L11a155

From Knot Atlas
Jump to: navigation, search

L11a154.gif

L11a154

L11a156.gif

L11a156

Contents

L11a155.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a155 at Knotilus!


Link Presentations

[edit Notes on L11a155's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X14,6,15,5 X20,11,21,12 X22,18,7,17 X18,22,19,21 X16,13,17,14 X12,19,13,20 X4,16,5,15 X2738 X6,9,1,10
Gauss code {1, -10, 2, -9, 3, -11}, {10, -1, 11, -2, 4, -8, 7, -3, 9, -7, 5, -6, 8, -4, 6, -5}
A Braid Representative
BraidPart1.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.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a155 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^4-t(2)^4+t(1)^2 t(2)^3-7 t(1) t(2)^3+5 t(2)^3-5 t(1)^2 t(2)^2+13 t(1) t(2)^2-5 t(2)^2+5 t(1)^2 t(2)-7 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1) t(2)^2} (db)
Jones polynomial -q^{7/2}+3 q^{5/2}-7 q^{3/2}+11 \sqrt{q}-\frac{15}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7+2 z^3 a^5+z a^5+a^5 z^{-1} -z^5 a^3-2 z a^3-2 a^3 z^{-1} -z^5 a+z^3 a+2 z a+2 a z^{-1} +2 z^3 a^{-1} - a^{-1} z^{-1} -z a^{-3} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-6 a^3 z^9-3 a z^9-4 a^6 z^8-7 a^4 z^8-8 a^2 z^8-5 z^8-3 a^7 z^7+a^5 z^7+5 a^3 z^7-4 a z^7-5 z^7 a^{-1} -a^8 z^6+9 a^6 z^6+19 a^4 z^6+17 a^2 z^6-3 z^6 a^{-2} +5 z^6+9 a^7 z^5+12 a^5 z^5+10 a^3 z^5+16 a z^5+8 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-4 a^6 z^4-14 a^4 z^4-13 a^2 z^4+5 z^4 a^{-2} -z^4-8 a^7 z^3-15 a^5 z^3-18 a^3 z^3-18 a z^3-5 z^3 a^{-1} +2 z^3 a^{-3} -2 a^8 z^2+3 a^4 z^2+4 a^2 z^2-2 z^2 a^{-2} +z^2+3 a^7 z+7 a^5 z+10 a^3 z+10 a z+3 z a^{-1} -z a^{-3} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} 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 \
-7-6-5-4-3-2-101234χ
8           11
6          2 -2
4         51 4
2        62  -4
0       95   4
-2      97    -2
-4     88     0
-6    710      3
-8   47       -3
-10  27        5
-12 14         -3
-14 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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.

L11a154.gif

L11a154

L11a156.gif

L11a156