L11a451

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L11a450.gif

L11a450

L11a452.gif

L11a452

Contents

L11a451.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a451's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X20,10,21,9 X16,8,17,7 X18,12,19,11 X22,16,13,15 X12,18,5,17 X10,20,11,19 X8,22,9,21 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 4, -9, 3, -8, 5, -7}, {11, -2, 6, -4, 7, -5, 8, -3, 9, -6}
A Braid Representative
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A Morse Link Presentation L11a451 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(2)^3+t(1) t(3) t(2)^3-2 t(3) t(2)^3+t(2)^3+t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+2 t(1) t(2)^2-3 t(1) t(3) t(2)^2+3 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-2 t(3)^3 t(2)-3 t(1) t(3)^2 t(2)+3 t(3)^2 t(2)+2 t(1) t(3) t(2)-t(3) t(2)-t(1) t(3)^3+t(3)^3+2 t(1) t(3)^2-t(3)^2}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial -q^7+3 q^6-4 q^5+7 q^4+ q^{-4} -8 q^3-2 q^{-3} +11 q^2+5 q^{-2} -10 q-7 q^{-1} +9 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4-2 z^2 a^2+z^4-z^2+ z^{-2} +z^4 a^{-2} -2 z^2 a^{-2} -2 a^{-2} z^{-2} -3 a^{-2} +z^4 a^{-4} +z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -z^2 a^{-6} (db)
Kauffman polynomial z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} +2 z^8 a^{-2} +2 z^8 a^{-4} +3 z^8 a^{-6} +3 z^8+3 a z^7-z^7 a^{-1} -17 z^7 a^{-3} -12 z^7 a^{-5} +z^7 a^{-7} +3 a^2 z^6-16 z^6 a^{-2} -22 z^6 a^{-4} -14 z^6 a^{-6} -5 z^6+2 a^3 z^5-2 a z^5-7 z^5 a^{-1} +12 z^5 a^{-3} +11 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-3 a^2 z^4+29 z^4 a^{-2} +35 z^4 a^{-4} +18 z^4 a^{-6} +8 z^4-2 a^3 z^3+14 z^3 a^{-1} +8 z^3 a^{-3} -z^3 a^{-5} +3 z^3 a^{-7} -2 a^4 z^2+a^2 z^2-28 z^2 a^{-2} -23 z^2 a^{-4} -7 z^2 a^{-6} -9 z^2-11 z a^{-1} -11 z a^{-3} +a^4+13 a^{-2} +8 a^{-4} +5+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-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
15           1-1
13          2 2
11         21 -1
9        52  3
7       54   -1
5      63    3
3     45     1
1    56      -1
-1   35       2
-3  24        -2
-5  3         3
-712          -1
-91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11a450

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L11a452