L11a362

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L11a361.gif

L11a361

L11a363.gif

L11a363

Contents

L11a362.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a362 at Knotilus!


Link Presentations

[edit Notes on L11a362's Link Presentations]

Planar diagram presentation X12,1,13,2 X14,4,15,3 X22,14,11,13 X2,11,3,12 X4,22,5,21 X20,10,21,9 X16,6,17,5 X18,8,19,7 X6,18,7,17 X8,20,9,19 X10,16,1,15
Gauss code {1, -4, 2, -5, 7, -9, 8, -10, 6, -11}, {4, -1, 3, -2, 11, -7, 9, -8, 10, -6, 5, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a362 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^3 t(1)^4-t(2)^2 t(1)^4+t(2)^4 t(1)^3-4 t(2)^3 t(1)^3+3 t(2)^2 t(1)^3-t(2) t(1)^3-t(2)^4 t(1)^2+3 t(2)^3 t(1)^2-5 t(2)^2 t(1)^2+3 t(2) t(1)^2-t(1)^2-t(2)^3 t(1)+3 t(2)^2 t(1)-4 t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^2 t(2)^2} (db)
Jones polynomial q^{23/2}-3 q^{21/2}+5 q^{19/2}-8 q^{17/2}+10 q^{15/2}-11 q^{13/2}+10 q^{11/2}-9 q^{9/2}+6 q^{7/2}-4 q^{5/2}+2 q^{3/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial -z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -4 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -3 z^3 a^{-5} -4 z^3 a^{-7} +3 z^3 a^{-9} +3 z a^{-3} +z a^{-5} -2 z a^{-7} +z a^{-9} + a^{-5} z^{-1} - a^{-7} z^{-1} (db)
Kauffman polynomial z^4 a^{-14} -z^2 a^{-14} +3 z^5 a^{-13} -4 z^3 a^{-13} +z a^{-13} +4 z^6 a^{-12} -4 z^4 a^{-12} +4 z^7 a^{-11} -3 z^5 a^{-11} -2 z^3 a^{-11} +2 z a^{-11} +4 z^8 a^{-10} -7 z^6 a^{-10} +6 z^4 a^{-10} -z^2 a^{-10} +3 z^9 a^{-9} -7 z^7 a^{-9} +9 z^5 a^{-9} -6 z^3 a^{-9} +z a^{-9} +z^{10} a^{-8} +2 z^8 a^{-8} -13 z^6 a^{-8} +16 z^4 a^{-8} -6 z^2 a^{-8} +5 z^9 a^{-7} -19 z^7 a^{-7} +25 z^5 a^{-7} -17 z^3 a^{-7} +6 z a^{-7} - a^{-7} z^{-1} +z^{10} a^{-6} -11 z^6 a^{-6} +16 z^4 a^{-6} -8 z^2 a^{-6} + a^{-6} +2 z^9 a^{-5} -7 z^7 a^{-5} +5 z^5 a^{-5} -2 z^3 a^{-5} +3 z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -9 z^6 a^{-4} +11 z^4 a^{-4} -4 z^2 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +7 z^3 a^{-3} -3 z a^{-3} (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         31 -2
18        52  3
16       64   -2
14      54    1
12     56     1
10    45      -1
8   25       3
6  24        -2
4 13         2
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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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).

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L11a361

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L11a363