L11n379

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L11n378.gif

L11n378

L11n380.gif

L11n380

Contents

L11n379.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n379 at Knotilus!


Link Presentations

[edit Notes on L11n379's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X22,18,19,17 X20,12,21,11 X12,20,13,19 X18,22,5,21 X9,16,10,17 X13,2,14,3
Gauss code {1, 11, -5, -3}, {8, -7, 9, -6}, {-4, -1, 2, 5, -10, 4, 7, -8, -11, -2, 3, 10, 6, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n379 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(3)-1) (t(3)+1) \left(t(1) t(3)^3+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} (db)
Jones polynomial q^2+1+ q^{-1} + q^{-2} + q^{-3} - q^{-4} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^6 z^{-2} -a^6+a^4 z^4+6 a^4 z^2+4 a^4 z^{-2} +8 a^4-a^2 z^6-7 a^2 z^4-15 a^2 z^2-5 a^2 z^{-2} -13 a^2+z^4+5 z^2+2 z^{-2} +6 (db)
Kauffman polynomial a^2 z^8+z^8+a^3 z^7+a z^7-a^4 z^6-9 a^2 z^6-8 z^6-a^5 z^5-9 a^3 z^5-8 a z^5+7 a^4 z^4+28 a^2 z^4+21 z^4+6 a^5 z^3+26 a^3 z^3+20 a z^3-15 a^4 z^2-37 a^2 z^2-22 z^2-9 a^5 z-27 a^3 z-18 a z+a^6+12 a^4+21 a^2+11+a^7 z^{-1} +5 a^5 z^{-1} +9 a^3 z^{-1} +5 a z^{-1} -a^6 z^{-2} -4 a^4 z^{-2} -5 a^2 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 \
-4-3-2-101234χ
5        11
3        11
1      1  1
-1    2    2
-3    31   2
-5  2      2
-7  11     0
-912       -1
-1111       0
Integral Khovanov Homology

(db, data source)

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