a^9+4 z^7 a^9+17 z^5 a^9-14 z^3 a^9+3 z a^9-a^9 z^{-1} -2 z^{10} a^8-19 z^8 a^8+52 z^6 a^8-36 z^4 a^8+5 z^2 a^8+3 a^8-14 z^9 a^7+13 z^7 a^7+24 z^5 a^7-33 z^3 a^7+13 z a^7-3 a^7 z^{-1} -2 z^{10} a^6-21 z^8 a^6+52 z^6 a^6-38 z^4 a^6+5 z^2 a^6+3 a^6-7 z^9 a^5-5 z^7 a^5+33 z^5 a^5-36 z^3 a^5+16 z a^5-4 a^5 z^{-1} -11 z^8 a^4+17 z^6 a^4-11 z^4 a^4+z^2 a^4+2 a^4-9 z^7 a^3+15 z^5 a^3-12 z^3 a^3+7 z a^3-2 a^3 z^{-1} -4 z^6 a^2+4 z^4 a^2-z^5 a+z^3 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          3 -3
-2         71 6
-4        94  -5
-6       136   7
-8      129    -3
-10     1313     0
-12    1013      3
-14   612       -6
-16  410        6
-18 16         -5
-20 4          4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-6 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-5 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-4 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
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.

L11a132.gif

L11a132

L11a134.gif

L11a134