L10a43

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L10a42.gif

L10a42

L10a44.gif

L10a44

Contents

L10a43.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a43's Link Presentations]

Planar diagram presentation X6172 X2,9,3,10 X12,3,13,4 X20,12,5,11 X18,6,19,5 X10,20,11,19 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15
Gauss code {1, -2, 3, -7}, {5, -1, 9, -10, 2, -6, 4, -3, 8, -9, 10, -8, 7, -5, 6, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a43 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^5-4 u v^4+8 u v^3-8 u v^2+4 u v+4 v^4-8 v^3+8 v^2-4 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -9 q^{9/2}+13 q^{7/2}-\frac{1}{q^{7/2}}-16 q^{5/2}+\frac{3}{q^{5/2}}+17 q^{3/2}-\frac{8}{q^{3/2}}-q^{13/2}+5 q^{11/2}-16 \sqrt{q}+\frac{11}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +a z^5-5 z^5 a^{-1} +2 z^5 a^{-3} +3 a z^3-12 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +5 a z-12 z a^{-1} +5 z a^{-3} +3 a z^{-1} -5 a^{-1} z^{-1} +2 a^{-3} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} +5 z^6 a^{-6} -6 z^4 a^{-6} + a^{-6} +9 z^7 a^{-5} -15 z^5 a^{-5} +5 z^3 a^{-5} +7 z^8 a^{-4} -4 z^6 a^{-4} -9 z^4 a^{-4} +4 z^2 a^{-4} +2 z^9 a^{-3} +15 z^7 a^{-3} +a^3 z^5-34 z^5 a^{-3} -2 a^3 z^3+18 z^3 a^{-3} +a^3 z-5 z a^{-3} +2 a^{-3} z^{-1} +12 z^8 a^{-2} +3 a^2 z^6-13 z^6 a^{-2} -4 a^2 z^4-10 z^4 a^{-2} +a^2 z^2+17 z^2 a^{-2} -5 a^{-2} +2 z^9 a^{-1} +6 a z^7+12 z^7 a^{-1} -12 a z^5-31 z^5 a^{-1} +14 a z^3+29 z^3 a^{-1} -10 a z-16 z a^{-1} +3 a z^{-1} +5 a^{-1} z^{-1} +5 z^8-z^6-11 z^4+14 z^2-5 (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 \
-4-3-2-10123456χ
14          11
12         4 -4
10        51 4
8       84  -4
6      85   3
4     98    -1
2    78     -1
0   510      5
-2  36       -3
-4  5        5
-613         -2
-81          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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}^{7}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\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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L10a42

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L10a44