L11n283

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L11n282.gif

L11n282

L11n284.gif

L11n284

Contents

L11n283.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n283 at Knotilus!


Link Presentations

[edit Notes on L11n283's Link Presentations]

Planar diagram presentation X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5
Gauss code {1, -4, 3, -10}, {2, -1, 5, -3, -6, 11}, {-8, -2, 4, -5, 10, 9, -7, 6, -11, 8, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L11n283 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1)^2 (w-1)^2}{\sqrt{u} v w} (db)
Jones polynomial q^6-4 q^5+6 q^4-9 q^3- q^{-3} +11 q^2+5 q^{-2} -10 q-6 q^{-1} +11 (db)
Signature 0 (db)
HOMFLY-PT polynomial z^4 a^{-4} +z^2 a^{-4} - a^{-4} z^{-2} - a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} -a^2 z^2-2 z^2 a^{-2} +2 a^2 z^{-2} +4 a^{-2} z^{-2} +a^2+4 a^{-2} +2 z^4+2 z^2-5 z^{-2} -4 (db)
Kauffman polynomial 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +5 z^8 a^{-4} +4 z^8+2 a z^7+2 z^7 a^{-3} +4 z^7 a^{-5} -28 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -12 z^6-a z^5-8 z^5 a^{-1} -19 z^5 a^{-3} -12 z^5 a^{-5} +5 a^2 z^4+29 z^4 a^{-2} +11 z^4 a^{-4} -2 z^4 a^{-6} +21 z^4+a^3 z^3+a z^3+6 z^3 a^{-1} +13 z^3 a^{-3} +7 z^3 a^{-5} -3 a^2 z^2-12 z^2 a^{-2} -4 z^2 a^{-4} -11 z^2+5 a z+9 z a^{-1} +5 z a^{-3} +z a^{-5} -3 a^2-2 a^{-2} -4-5 a z^{-1} -9 a^{-1} z^{-1} -5 a^{-3} z^{-1} - a^{-5} z^{-1} +2 a^2 z^{-2} +4 a^{-2} z^{-2} + a^{-4} z^{-2} +5 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        3 -3
9       31 2
7      63  -3
5     53   2
3    56    1
1   65     1
-1  27      5
-3 34       -1
-5 4        4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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L11n282.gif

L11n282

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L11n284