L11a272

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L11a271.gif

L11a271

L11a273.gif

L11a273

Contents

L11a272.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a272 at Knotilus!


Link Presentations

[edit Notes on L11a272's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,6,15,5 X22,14,9,13 X18,8,19,7 X20,18,21,17 X16,22,17,21 X6,16,7,15 X8,20,1,19
Gauss code {1, -2, 3, -4, 5, -10, 7, -11}, {4, -1, 2, -3, 6, -5, 10, -9, 8, -7, 11, -8, 9, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a272 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+5 t(2) t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -10 q^{9/2}+14 q^{7/2}-\frac{1}{q^{7/2}}-17 q^{5/2}+\frac{3}{q^{5/2}}+16 q^{3/2}-\frac{7}{q^{3/2}}+q^{15/2}-3 q^{13/2}+6 q^{11/2}-15 \sqrt{q}+\frac{11}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} -z^7 a^{-3} +a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} +3 a z^3-6 z^3 a^{-1} -5 z^3 a^{-3} +3 z^3 a^{-5} +3 a z-4 z a^{-1} -z a^{-3} +2 z a^{-5} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^6 a^{-8} -3 z^4 a^{-8} +2 z^2 a^{-8} +3 z^7 a^{-7} -9 z^5 a^{-7} +7 z^3 a^{-7} -z a^{-7} +4 z^8 a^{-6} -10 z^6 a^{-6} +7 z^4 a^{-6} -2 z^2 a^{-6} +3 z^9 a^{-5} -3 z^7 a^{-5} -3 z^5 a^{-5} +z^3 a^{-5} +z a^{-5} +z^{10} a^{-4} +6 z^8 a^{-4} -16 z^6 a^{-4} +12 z^4 a^{-4} -3 z^2 a^{-4} +6 z^9 a^{-3} -8 z^7 a^{-3} +a^3 z^5+3 z^5 a^{-3} -2 a^3 z^3-z^3 a^{-3} +a^3 z+z^{10} a^{-2} +7 z^8 a^{-2} +3 a^2 z^6-13 z^6 a^{-2} -5 a^2 z^4+7 z^4 a^{-2} +2 a^2 z^2+3 z^9 a^{-1} +5 a z^7+3 z^7 a^{-1} -8 a z^5-12 z^5 a^{-1} +5 a z^3+12 z^3 a^{-1} -3 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +5 z^8-5 z^6+z^2-1 (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 \
-4-3-2-101234567χ
16           1-1
14          2 2
12         41 -3
10        62  4
8       84   -4
6      96    3
4     78     1
2    89      -1
0   59       4
-2  26        -4
-4 15         4
-6 2          -2
-81           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11a271.gif

L11a271

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L11a273