L11a127

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L11a126.gif

L11a126

L11a128.gif

L11a128

Contents

L11a127.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a127 at Knotilus!


Link Presentations

[edit Notes on L11a127's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X22,10,5,9 X18,7,19,8 X8,17,9,18 X12,19,13,20 X20,11,21,12 X10,16,11,15 X16,22,17,21 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 4, -5, 3, -8, 7, -6, 11, -2, 8, -9, 5, -4, 6, -7, 9, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a127 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 u v^4-10 u v^3+14 u v^2-7 u v+u+v^5-7 v^4+14 v^3-10 v^2+3 v}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -\frac{19}{q^{9/2}}+\frac{22}{q^{7/2}}+q^{5/2}-\frac{23}{q^{5/2}}-4 q^{3/2}+\frac{20}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{14}{q^{11/2}}+9 \sqrt{q}-\frac{16}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^9 z^{-1} -4 z a^7-3 a^7 z^{-1} +6 z^3 a^5+9 z a^5+4 a^5 z^{-1} -3 z^5 a^3-7 z^3 a^3-8 z a^3-2 a^3 z^{-1} -z^5 a+z^3 a+z a+z^3 a^{-1} (db)
Kauffman polynomial a^9 z^7-4 a^9 z^5+6 a^9 z^3-4 a^9 z+a^9 z^{-1} +3 a^8 z^8-10 a^8 z^6+12 a^8 z^4-6 a^8 z^2+a^8+4 a^7 z^9-7 a^7 z^7-8 a^7 z^5+23 a^7 z^3-15 a^7 z+3 a^7 z^{-1} +2 a^6 z^{10}+9 a^6 z^8-42 a^6 z^6+44 a^6 z^4-17 a^6 z^2+3 a^6+13 a^5 z^9-22 a^5 z^7-19 a^5 z^5+44 a^5 z^3-24 a^5 z+4 a^5 z^{-1} +2 a^4 z^{10}+22 a^4 z^8-68 a^4 z^6+53 a^4 z^4-15 a^4 z^2+2 a^4+9 a^3 z^9+a^3 z^7-41 a^3 z^5+42 a^3 z^3-16 a^3 z+2 a^3 z^{-1} +16 a^2 z^8-27 a^2 z^6+14 a^2 z^4+z^4 a^{-2} -3 a^2 z^2+a^2+15 a z^7-22 a z^5+4 z^5 a^{-1} +14 a z^3-z^3 a^{-1} -3 a z+9 z^6-6 z^4+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 \
-8-7-6-5-4-3-2-10123χ
6           1-1
4          3 3
2         61 -5
0        103  7
-2       117   -4
-4      129    3
-6     1011     1
-8    912      -3
-10   611       5
-12  28        -6
-14 16         5
-16 2          -2
-181           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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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