L11a43

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L11a42.gif

L11a42

L11a44.gif

L11a44

Contents

L11a43.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a43 at Knotilus!


Link Presentations

[edit Notes on L11a43's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X18,8,19,7 X22,13,5,14 X20,15,21,16 X16,19,17,20 X14,21,15,22 X12,10,13,9 X8,18,9,17 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 8, -2, 11, -8, 4, -7, 5, -6, 9, -3, 6, -5, 7, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11a43 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -3 q^{9/2}+\frac{2}{q^{9/2}}+5 q^{7/2}-\frac{5}{q^{7/2}}-9 q^{5/2}+\frac{8}{q^{5/2}}+11 q^{3/2}-\frac{10}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-13 \sqrt{q}+\frac{12}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} +z a^{-5} -2 a^3 z^3-2 z^3 a^{-3} -3 a^3 z-2 a^3 z^{-1} -2 z a^{-3} - a^{-3} z^{-1} +a z^5+z^5 a^{-1} +a z^3+z^3 a^{-1} +a z+a z^{-1} +2 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-2 a^3 z^9-5 a z^9-3 z^9 a^{-1} -2 a^4 z^8-2 a^2 z^8-4 z^8 a^{-2} -4 z^8-a^5 z^7+4 a^3 z^7+12 a z^7+3 z^7 a^{-1} -4 z^7 a^{-3} +8 a^4 z^6+16 a^2 z^6+3 z^6 a^{-2} -4 z^6 a^{-4} +15 z^6+5 a^5 z^5+8 a^3 z^5-3 z^5 a^{-5} -9 a^4 z^4-18 a^2 z^4+z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -12 z^4-8 a^5 z^3-20 a^3 z^3-13 a z^3+2 z^3 a^{-1} +7 z^3 a^{-3} +4 z^3 a^{-5} +4 a^4 z^2+8 a^2 z^2+z^2 a^{-6} +5 z^2+5 a^5 z+13 a^3 z+8 a z-3 z a^{-1} -5 z a^{-3} -2 z a^{-5} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -2 a^3 z^{-1} -a z^{-1} + 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 \
-6-5-4-3-2-1012345χ
12           1-1
10          2 2
8         31 -2
6        62  4
4       53   -2
2      86    2
0     67     1
-2    46      -2
-4   46       2
-6  14        -3
-8 14         3
-10 1          -1
-121           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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L11a42

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L11a44