L11n51

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L11n50.gif

L11n50

L11n52.gif

L11n52

Contents

L11n51.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n51 at Knotilus!


Link Presentations

[edit Notes on L11n51's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X18,14,19,13 X9,17,10,16 X17,9,18,8 X22,20,5,19 X20,15,21,16 X14,21,15,22 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, 6, -5, -2, 11, -3, 4, -9, 8, 5, -6, -4, 7, -8, 9, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n51 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^4-2 v^3-2 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{13/2}-4 q^{11/2}+6 q^{9/2}-7 q^{7/2}+8 q^{5/2}-8 q^{3/2}+6 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-8 z^3 a^{-1} +9 z^3 a^{-3} -4 z^3 a^{-5} +3 a z-9 z a^{-1} +9 z a^{-3} -4 z a^{-5} +z a^{-7} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial z^2 a^{-8} +4 z^3 a^{-7} -z a^{-7} +2 z^6 a^{-6} -z^4 a^{-6} +z^2 a^{-6} - a^{-6} +5 z^7 a^{-5} -15 z^5 a^{-5} +16 z^3 a^{-5} -6 z a^{-5} + a^{-5} z^{-1} +4 z^8 a^{-4} -10 z^6 a^{-4} +3 z^4 a^{-4} +4 z^2 a^{-4} -3 a^{-4} +z^9 a^{-3} +6 z^7 a^{-3} -32 z^5 a^{-3} +37 z^3 a^{-3} -17 z a^{-3} +3 a^{-3} z^{-1} +6 z^8 a^{-2} -20 z^6 a^{-2} +12 z^4 a^{-2} +4 z^2 a^{-2} -3 a^{-2} +z^9 a^{-1} +a z^7+2 z^7 a^{-1} -5 a z^5-22 z^5 a^{-1} +9 a z^3+34 z^3 a^{-1} -7 a z-19 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +2 z^8-8 z^6+8 z^4-2 (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
14         1-1
12        3 3
10       31 -2
8      43  1
6     43   -1
4    44    0
2   46     2
0  12      -1
-2 14       3
-4 1        -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}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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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L11n52