L11n296

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L11n295.gif

L11n295

L11n297.gif

L11n297

Contents

L11n296.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n296 at Knotilus!


Link Presentations

[edit Notes on L11n296's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X15,20,16,21 X14,8,15,7 X10,22,5,21 X18,11,19,12 X9,17,10,16 X22,17,11,18 X19,9,20,8 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 4, 9, -7, -5}, {6, -2, 11, -4, -3, 7, 8, -6, -9, 3, 5, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n296 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^2 w+u v w^4-u v w^3+u v w^2-2 u v w+u v-u w^4+u w^3-v^2 w+v^2-v w^4+2 v w^3-v w^2+v w-v-w^3}{\sqrt{u} v w^2} (db)
Jones polynomial q^6-2 q^5+4 q^4-4 q^3+7 q^2-5 q+6-4 q^{-1} +2 q^{-2} - q^{-3} (db)
Signature 0 (db)
HOMFLY-PT polynomial z^4 a^{-4} +3 z^2 a^{-4} +2 a^{-4} z^{-2} +3 a^{-4} -z^6 a^{-2} -5 z^4 a^{-2} -a^2 z^2-10 z^2 a^{-2} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^2-10 a^{-2} +2 z^4+7 z^2+4 z^{-2} +9 (db)
Kauffman polynomial z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+a z^7-z^7 a^{-1} +2 z^7 a^{-5} -25 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -10 z^6-4 a z^5-10 z^5 a^{-1} -13 z^5 a^{-3} -7 z^5 a^{-5} +2 a^2 z^4+48 z^4 a^{-2} +23 z^4 a^{-4} -4 z^4 a^{-6} +23 z^4+a^3 z^3+12 a z^3+30 z^3 a^{-1} +23 z^3 a^{-3} +4 z^3 a^{-5} -3 a^2 z^2-44 z^2 a^{-2} -22 z^2 a^{-4} +3 z^2 a^{-6} -22 z^2-2 a^3 z-13 a z-27 z a^{-1} -16 z a^{-3} +2 a^2+20 a^{-2} +10 a^{-4} +13+a^3 z^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +5 a^{-3} z^{-1} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^{-4} z^{-2} -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 \
-3-2-10123456χ
13         11
11        21-1
9       2  2
7      22  0
5     52   3
3    13    2
1   54     1
-1  13      2
-3 13       -2
-5 1        1
-71         -1
Integral Khovanov Homology

(db, data source)

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

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L11n295.gif

L11n295

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L11n297