L11a60

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L11a59.gif

L11a59

L11a61.gif

L11a61

Contents

L11a60.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a60 at Knotilus!


Link Presentations

[edit Notes on L11a60's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X14,8,15,7 X20,18,21,17 X18,9,19,10 X8,19,9,20 X22,16,5,15 X16,22,17,21 X10,14,11,13 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 11, -2, 9, -3, 7, -8, 4, -5, 6, -4, 8, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a60 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^3-4 t(2)^3-8 t(1) t(2)^2+11 t(2)^2+11 t(1) t(2)-8 t(2)-4 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{9/2}-\frac{6}{q^{9/2}}-4 q^{7/2}+\frac{9}{q^{7/2}}+7 q^{5/2}-\frac{13}{q^{5/2}}-11 q^{3/2}+\frac{16}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+14 \sqrt{q}-\frac{16}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^{-1} -3 a^5 z-a^5 z^{-1} +3 a^3 z^3+z^3 a^{-3} +a^3 z-a^3 z^{-1} -a z^5-z^5 a^{-1} +a z^3-z^3 a^{-1} +2 a z+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-2 a^3 z^9-6 a z^9-4 z^9 a^{-1} -3 a^4 z^8-5 a^2 z^8-6 z^8 a^{-2} -8 z^8-3 a^5 z^7-4 a^3 z^7+8 a z^7+5 z^7 a^{-1} -4 z^7 a^{-3} -2 a^6 z^6+a^4 z^6+10 a^2 z^6+17 z^6 a^{-2} -z^6 a^{-4} +25 z^6-a^7 z^5+3 a^5 z^5+12 a^3 z^5+5 a z^5+8 z^5 a^{-1} +11 z^5 a^{-3} +3 a^6 z^4+a^4 z^4-9 a^2 z^4-12 z^4 a^{-2} +2 z^4 a^{-4} -21 z^4+3 a^7 z^3+a^5 z^3-14 a^3 z^3-18 a z^3-12 z^3 a^{-1} -6 z^3 a^{-3} +2 a^4 z^2+6 a^2 z^2+2 z^2 a^{-2} +6 z^2-3 a^7 z-2 a^5 z+8 a^3 z+12 a z+5 z a^{-1} -a^6-2 a^4-3 a^2-1+a^7 z^{-1} +a^5 z^{-1} -a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1} (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          3 3
6         41 -3
4        73  4
2       74   -3
0      97    2
-2     88     0
-4    58      -3
-6   48       4
-8  25        -3
-10 15         4
-12 1          -1
-141           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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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