L11a453

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L11a452.gif

L11a452

L11a454.gif

L11a454

Contents

L11a453.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a453 at Knotilus!


Link Presentations

[edit Notes on L11a453's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X18,11,19,12 X16,8,17,7 X20,17,21,18 X12,19,5,20 X8,22,9,21 X22,10,13,9 X10,14,11,13 X2536 X4,16,1,15
Gauss code {1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, 3, -6}, {9, -2, 11, -4, 5, -3, 6, -5, 7, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a453 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(v w-v+1) (v w-w+1) (u v w-u w+u-v w+v-1)}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial q^7-3 q^6+7 q^5-12 q^4+16 q^3-16 q^2+18 q-14+11 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4} (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-4} +4 z^4 a^{-4} +6 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -z^8 a^{-2} -6 z^6 a^{-2} -a^2 z^4-15 z^4 a^{-2} -3 a^2 z^2-18 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2-9 a^{-2} +2 z^6+9 z^4+14 z^2+ z^{-2} +8 (db)
Kauffman polynomial z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -6 z^4 a^{-6} +4 z^2 a^{-6} - a^{-6} +9 z^7 a^{-5} -14 z^5 a^{-5} +11 z^3 a^{-5} -z a^{-5} +9 z^8 a^{-4} -15 z^6 a^{-4} +11 z^4 a^{-4} -6 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +5 z^9 a^{-3} +a^3 z^7+z^7 a^{-3} -4 a^3 z^5-22 z^5 a^{-3} +5 a^3 z^3+22 z^3 a^{-3} -2 a^3 z-8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +3 a^2 z^8+15 z^8 a^{-2} -12 a^2 z^6-51 z^6 a^{-2} +16 a^2 z^4+58 z^4 a^{-2} -9 a^2 z^2-37 z^2 a^{-2} -2 a^{-2} z^{-2} +3 a^2+11 a^{-2} +3 a z^9+8 z^9 a^{-1} -6 a z^7-15 z^7 a^{-1} -6 a z^5-7 z^5 a^{-1} +16 a z^3+20 z^3 a^{-1} -7 a z-12 z a^{-1} +2 a^{-1} z^{-1} +z^{10}+9 z^8-42 z^6+56 z^4-35 z^2- z^{-2} +11 (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 \
-5-4-3-2-10123456χ
15           11
13          2 -2
11         51 4
9        83  -5
7       84   4
5      88    0
3     108     2
1    711      4
-1   47       -3
-3  27        5
-5 14         -3
-7 2          2
-91           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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

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See/edit the Link Page master template (intermediate).

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L11a452.gif

L11a452

L11a454.gif

L11a454