L11a7

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L11a6.gif

L11a6

L11a8.gif

L11a8

Contents

L11a7.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a7 at Knotilus!


Link Presentations

[edit Notes on L11a7's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X22,14,5,13 X14,22,15,21 X18,10,19,9 X20,12,21,11 X10,20,11,19 X2,16,3,15
Gauss code {1, -11, 5, -3}, {4, -1, 2, -5, 8, -10, 9, -4, 6, -7, 11, -2, 3, -8, 10, -9, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a7 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(2 v^4-3 v^3+3 v^2-3 v+2\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial 16 q^{9/2}-17 q^{7/2}+14 q^{5/2}-12 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+3 q^{17/2}-5 q^{15/2}+10 q^{13/2}-14 q^{11/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-7} -3 z^3 a^{-7} -z a^{-7} + a^{-7} z^{-1} +z^7 a^{-5} +4 z^5 a^{-5} +5 z^3 a^{-5} +z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +z^3 a^{-3} -z a^{-3} -z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-11} -2 z^3 a^{-11} +3 z^6 a^{-10} -7 z^4 a^{-10} +4 z^2 a^{-10} +4 z^7 a^{-9} -7 z^5 a^{-9} +3 z^3 a^{-9} +4 z^8 a^{-8} -4 z^6 a^{-8} -3 z^4 a^{-8} +7 z^2 a^{-8} -2 a^{-8} +4 z^9 a^{-7} -8 z^7 a^{-7} +13 z^5 a^{-7} -9 z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} -2 z^6 a^{-6} +z^4 a^{-6} +6 z^2 a^{-6} -5 a^{-6} +9 z^9 a^{-5} -27 z^7 a^{-5} +38 z^5 a^{-5} -24 z^3 a^{-5} +3 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +2 z^8 a^{-4} -12 z^6 a^{-4} +8 z^4 a^{-4} +2 z^2 a^{-4} -3 a^{-4} +5 z^9 a^{-3} -11 z^7 a^{-3} +6 z^5 a^{-3} -4 z^3 a^{-3} +2 z a^{-3} +6 z^8 a^{-2} -16 z^6 a^{-2} +9 z^4 a^{-2} -z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -11 z^5 a^{-1} +6 z^3 a^{-1} +z a^{-1} - a^{-1} z^{-1} +z^6-2 z^4 (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 \
-3-2-1012345678χ
20           11
18          2 -2
16         31 2
14        72  -5
12       73   4
10      97    -2
8     87     1
6    69      3
4   68       -2
2  38        5
0 14         -3
-2 3          3
-41           -1
Integral Khovanov Homology

(db, data source)

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