L11n303

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L11n302.gif

L11n302

L11n304.gif

L11n304

Contents

L11n303.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n303's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X13,21,14,20 X19,11,20,22 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X21,15,22,14 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, {11, -2, -3, 9, -5, 7, -6, 8, -4, 3, -9, 4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n303 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(2)^2 t(3)^4+t(1) t(2)^2 t(3)^3-t(1) t(2) t(3)^3+t(2) t(3)^3-t(1) t(2)^2 t(3)^2+2 t(1) t(2) t(3)^2-2 t(2) t(3)^2+t(3)^2-t(1) t(2) t(3)+t(2) t(3)-t(3)+t(1)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial -q^8+2 q^7-3 q^6+4 q^5-4 q^4+4 q^3-2 q^2+2 q+1+ q^{-2} (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^6 a^{-2} +z^6 a^{-4} -8 z^4 a^{-2} +6 z^4 a^{-4} -z^4 a^{-6} +z^4-19 z^2 a^{-2} +14 z^2 a^{-4} -3 z^2 a^{-6} +5 z^2-17 a^{-2} +13 a^{-4} -3 a^{-6} +7-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} (db)
Kauffman polynomial z^8 a^{-2} +z^8 a^{-4} +z^8 a^{-6} +z^8+z^7 a^{-1} +2 z^7 a^{-3} +3 z^7 a^{-5} +2 z^7 a^{-7} -10 z^6 a^{-2} -6 z^6 a^{-4} -2 z^6 a^{-6} +2 z^6 a^{-8} -8 z^6-9 z^5 a^{-1} -16 z^5 a^{-3} -14 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} +31 z^4 a^{-2} +15 z^4 a^{-4} -z^4 a^{-6} -6 z^4 a^{-8} +21 z^4+22 z^3 a^{-1} +41 z^3 a^{-3} +26 z^3 a^{-5} +4 z^3 a^{-7} -3 z^3 a^{-9} -39 z^2 a^{-2} -16 z^2 a^{-4} +3 z^2 a^{-6} +3 z^2 a^{-8} -23 z^2-19 z a^{-1} -35 z a^{-3} -19 z a^{-5} -2 z a^{-7} +z a^{-9} +22 a^{-2} +13 a^{-4} - a^{-8} +11+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -2 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 \
-4-3-2-101234567χ
17           1-1
15          1 1
13         21 -1
11        21  1
9       22   0
7     132    0
5     24     2
3   121      0
1    3       3
-1  1         1
-31           1
-51           1
Integral Khovanov Homology

(db, data source)

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