L11a299

From Knot Atlas
Jump to: navigation, search

L11a298.gif

L11a298

L11a300.gif

L11a300

Contents

L11a299.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a299 at Knotilus!


Link Presentations

[edit Notes on L11a299's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X22,12,9,11 X16,6,17,5 X20,16,21,15 X14,22,15,21 X18,8,19,7 X2,9,3,10 X4,14,5,13 X6,18,7,17 X8,20,1,19
Gauss code {1, -8, 2, -9, 4, -10, 7, -11}, {8, -1, 3, -2, 9, -6, 5, -4, 10, -7, 11, -5, 6, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a299 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(2 t(2) t(1)-t(1)-t(2)+2) \left(t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+t(2) t(1)-t(1)-t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial q^{23/2}-3 q^{21/2}+6 q^{19/2}-10 q^{17/2}+12 q^{15/2}-13 q^{13/2}+13 q^{11/2}-11 q^{9/2}+7 q^{7/2}-5 q^{5/2}+2 q^{3/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial z^5 a^{-9} +3 z^3 a^{-9} +2 z a^{-9} -z^7 a^{-7} -4 z^5 a^{-7} -5 z^3 a^{-7} -3 z a^{-7} -z^7 a^{-5} -4 z^5 a^{-5} -4 z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +4 z^3 a^{-3} +4 z a^{-3} + a^{-3} z^{-1} (db)
Kauffman polynomial z^4 a^{-14} -z^2 a^{-14} +3 z^5 a^{-13} -3 z^3 a^{-13} +z a^{-13} +5 z^6 a^{-12} -5 z^4 a^{-12} +2 z^2 a^{-12} +6 z^7 a^{-11} -7 z^5 a^{-11} +3 z^3 a^{-11} +z a^{-11} +5 z^8 a^{-10} -5 z^6 a^{-10} +z^4 a^{-10} +3 z^9 a^{-9} -z^7 a^{-9} -3 z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +z^{10} a^{-8} +4 z^8 a^{-8} -12 z^6 a^{-8} +8 z^4 a^{-8} -5 z^2 a^{-8} +5 z^9 a^{-7} -13 z^7 a^{-7} +10 z^5 a^{-7} -6 z^3 a^{-7} +2 z a^{-7} +z^{10} a^{-6} +z^8 a^{-6} -10 z^6 a^{-6} +9 z^4 a^{-6} -2 z^2 a^{-6} +2 z^9 a^{-5} -5 z^7 a^{-5} -2 z^5 a^{-5} +9 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} -8 z^6 a^{-4} +8 z^4 a^{-4} - a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +8 z^3 a^{-3} -5 z a^{-3} + a^{-3} 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 \
-2-10123456789χ
24           1-1
22          2 2
20         41 -3
18        62  4
16       64   -2
14      76    1
12     77     0
10    46      -2
8   37       4
6  24        -2
4 14         3
2 1          -1
01           1
Integral Khovanov Homology

(db, data source)

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

L11a298.gif

L11a298

L11a300.gif

L11a300