L11a10

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L11a9.gif

L11a9

L11a11.gif

L11a11

Contents

L11a10.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a10's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X8493 X16,10,17,9 X22,14,5,13 X10,16,11,15 X14,22,15,21 X20,12,21,11 X2,18,3,17
Gauss code {1, -11, 5, -3}, {4, -1, 2, -5, 6, -8, 10, -4, 7, -9, 8, -6, 11, -2, 3, -10, 9, -7}
A Braid Representative
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A Morse Link Presentation L11a10 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(4 t(2)^2-7 t(2)+4\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -q^{17/2}+4 q^{15/2}-7 q^{13/2}+12 q^{11/2}-16 q^{9/2}+18 q^{7/2}-20 q^{5/2}+16 q^{3/2}-13 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{1}{q^{5/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-7} + a^{-7} z^{-1} +z^5 a^{-5} -2 z a^{-5} -2 a^{-5} z^{-1} +2 z^5 a^{-3} +3 z^3 a^{-3} +2 z a^{-3} +z^5 a^{-1} -a z^3+ a^{-1} z^{-1} (db)
Kauffman polynomial z^7 a^{-9} -3 z^5 a^{-9} +2 z^3 a^{-9} +4 z^8 a^{-8} -15 z^6 a^{-8} +16 z^4 a^{-8} -3 z^2 a^{-8} -2 a^{-8} +5 z^9 a^{-7} -16 z^7 a^{-7} +13 z^5 a^{-7} -2 z^3 a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} +5 z^8 a^{-6} -32 z^6 a^{-6} +32 z^4 a^{-6} -3 z^2 a^{-6} -5 a^{-6} +11 z^9 a^{-5} -27 z^7 a^{-5} +11 z^5 a^{-5} +5 z^3 a^{-5} -3 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +11 z^8 a^{-4} -37 z^6 a^{-4} +25 z^4 a^{-4} -z^2 a^{-4} -3 a^{-4} +6 z^9 a^{-3} +z^7 a^{-3} -24 z^5 a^{-3} +19 z^3 a^{-3} -4 z a^{-3} +10 z^8 a^{-2} -12 z^6 a^{-2} +a^2 z^4+z^4 a^{-2} + a^{-2} +11 z^7 a^{-1} +4 a z^5-15 z^5 a^{-1} -2 a z^3+8 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +8 z^6-7 z^4+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
18           11
16          3 -3
14         41 3
12        83  -5
10       84   4
8      108    -2
6     108     2
4    610      4
2   710       -3
0  38        5
-2 15         -4
-4 3          3
-61           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L11a9

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L11a11