L11n86

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L11n85.gif

L11n85

L11n87.gif

L11n87

Contents

L11n86.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n86 at Knotilus!


Link Presentations

[edit Notes on L11n86's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X14,8,15,7 X18,10,19,9 X19,5,20,22 X15,21,16,20 X21,17,22,16 X8,18,9,17 X10,14,11,13 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 11, -2, 9, -3, -6, 7, 8, -4, -5, 6, -7, 5}
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n86 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^5-2 u v^4+u v^3-u v^2+2 u v-2 u-2 v^5+2 v^4-v^3+v^2-2 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{13/2}-3 q^{11/2}+5 q^{9/2}-5 q^{7/2}+6 q^{5/2}-6 q^{3/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z a^{-7} -z^5 a^{-5} -4 z^3 a^{-5} -3 z a^{-5} +z^7 a^{-3} +5 z^5 a^{-3} +8 z^3 a^{-3} +7 z a^{-3} +2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-9 z^3 a^{-1} +4 a z-11 z a^{-1} +3 a z^{-1} -5 a^{-1} z^{-1} (db)
Kauffman polynomial -z^9 a^{-1} -z^9 a^{-3} -4 z^8 a^{-2} -3 z^8 a^{-4} -z^8-a z^7+2 z^7 a^{-1} -3 z^7 a^{-5} +15 z^6 a^{-2} +11 z^6 a^{-4} -z^6 a^{-6} +3 z^6+6 a z^5+6 z^5 a^{-1} +9 z^5 a^{-3} +9 z^5 a^{-5} -11 z^4 a^{-2} -14 z^4 a^{-4} -z^4 a^{-6} +2 z^4-12 a z^3-15 z^3 a^{-1} -9 z^3 a^{-3} -9 z^3 a^{-5} -3 z^3 a^{-7} -4 z^2 a^{-2} +8 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} -9 z^2+10 a z+13 z a^{-1} +4 z a^{-3} +2 z a^{-5} +z a^{-7} +5 a^{-2} - a^{-6} +5-3 a z^{-1} -5 a^{-1} z^{-1} -2 a^{-3} 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 \
-4-3-2-1012345χ
14         1-1
12        2 2
10       31 -2
8      22  0
6     43   -1
4    22    0
2   35     2
0  11      0
-2  3       3
-411        0
-61         1
Integral Khovanov Homology

(db, data source)

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

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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L11n87