L11n448

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L11n447.gif

L11n447

L11n449.gif

L11n449

Contents

L11n448.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n448 at Knotilus!


Link Presentations

[edit Notes on L11n448's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X13,20,14,21 X16,12,17,11 X19,12,20,13 X8,16,5,15 X14,8,15,7 X17,19,18,22 X21,9,22,18 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 7, -6}, {-5, 3, -9, 8}, {11, -2, 4, 5, -3, -7, 6, -4, -8, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n448 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) (t(3)-1) (t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} (db)
Jones polynomial q^{9/2}-\frac{1}{q^{9/2}}-3 q^{7/2}+4 q^{5/2}-\frac{4}{q^{5/2}}-5 q^{3/2}+\frac{3}{q^{3/2}}+4 \sqrt{q}-\frac{7}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^{-3} +a^5 z^{-1} -3 a^3 z^{-3} +z^3 a^{-3} -2 a^3 z-4 a^3 z^{-1} +z a^{-3} -z^5 a^{-1} +2 a z^3+3 a z^{-3} -3 z^3 a^{-1} - a^{-1} z^{-3} +4 a z+5 a z^{-1} -3 z a^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial a^5 z^3-a^5 z^{-3} -3 a^5 z+3 a^5 z^{-1} +z^6 a^{-4} -3 z^4 a^{-4} +3 a^4 z^2+z^2 a^{-4} +3 a^4 z^{-2} -6 a^4+3 z^7 a^{-3} +a^3 z^5-11 z^5 a^{-3} +8 z^3 a^{-3} -3 a^3 z^{-3} -5 a^3 z-3 z a^{-3} +6 a^3 z^{-1} +a^2 z^8+3 z^8 a^{-2} -5 a^2 z^6-11 z^6 a^{-2} +9 a^2 z^4+7 z^4 a^{-2} +a^2 z^2+z^2 a^{-2} +6 a^2 z^{-2} -11 a^2+a z^9+z^9 a^{-1} -3 a z^7-a z^5-13 z^5 a^{-1} +10 a z^3+19 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -7 a z-8 z a^{-1} +6 a z^{-1} +3 a^{-1} z^{-1} +4 z^8-17 z^6+19 z^4-2 z^2+3 z^{-2} -6 (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    232   1
0   173    3
-2   37     4
-4  21      1
-61 3       4
-821        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=-4 {\mathbb Z} {\mathbb Z}^{2} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2 {\mathbb Z}^{7} {\mathbb Z}^{2}
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

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See/edit the Link Page master template (intermediate).

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L11n447.gif

L11n447

L11n449.gif

L11n449