L10a17

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L10a16.gif

L10a16

L10a18.gif

L10a18

Contents

L10a17.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a17's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X16,8,17,7 X20,18,5,17 X18,14,19,13 X14,20,15,19 X12,10,13,9 X8,16,9,15 X2536 X4,12,1,11
Gauss code {1, -9, 2, -10}, {9, -1, 3, -8, 7, -2, 10, -7, 5, -6, 8, -3, 4, -5, 6, -4}
A Braid Representative
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A Morse Link Presentation L10a17 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(3 v^2-4 v+3\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial 13 q^{9/2}-13 q^{7/2}+9 q^{5/2}-6 q^{3/2}-q^{19/2}+4 q^{17/2}-7 q^{15/2}+11 q^{13/2}-13 q^{11/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z a^{-9} +3 z^3 a^{-7} +4 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -5 z^3 a^{-5} -5 z a^{-5} -2 a^{-5} z^{-1} -z^5 a^{-3} -z^3 a^{-3} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-11} -z^3 a^{-11} +4 z^6 a^{-10} -7 z^4 a^{-10} +3 z^2 a^{-10} +6 z^7 a^{-9} -10 z^5 a^{-9} +4 z^3 a^{-9} -2 z a^{-9} +4 z^8 a^{-8} +2 z^6 a^{-8} -17 z^4 a^{-8} +12 z^2 a^{-8} -2 a^{-8} +z^9 a^{-7} +12 z^7 a^{-7} -29 z^5 a^{-7} +23 z^3 a^{-7} -8 z a^{-7} + a^{-7} z^{-1} +7 z^8 a^{-6} -5 z^6 a^{-6} -10 z^4 a^{-6} +14 z^2 a^{-6} -5 a^{-6} +z^9 a^{-5} +9 z^7 a^{-5} -22 z^5 a^{-5} +21 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +3 z^8 a^{-4} -z^6 a^{-4} -3 z^4 a^{-4} +5 z^2 a^{-4} -3 a^{-4} +3 z^7 a^{-3} -3 z^5 a^{-3} +2 z^6 a^{-2} -3 z^4 a^{-2} + a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +3 z a^{-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 \
-2-1012345678χ
20          11
18         3 -3
16        41 3
14       73  -4
12      64   2
10     77    0
8    66     0
6   37      4
4  36       -3
2 15        4
0 1         -1
-21          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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L10a16

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L10a18