L10a62

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L10a61.gif

L10a61

L10a63.gif

L10a63

Contents

L10a62.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a62's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X20,10,7,9 X2738 X16,12,17,11 X14,5,15,6 X4,15,5,16 X18,14,19,13 X12,18,13,17 X6,20,1,19
Gauss code {1, -4, 2, -7, 6, -10}, {4, -1, 3, -2, 5, -9, 8, -6, 7, -5, 9, -8, 10, -3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a62 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-2 u^2 v^3+2 u^2 v^2-2 u^2 v+u^2-2 u v^4+3 u v^3-3 u v^2+3 u v-2 u+v^4-2 v^3+2 v^2-2 v+1}{u v^2} (db)
Jones polynomial 8 q^{9/2}-9 q^{7/2}+9 q^{5/2}-\frac{1}{q^{5/2}}-9 q^{3/2}+\frac{2}{q^{3/2}}-q^{15/2}+3 q^{13/2}-5 q^{11/2}+6 \sqrt{q}-\frac{5}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-5} -3 z^3 a^{-5} -2 z a^{-5} +z^7 a^{-3} +5 z^5 a^{-3} +9 z^3 a^{-3} +7 z a^{-3} + a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-9 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} (db)
Kauffman polynomial z^3 a^{-9} +3 z^4 a^{-8} -z^2 a^{-8} +5 z^5 a^{-7} -3 z^3 a^{-7} +7 z^6 a^{-6} -10 z^4 a^{-6} +4 z^2 a^{-6} +7 z^7 a^{-5} -14 z^5 a^{-5} +9 z^3 a^{-5} -4 z a^{-5} +4 z^8 a^{-4} -3 z^6 a^{-4} -13 z^4 a^{-4} +11 z^2 a^{-4} - a^{-4} +z^9 a^{-3} +8 z^7 a^{-3} -35 z^5 a^{-3} +36 z^3 a^{-3} -13 z a^{-3} + a^{-3} z^{-1} +6 z^8 a^{-2} -18 z^6 a^{-2} +8 z^4 a^{-2} +7 z^2 a^{-2} -3 a^{-2} +z^9 a^{-1} +a z^7+2 z^7 a^{-1} -5 a z^5-21 z^5 a^{-1} +9 a z^3+32 z^3 a^{-1} -7 a z-16 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +2 z^8-8 z^6+8 z^4+z^2-3 (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
16          11
14         2 -2
12        31 2
10       52  -3
8      43   1
6     55    0
4    44     0
2   36      3
0  23       -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}^{2}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L10a61

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L10a63