L11a25

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L11a24

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L11a26

Contents

L11a25.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a25 at Knotilus!


Link Presentations

[edit Notes on L11a25's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,8,15,7 X20,16,21,15 X18,12,19,11 X12,20,13,19 X22,18,5,17 X16,22,17,21 X8,14,9,13 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -6, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a25 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^5-4 u v^4+6 u v^3-7 u v^2+6 u v-3 u-3 v^5+6 v^4-7 v^3+6 v^2-4 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial 16 q^{9/2}-17 q^{7/2}+17 q^{5/2}-\frac{1}{q^{5/2}}-15 q^{3/2}+\frac{2}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 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} +2 z a^{-7} -2 z^5 a^{-5} -5 z^3 a^{-5} -2 z a^{-5} + a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +2 z^3 a^{-3} - a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +3 a z-5 z a^{-1} +2 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -8 z^9 a^{-3} -5 z^9 a^{-5} -8 z^8 a^{-2} -15 z^8 a^{-4} -9 z^8 a^{-6} -2 z^8-a z^7+8 z^7 a^{-1} +16 z^7 a^{-3} -2 z^7 a^{-5} -9 z^7 a^{-7} +35 z^6 a^{-2} +49 z^6 a^{-4} +14 z^6 a^{-6} -6 z^6 a^{-8} +6 z^6+5 a z^5-4 z^5 a^{-1} -z^5 a^{-3} +25 z^5 a^{-5} +14 z^5 a^{-7} -3 z^5 a^{-9} -38 z^4 a^{-2} -49 z^4 a^{-4} -7 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} -3 z^4-9 a z^3-2 z^3 a^{-1} -z^3 a^{-3} -22 z^3 a^{-5} -12 z^3 a^{-7} +2 z^3 a^{-9} +14 z^2 a^{-2} +23 z^2 a^{-4} -4 z^2 a^{-8} +z^2 a^{-10} -4 z^2+7 a z+3 z a^{-1} -4 z a^{-3} +3 z a^{-5} +3 z a^{-7} -3 a^{-4} + a^{-8} +3-2 a z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} + a^{-5} z^{-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          2 2
14         51 -4
12        72  5
10       95   -4
8      87    1
6     99     0
4    68      -2
2   510       5
0  25        -3
-2  5         5
-412          -1
-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} {\mathbb Z}
r=-3 {\mathbb Z}^{2}
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}^{6}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L11a24

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