L11n57

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L11n56.gif

L11n56

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L11n58

Contents

L11n57.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n57's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,14,8,15 X11,19,12,18 X19,5,20,22 X15,21,16,20 X21,17,22,16 X17,13,18,12 X13,8,14,9 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -4, 8, -9, 3, -6, 7, -8, 4, -5, 6, -7, 5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11n57 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^5-t(1) t(2)^4-3 t(2)^4+2 t(2)^3+2 t(1) t(2)^2-3 t(1) t(2)-t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{9/2}+\frac{2}{q^{9/2}}-q^{7/2}-\frac{4}{q^{7/2}}+\frac{4}{q^{5/2}}+q^{3/2}-\frac{4}{q^{3/2}}-\frac{1}{q^{11/2}}-3 \sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} -2 a^3 z^3+z^3 a^{-3} -4 a^3 z-2 a^3 z^{-1} +3 z a^{-3} + a^{-3} z^{-1} +a z^5-z^5 a^{-1} +4 a z^3-6 z^3 a^{-1} +6 a z+3 a z^{-1} -8 z a^{-1} -3 a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-2 a^4 z^8-4 a^2 z^8-2 z^8-a^5 z^7+a^3 z^7+2 a z^7-z^7 a^{-1} -z^7 a^{-3} +9 a^4 z^6+19 a^2 z^6-z^6 a^{-4} +11 z^6+5 a^5 z^5+12 a^3 z^5+10 a z^5+9 z^5 a^{-1} +6 z^5 a^{-3} -10 a^4 z^4-23 a^2 z^4+4 z^4 a^{-2} +5 z^4 a^{-4} -14 z^4-8 a^5 z^3-20 a^3 z^3-22 a z^3-18 z^3 a^{-1} -8 z^3 a^{-3} +3 a^4 z^2+10 a^2 z^2-7 z^2 a^{-2} -5 z^2 a^{-4} +5 z^2+5 a^5 z+10 a^3 z+14 a z+13 z a^{-1} +4 z a^{-3} -2 a^2+2 a^{-2} + a^{-4} -a^5 z^{-1} -2 a^3 z^{-1} -3 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 \
-6-5-4-3-2-1012345χ
10           1-1
8            0
6        111 1
4       21   -1
2      211   2
0     341    0
-2    221     1
-4   231      0
-6  22        0
-8 13         2
-10 1          -1
-121           1
Integral Khovanov Homology

(db, data source)

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