L11a14

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L11a13

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L11a15

Contents

L11a14.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a14 at Knotilus!


Link Presentations

[edit Notes on L11a14's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X18,10,19,9 X8493 X14,6,15,5 X20,16,21,15 X22,18,5,17 X16,22,17,21 X10,20,11,19 X2,12,3,11
Gauss code {1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 11, -2, 3, -6, 7, -9, 8, -4, 10, -7, 9, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a14 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(2 v^4-4 v^3+5 v^2-4 v+2\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial 22 q^{9/2}-22 q^{7/2}+17 q^{5/2}-13 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+5 q^{17/2}-10 q^{15/2}+16 q^{13/2}-20 q^{11/2}+6 \sqrt{q}-\frac{3}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-7} -z^3 a^{-7} +2 z a^{-7} +2 a^{-7} z^{-1} +z^7 a^{-5} +2 z^5 a^{-5} -3 z^3 a^{-5} -9 z a^{-5} -5 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +9 z a^{-3} +3 a^{-3} z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -2 z a^{-1} (db)
Kauffman polynomial z^5 a^{-11} +5 z^6 a^{-10} -5 z^4 a^{-10} + a^{-10} +10 z^7 a^{-9} -14 z^5 a^{-9} +3 z^3 a^{-9} +11 z^8 a^{-8} -14 z^6 a^{-8} +z^4 a^{-8} +7 z^9 a^{-7} -z^7 a^{-7} -12 z^5 a^{-7} +6 z^3 a^{-7} -3 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +13 z^8 a^{-6} -28 z^6 a^{-6} +12 z^4 a^{-6} +6 z^2 a^{-6} -5 a^{-6} +11 z^9 a^{-5} -17 z^7 a^{-5} +2 z^5 a^{-5} +14 z^3 a^{-5} -13 z a^{-5} +5 a^{-5} z^{-1} +2 z^{10} a^{-4} +6 z^8 a^{-4} -17 z^6 a^{-4} +9 z^4 a^{-4} +7 z^2 a^{-4} -5 a^{-4} +4 z^9 a^{-3} -3 z^7 a^{-3} -10 z^5 a^{-3} +20 z^3 a^{-3} -14 z a^{-3} +3 a^{-3} z^{-1} +4 z^8 a^{-2} -7 z^6 a^{-2} +3 z^2 a^{-2} +3 z^7 a^{-1} -9 z^5 a^{-1} +9 z^3 a^{-1} -4 z a^{-1} +z^6-3 z^4+2 z^2 (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          4 -4
16         61 5
14        104  -6
12       106   4
10      1210    -2
8     1010     0
6    712      5
4   610       -4
2  29        7
0 14         -3
-2 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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