L11a450

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L11a449.gif

L11a449

L11a451.gif

L11a451

Contents

L11a450.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a450 at Knotilus!


Link Presentations

[edit Notes on L11a450's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X20,18,21,17 X16,8,17,7 X10,14,11,13 X22,12,13,11 X12,22,5,21 X8,19,9,20 X18,9,19,10 X2536 X4,16,1,15
Gauss code {1, -10, 2, -11}, {10, -1, 4, -8, 9, -5, 6, -7}, {5, -2, 11, -4, 3, -9, 8, -3, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a450 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(v+w-1) (v w-v-w) (u v w-u w+u-v w+v-1)}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial -q^8+3 q^7-6 q^6+11 q^5-15 q^4+18 q^3+ q^{-3} -16 q^2-3 q^{-2} +16 q+7 q^{-1} -11 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-2 z^2 a^{-2} +5 z^2 a^{-4} -2 z^2 a^{-6} -3 z^2+a^2-3 a^{-2} +3 a^{-4} - a^{-6} -2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} (db)
Kauffman polynomial z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -6 z^4 a^{-8} +2 z^2 a^{-8} +5 z^7 a^{-7} -10 z^5 a^{-7} +7 z^3 a^{-7} -2 z a^{-7} +6 z^8 a^{-6} -13 z^6 a^{-6} +15 z^4 a^{-6} -8 z^2 a^{-6} +3 a^{-6} +4 z^9 a^{-5} -2 z^7 a^{-5} -9 z^5 a^{-5} +19 z^3 a^{-5} -7 z a^{-5} +z^{10} a^{-4} +11 z^8 a^{-4} -36 z^6 a^{-4} +52 z^4 a^{-4} -36 z^2 a^{-4} - a^{-4} z^{-2} +11 a^{-4} +7 z^9 a^{-3} -9 z^7 a^{-3} -6 z^5 a^{-3} +19 z^3 a^{-3} -12 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +9 z^8 a^{-2} +a^2 z^6-28 z^6 a^{-2} -3 a^2 z^4+34 z^4 a^{-2} +3 a^2 z^2-29 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2+11 a^{-2} +3 z^9 a^{-1} +3 a z^7+z^7 a^{-1} -8 a z^5-16 z^5 a^{-1} +6 a z^3+15 z^3 a^{-1} -a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-7 z^6- 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 \
-4-3-2-101234567χ
17           1-1
15          2 2
13         41 -3
11        72  5
9       95   -4
7      96    3
5     79     2
3    99      0
1   510       5
-1  26        -4
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
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}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
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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L11a449.gif

L11a449

L11a451.gif

L11a451