L11a219

From Knot Atlas
Jump to: navigation, search

L11a218.gif

L11a218

L11a220.gif

L11a220

Contents

L11a219.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a219 at Knotilus!


Link Presentations

[edit Notes on L11a219's Link Presentations]

Planar diagram presentation X8192 X12,4,13,3 X20,10,21,9 X22,12,7,11 X10,22,11,21 X2738 X18,15,19,16 X6,14,1,13 X4,18,5,17 X16,6,17,5 X14,19,15,20
Gauss code {1, -6, 2, -9, 10, -8}, {6, -1, 3, -5, 4, -2, 8, -11, 7, -10, 9, -7, 11, -3, 5, -4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a219 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-3 t(1) t(2)^4+2 t(2)^4-3 t(1)^2 t(2)^3+7 t(1) t(2)^3-3 t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-3 t(1)^2 t(2)+7 t(1) t(2)-3 t(2)+2 t(1)^2-3 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{17/2}-3 q^{15/2}+7 q^{13/2}-11 q^{11/2}+14 q^{9/2}-17 q^{7/2}+15 q^{5/2}-14 q^{3/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -6 z a^{-5} -3 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +7 z^3 a^{-3} +7 z a^{-3} +2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +2 a z-4 z a^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -7 z^9 a^{-3} -4 z^9 a^{-5} -7 z^8 a^{-2} -11 z^8 a^{-4} -7 z^8 a^{-6} -3 z^8-a z^7+7 z^7 a^{-1} +13 z^7 a^{-3} -3 z^7 a^{-5} -8 z^7 a^{-7} +32 z^6 a^{-2} +33 z^6 a^{-4} +7 z^6 a^{-6} -6 z^6 a^{-8} +12 z^6+4 a z^5+2 z^5 a^{-1} +5 z^5 a^{-3} +22 z^5 a^{-5} +12 z^5 a^{-7} -3 z^5 a^{-9} -34 z^4 a^{-2} -24 z^4 a^{-4} +3 z^4 a^{-6} +7 z^4 a^{-8} -z^4 a^{-10} -15 z^4-5 a z^3-9 z^3 a^{-1} -12 z^3 a^{-3} -21 z^3 a^{-5} -11 z^3 a^{-7} +2 z^3 a^{-9} +11 z^2 a^{-2} +4 z^2 a^{-4} -7 z^2 a^{-6} -5 z^2 a^{-8} +z^2 a^{-10} +6 z^2+2 a z+2 z a^{-1} +5 z a^{-3} +10 z a^{-5} +5 z a^{-7} +3 a^{-4} +3 a^{-6} + a^{-8} -2 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} 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 \
-4-3-2-101234567χ
18           1-1
16          2 2
14         51 -4
12        62  4
10       85   -3
8      96    3
6     79     2
4    78      -1
2   48       4
0  26        -4
-2 14         3
-4 2          -2
-61           1
Integral Khovanov Homology

(db, data source)

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

L11a218.gif

L11a218

L11a220.gif

L11a220