L11a141

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L11a140

L11a142.gif

L11a142

Contents

L11a141.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a141's Link Presentations]

Planar diagram presentation X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21
Gauss code {1, -4, 3, -10, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -11, 9, -7, 4, -2, 10, -8, 11, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11a141 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^6-t(1) t(2)^6-3 t(1)^2 t(2)^5+4 t(1) t(2)^5-t(2)^5+4 t(1)^2 t(2)^4-8 t(1) t(2)^4+3 t(2)^4-4 t(1)^2 t(2)^3+9 t(1) t(2)^3-4 t(2)^3+3 t(1)^2 t(2)^2-8 t(1) t(2)^2+4 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^3} (db)
Jones polynomial -q^{13/2}+4 q^{11/2}-8 q^{9/2}+14 q^{7/2}-19 q^{5/2}+21 q^{3/2}-22 \sqrt{q}+\frac{18}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^9 a^{-1} -a z^7+6 z^7 a^{-1} -z^7 a^{-3} -4 a z^5+12 z^5 a^{-1} -4 z^5 a^{-3} -4 a z^3+7 z^3 a^{-1} -4 z^3 a^{-3} +a z-3 z a^{-1} +z a^{-3} +2 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial -3 z^{10} a^{-2} -3 z^{10}-7 a z^9-14 z^9 a^{-1} -7 z^9 a^{-3} -7 a^2 z^8-5 z^8 a^{-2} -8 z^8 a^{-4} -4 z^8-4 a^3 z^7+16 a z^7+36 z^7 a^{-1} +9 z^7 a^{-3} -7 z^7 a^{-5} -a^4 z^6+18 a^2 z^6+16 z^6 a^{-2} +9 z^6 a^{-4} -4 z^6 a^{-6} +22 z^6+10 a^3 z^5-11 a z^5-38 z^5 a^{-1} -6 z^5 a^{-3} +10 z^5 a^{-5} -z^5 a^{-7} +2 a^4 z^4-12 a^2 z^4-10 z^4 a^{-2} +2 z^4 a^{-4} +6 z^4 a^{-6} -20 z^4-5 a^3 z^3+2 a z^3+14 z^3 a^{-1} +3 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +2 a^2 z^2-3 z^2 a^{-2} -5 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2+3 a z+4 z a^{-1} +z a^{-3} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 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).   
\ r
  \  
j \
-5-4-3-2-10123456χ
14           11
12          3 -3
10         51 4
8        93  -6
6       105   5
4      119    -2
2     1110     1
0    812      4
-2   610       -4
-4  39        6
-6 15         -4
-8 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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).

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L11a140.gif

L11a140

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L11a142