L11n446

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L11n445.gif

L11n445

L11n447.gif

L11n447

Contents

L11n446.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n446 at Knotilus!


Link Presentations

[edit Notes on L11n446's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X22,19,17,20 X16,21,9,22 X20,15,21,16 X12,18,13,17 X2536 X9,1,10,4
Gauss code {1, -10, -2, 11}, {10, -1, -3, 4}, {9, -5, 6, -8, 7, -6}, {-11, 2, 5, -9, -4, 3, 8, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11n446 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(w-1) (x-1) (u-x) (x-v)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} (db)
Jones polynomial q^{9/2}+\frac{1}{q^{9/2}}-3 q^{7/2}-\frac{2}{q^{7/2}}+q^{5/2}+\frac{2}{q^{5/2}}-2 q^{3/2}-\frac{2}{q^{3/2}}-q^{11/2}-2 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 0 (db)
HOMFLY-PT polynomial - a^{-5} z^{-3} -z a^{-5} -2 a^{-5} z^{-1} -a^3 z^3+2 z^3 a^{-3} +3 a^{-3} z^{-3} -2 a^3 z+7 z a^{-3} -a^3 z^{-1} +8 a^{-3} z^{-1} +a z^5-z^5 a^{-1} +5 a z^3-6 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +8 a z-12 z a^{-1} +6 a z^{-1} -11 a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^8-z^8 a^{-2} -z^8 a^{-4} -z^8-2 a^3 z^7-3 a z^7-4 z^7 a^{-1} -4 z^7 a^{-3} -z^7 a^{-5} -a^4 z^6+3 a^2 z^6+4 z^6 a^{-2} +4 z^6 a^{-4} +4 z^6+9 a^3 z^5+20 a z^5+29 z^5 a^{-1} +24 z^5 a^{-3} +6 z^5 a^{-5} +4 a^4 z^4+6 a^2 z^4+6 z^4 a^{-2} +8 z^4-9 a^3 z^3-36 a z^3-57 z^3 a^{-1} -42 z^3 a^{-3} -12 z^3 a^{-5} -3 a^4 z^2-14 a^2 z^2-24 z^2 a^{-2} -9 z^2 a^{-4} -26 z^2+6 a^3 z+27 a z+44 z a^{-1} +34 z a^{-3} +11 z a^{-5} +a^4+6 a^2+21 a^{-2} +9 a^{-4} +18-2 a^3 z^{-1} -11 a z^{-1} -18 a^{-1} z^{-1} -14 a^{-3} z^{-1} -5 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3} (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
12           11
10            0
8         31 2
6       112  2
4       21   1
2     521    4
0    272     3
-2   113      3
-4  121       0
-6 11         0
-8 1          1
-101           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2 {\mathbb Z}^{5}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=5 {\mathbb Z}
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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L11n445.gif

L11n445

L11n447.gif

L11n447