L11n245

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L11n244.gif

L11n244

L11n246.gif

L11n246

Contents

L11n245.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n245 at Knotilus!


Link Presentations

[edit Notes on L11n245's Link Presentations]

Planar diagram presentation X12,1,13,2 X10,11,1,12 X14,5,15,6 X9,19,10,18 X17,3,18,2 X16,8,17,7 X3849 X20,16,21,15 X22,13,11,14 X4,20,5,19 X6,21,7,22
Gauss code {1, 5, -7, -10, 3, -11, 6, 7, -4, -2}, {2, -1, 9, -3, 8, -6, -5, 4, 10, -8, 11, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n245 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(2 u^2 v-u^2+u v^2-u v+u-v^2+2 v\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{9/2}-\frac{2}{q^{9/2}}-3 q^{7/2}+\frac{6}{q^{7/2}}+6 q^{5/2}-\frac{9}{q^{5/2}}-10 q^{3/2}+\frac{11}{q^{3/2}}+11 \sqrt{q}-\frac{13}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -2 a z^5-z^5 a^{-1} +2 a^3 z^3-5 a z^3-z^3 a^{-1} +z^3 a^{-3} +2 a^3 z-3 a z+z a^{-3} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 a z^9-2 z^9 a^{-1} -5 a^2 z^8-4 z^8 a^{-2} -9 z^8-4 a^3 z^7-3 a z^7-2 z^7 a^{-1} -3 z^7 a^{-3} -a^4 z^6+10 a^2 z^6+10 z^6 a^{-2} -z^6 a^{-4} +22 z^6+3 a^3 z^5+10 a z^5+16 z^5 a^{-1} +9 z^5 a^{-3} -6 a^4 z^4-14 a^2 z^4-5 z^4 a^{-2} +3 z^4 a^{-4} -16 z^4-3 a^5 z^3-2 a^3 z^3-4 a z^3-13 z^3 a^{-1} -8 z^3 a^{-3} +5 a^4 z^2+8 a^2 z^2-2 z^2 a^{-4} +5 z^2+a^5 z+4 z a^{-1} +3 z a^{-3} +1-a z^{-1} - a^{-1} 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 \
-4-3-2-1012345χ
10         1-1
8        2 2
6       41 -3
4      62  4
2     54   -1
0    86    2
-2   57     2
-4  46      -2
-6 25       3
-8 4        -4
-102         2
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L11n244.gif

L11n244

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L11n246