L11n431

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L11n430.gif

L11n430

L11n432.gif

L11n432

Contents

L11n431.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n431 at Knotilus!


Link Presentations

[edit Notes on L11n431's Link Presentations]

Planar diagram presentation X8192 X20,10,21,9 X5,15,6,14 X12,14,7,13 X16,8,17,7 X22,18,13,17 X10,4,11,3 X18,11,19,12 X15,1,16,6 X4,20,5,19 X2,21,3,22
Gauss code {1, -11, 7, -10, -3, 9}, {5, -1, 2, -7, 8, -4}, {4, 3, -9, -5, 6, -8, 10, -2, 11, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n431 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(w-1) \left(-u^2 v^2 w+u^2 v^2+u^2 v w-u^2 v+u v^2 w-u v^2+u v w^2-3 u v w+u v-u w^2+u w-v w^2+v w+w^2-w\right)}{u v w^{3/2}} (db)
Jones polynomial q^9-4 q^8+7 q^7-9 q^6+12 q^5-11 q^4+11 q^3-7 q^2+5 q-1 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^2 a^{-8} -2 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} z^{-2} +z^6 a^{-4} +3 z^4 a^{-4} +3 z^2 a^{-4} -2 a^{-4} z^{-2} -2 a^{-4} -z^4 a^{-2} + a^{-2} z^{-2} +2 a^{-2} (db)
Kauffman polynomial z^6 a^{-10} -2 z^4 a^{-10} +4 z^7 a^{-9} -11 z^5 a^{-9} +5 z^3 a^{-9} +6 z^8 a^{-8} -19 z^6 a^{-8} +15 z^4 a^{-8} -4 z^2 a^{-8} +3 z^9 a^{-7} -3 z^7 a^{-7} -10 z^5 a^{-7} +8 z^3 a^{-7} +11 z^8 a^{-6} -36 z^6 a^{-6} +37 z^4 a^{-6} -11 z^2 a^{-6} + a^{-6} z^{-2} -2 a^{-6} +3 z^9 a^{-5} -5 z^7 a^{-5} +z^5 a^{-5} +2 z^3 a^{-5} +2 z a^{-5} -2 a^{-5} z^{-1} +5 z^8 a^{-4} -16 z^6 a^{-4} +25 z^4 a^{-4} -10 z^2 a^{-4} +2 a^{-4} z^{-2} -3 a^{-4} +2 z^7 a^{-3} +2 z a^{-3} -2 a^{-3} z^{-1} +5 z^4 a^{-2} -3 z^2 a^{-2} + a^{-2} z^{-2} -2 a^{-2} +z^3 a^{-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 \
-1012345678χ
19         11
17        3 -3
15       41 3
13      53  -2
11     74   3
9    56    1
7   66     0
5  37      4
3 24       -2
1 4        4
-11         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-1 {\mathbb Z}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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L11n430.gif

L11n430

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L11n432