L9a37

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L9a36.gif

L9a36

L9a38.gif

L9a38

Contents

L9a37.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a37 at Knotilus!

L9a37 is 9^2_{7} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a37's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X18,12,9,11 X2,9,3,10 X4,18,5,17 X16,8,17,7 X14,6,15,5 X6,16,7,15 X8,14,1,13
Gauss code {1, -4, 2, -5, 7, -8, 6, -9}, {4, -1, 3, -2, 9, -7, 8, -6, 5, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9a37 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^2-u^3 v+u^2 v^3-4 u^2 v^2+3 u^2 v-u^2-u v^3+3 u v^2-4 u v+u-v^2+v}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{17/2}-3 q^{15/2}+5 q^{13/2}-7 q^{11/2}+7 q^{9/2}-8 q^{7/2}+6 q^{5/2}-4 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +z a^{-7} -z^5 a^{-5} -2 z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} -z^5 a^{-3} -2 z^3 a^{-3} + a^{-3} z^{-1} +z^3 a^{-1} +2 z a^{-1} (db)
Kauffman polynomial z^4 a^{-10} -z^2 a^{-10} +3 z^5 a^{-9} -4 z^3 a^{-9} +z a^{-9} +4 z^6 a^{-8} -5 z^4 a^{-8} +z^2 a^{-8} +3 z^7 a^{-7} -2 z^5 a^{-7} +z^8 a^{-6} +4 z^6 a^{-6} -7 z^4 a^{-6} +3 z^2 a^{-6} +5 z^7 a^{-5} -9 z^5 a^{-5} +8 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +z^8 a^{-4} +2 z^6 a^{-4} -6 z^4 a^{-4} +4 z^2 a^{-4} - a^{-4} +2 z^7 a^{-3} -3 z^5 a^{-3} +z^3 a^{-3} -2 z a^{-3} + a^{-3} z^{-1} +2 z^6 a^{-2} -5 z^4 a^{-2} +3 z^2 a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +2 z a^{-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 \
-2-101234567χ
18         1-1
16        2 2
14       31 -2
12      42  2
10     44   0
8    43    1
6   24     2
4  24      -2
2 13       2
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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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L9a36.gif

L9a36

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L9a38