L7a4

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L7a3.gif

L7a3

L7a5.gif

L7a5

Contents

L7a4.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L7a4 at Knotilus!

L7a4 is 7^2_3 in the Rolfsen table of links.


Link Presentations

[edit Notes on L7a4's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X14,8,5,7 X12,10,13,9 X8,14,9,13 X2536 X4,12,1,11
Gauss code {1, -6, 2, -7}, {6, -1, 3, -5, 4, -2, 7, -4, 5, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L7a4 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 (u-1) (v-1)}{\sqrt{u} \sqrt{v}} (db)
Jones polynomial -2 q^{9/2}+2 q^{7/2}-3 q^{5/2}+3 q^{3/2}-\frac{1}{q^{3/2}}+q^{11/2}-3 \sqrt{q}+\frac{1}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-1} -z^3 a^{-3} +a z-z a^{-1} -z a^{-3} +z a^{-5} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -z^6 a^{-2} -z^6 a^{-4} -z^5 a^{-1} -3 z^5 a^{-3} -2 z^5 a^{-5} +z^4 a^{-2} +z^4 a^{-4} -z^4 a^{-6} -z^4-a z^3-z^3 a^{-1} +5 z^3 a^{-3} +5 z^3 a^{-5} -2 z^2 a^{-2} +2 z^2 a^{-6} +2 a z+2 z a^{-1} -2 z a^{-3} -2 z a^{-5} +1-a z^{-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 \
-2-1012345χ
12       1-1
10      1 1
8     11 0
6    21  1
4   11   0
2  22    0
0 13     2
-2        0
-41       1
Integral Khovanov Homology

(db, data source)

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