Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

L11a1

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L10n113.gif

L10n113

L11a2.gif

L11a2

Contents

L11a1.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a1's page at Knotilus.

Visit L11a1's page at the original Knot Atlas.


Link L11a1.
A graph, link L11a1.
A part of a link and a part of a graph.

Link Presentations

[edit Notes on L11a1's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X20,12,21,11 X22,17,5,18 X18,21,19,22 X12,20,13,19 X16,10,17,9 X2,14,3,13
Gauss code {1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 11, -2, 3, -10, 7, -8, 9, -6, 8, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a1 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+3 a z^3-4 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a z+2 z a^{-3} -z a^{-5} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10}-7 a z^9-13 z^9 a^{-1} -6 z^9 a^{-3} -9 a^2 z^8-17 z^8 a^{-2} -9 z^8 a^{-4} -17 z^8-5 a^3 z^7+5 a z^7+12 z^7 a^{-1} -6 z^7 a^{-3} -8 z^7 a^{-5} -a^4 z^6+20 a^2 z^6+36 z^6 a^{-2} +9 z^6 a^{-4} -4 z^6 a^{-6} +44 z^6+10 a^3 z^5+14 a z^5+16 z^5 a^{-1} +25 z^5 a^{-3} +12 z^5 a^{-5} -z^5 a^{-7} +a^4 z^4-11 a^2 z^4-19 z^4 a^{-2} -z^4 a^{-4} +5 z^4 a^{-6} -25 z^4-4 a^3 z^3-9 a z^3-14 z^3 a^{-1} -18 z^3 a^{-3} -8 z^3 a^{-5} +z^3 a^{-7} +2 z^2 a^{-2} -z^2 a^{-4} -2 z^2 a^{-6} +z^2-a^3 z-4 a z-2 z a^{-1} +3 z a^{-3} +2 z a^{-5} +1+a^3 z^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} + 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). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=1 is the signature of L11a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
14           11
12          3 -3
10         61 5
8        93  -6
6       126   6
4      129    -3
2     1312     1
0    1014      4
-2   611       -5
-4  410        6
-6 16         -5
-8 4          4
-101           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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.

L10n113.gif

L10n113

L11a2.gif

L11a2