L11a289

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L11a288.gif

L11a288

L11a290.gif

L11a290

Contents

L11a289.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a289 at Knotilus!

Celtic or pseudo-Celtic linear decorative knot
Decorative variant with big loops at ends

(Also see Detecting a Link Using the Multivariable Alexander Polynomial.)

Link Presentations

[edit Notes on L11a289's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X20,14,21,13 X14,7,15,8 X18,16,19,15 X16,6,17,5 X6,18,7,17 X4,19,5,20 X8,22,1,21
Gauss code {1, -4, 2, -10, 8, -9, 6, -11}, {4, -1, 3, -2, 5, -6, 7, -8, 9, -7, 10, -5, 11, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a289 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+2 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial 16 q^{9/2}-18 q^{7/2}+17 q^{5/2}-\frac{1}{q^{5/2}}-15 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-4 q^{15/2}+8 q^{13/2}-12 q^{11/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +z a^{-7} -2 z^5 a^{-5} -5 z^3 a^{-5} -3 z a^{-5} +z^7 a^{-3} +4 z^5 a^{-3} +7 z^3 a^{-3} +5 z a^{-3} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +2 a z-5 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-10} +4 z^5 a^{-9} -2 z^3 a^{-9} +8 z^6 a^{-8} -8 z^4 a^{-8} +3 z^2 a^{-8} +10 z^7 a^{-7} -12 z^5 a^{-7} +6 z^3 a^{-7} -2 z a^{-7} +8 z^8 a^{-6} -4 z^6 a^{-6} -9 z^4 a^{-6} +4 z^2 a^{-6} +4 z^9 a^{-5} +9 z^7 a^{-5} -34 z^5 a^{-5} +24 z^3 a^{-5} -6 z a^{-5} +z^{10} a^{-4} +13 z^8 a^{-4} -33 z^6 a^{-4} +16 z^4 a^{-4} +7 z^9 a^{-3} -7 z^7 a^{-3} -24 z^5 a^{-3} +32 z^3 a^{-3} -10 z a^{-3} +z^{10} a^{-2} +8 z^8 a^{-2} -32 z^6 a^{-2} +28 z^4 a^{-2} -4 z^2 a^{-2} +3 z^9 a^{-1} +a z^7-5 z^7 a^{-1} -4 a z^5-10 z^5 a^{-1} +6 a z^3+22 z^3 a^{-1} -4 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +3 z^8-11 z^6+12 z^4-3 z^2-1 (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 \
-4-3-2-101234567χ
18           1-1
16          3 3
14         51 -4
12        73  4
10       95   -4
8      97    2
6     89     1
4    79      -2
2   510       5
0  25        -3
-2 15         4
-4 2          -2
-61           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
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}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L11a288.gif

L11a288

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L11a290