L11n423

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L11n422.gif

L11n422

L11n424.gif

L11n424

Contents

L11n423.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n423 at Knotilus!


Link Presentations

[edit Notes on L11n423's Link Presentations]

Planar diagram presentation X8192 X16,8,17,7 X10,4,11,3 X2,18,3,17 X18,9,19,10 X20,12,21,11 X5,14,6,15 X15,13,16,22 X13,6,14,1 X4,19,5,20 X12,22,7,21
Gauss code {1, -4, 3, -10, -7, 9}, {2, -1, 5, -3, 6, -11}, {-9, 7, -8, -2, 4, -5, 10, -6, 11, 8}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n423 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) \left(u^2 v-u^2+u v^2 w^2-u v w^2-u v w-u v+u-v^2 w^2+v w^2\right)}{u v w^{3/2}} (db)
Jones polynomial -q^4- q^{-4} +3 q^3+3 q^{-3} -3 q^2-3 q^{-2} +5 q+5 q^{-1} -4 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6-a^2 z^4-z^4 a^{-2} +4 z^4-2 a^2 z^2-2 z^2 a^{-2} +3 z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial a^3 z^7+z^7 a^{-3} -4 a^3 z^5-4 z^5 a^{-3} +3 a^3 z^3+3 z^3 a^{-3} +3 a^2 z^8+3 z^8 a^{-2} -15 a^2 z^6-15 z^6 a^{-2} +20 a^2 z^4+20 z^4 a^{-2} -7 a^2 z^2-7 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +2 a z^9+2 z^9 a^{-1} -8 a z^7-8 z^7 a^{-1} +5 a z^5+5 z^5 a^{-1} +a z^3+z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +6 z^8-30 z^6+40 z^4-14 z^2+2 z^{-2} -3 (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 \
-5-4-3-2-10123χ
9        1-1
7       2 2
5      22 0
3     321 2
1    34   1
-1   221   1
-3  24     2
-5 11      0
-7 2       2
-91        -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z} {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11n422.gif

L11n422

L11n424.gif

L11n424