L11n84

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L11n83.gif

L11n83

L11n85.gif

L11n85

Contents

L11n84.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n84's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X14,8,15,7 X9,18,10,19 X19,5,20,22 X15,21,16,20 X21,17,22,16 X17,8,18,9 X10,14,11,13 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, 8, -4, -9, 11, -2, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5}
A Braid Representative
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n84 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^3-t(2)^3-3 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-3 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+4 q^{5/2}-5 q^{3/2}+6 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^{-1} -2 z a^3-a^3 z^{-1} +2 z^3 a+2 z a-z^5 a^{-1} -3 z^3 a^{-1} -3 z a^{-1} +z^3 a^{-3} +z a^{-3} (db)
Kauffman polynomial a^5 z^3-2 a^5 z+a^5 z^{-1} +z^6 a^{-4} +a^4 z^4-3 z^4 a^{-4} +z^2 a^{-4} -a^4+3 z^7 a^{-3} +2 a^3 z^5-11 z^5 a^{-3} -2 a^3 z^3+8 z^3 a^{-3} -z a^{-3} +a^3 z^{-1} +a^2 z^8+3 z^8 a^{-2} -4 a^2 z^6-11 z^6 a^{-2} +9 a^2 z^4+9 z^4 a^{-2} -5 a^2 z^2-2 z^2 a^{-2} +a z^9+z^9 a^{-1} -3 a z^7+3 a z^5-10 z^5 a^{-1} +11 z^3 a^{-1} +a z-2 z a^{-1} +4 z^8-16 z^6+20 z^4-8 z^2 (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 \
-4-3-2-1012345χ
10         1-1
8        2 2
6       21 -1
4      32  1
2     32   -1
0    33    0
-2   34     1
-4  12      -1
-6  3       3
-811        0
-101         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11n83.gif

L11n83

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L11n85