L11a461

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L11a460.gif

L11a460

L11a462.gif

L11a462

Contents

L11a461.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a461's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X18,15,19,16 X16,6,17,5 X12,18,5,17 X8,22,9,21 X20,8,21,7 X22,10,13,9 X10,14,11,13 X2,11,3,12 X4,20,1,19
Gauss code {1, -10, 2, -11}, {4, -1, 7, -6, 8, -9, 10, -5}, {9, -2, 3, -4, 5, -3, 11, -7, 6, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11a461 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3-2 t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3-t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+2 t(3)^3 t(2)^2+4 t(1) t(3)^2 t(2)^2-4 t(3)^2 t(2)^2-2 t(1) t(3) t(2)^2+3 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-3 t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-2 t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial q^{10}-3 q^9+7 q^8-10 q^7+14 q^6-15 q^5+16 q^4-13 q^3+10 q^2-6 q+4- q^{-1} (db)
Signature 4 (db)
HOMFLY-PT polynomial z^4 a^{-8} +3 z^2 a^{-8} + a^{-8} z^{-2} +2 a^{-8} -2 z^6 a^{-6} -8 z^4 a^{-6} -9 z^2 a^{-6} -2 a^{-6} z^{-2} -5 a^{-6} +z^8 a^{-4} +5 z^6 a^{-4} +8 z^4 a^{-4} +5 z^2 a^{-4} + a^{-4} z^{-2} + a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} +2 a^{-2} (db)
Kauffman polynomial z^4 a^{-12} -z^2 a^{-12} +3 z^5 a^{-11} -2 z^3 a^{-11} +6 z^6 a^{-10} -8 z^4 a^{-10} +6 z^2 a^{-10} -2 a^{-10} +7 z^7 a^{-9} -8 z^5 a^{-9} +2 z^3 a^{-9} +z a^{-9} +7 z^8 a^{-8} -11 z^6 a^{-8} +8 z^4 a^{-8} -6 z^2 a^{-8} - a^{-8} z^{-2} +3 a^{-8} +5 z^9 a^{-7} -6 z^7 a^{-7} -4 z^5 a^{-7} +9 z^3 a^{-7} -8 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +5 z^8 a^{-6} -29 z^6 a^{-6} +37 z^4 a^{-6} -24 z^2 a^{-6} -2 a^{-6} z^{-2} +9 a^{-6} +10 z^9 a^{-5} -33 z^7 a^{-5} +29 z^5 a^{-5} -z^3 a^{-5} -8 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +2 z^8 a^{-4} -28 z^6 a^{-4} +36 z^4 a^{-4} -13 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +5 z^9 a^{-3} -19 z^7 a^{-3} +19 z^5 a^{-3} -5 z^3 a^{-3} +z a^{-3} +4 z^8 a^{-2} -16 z^6 a^{-2} +16 z^4 a^{-2} -2 z^2 a^{-2} -2 a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +z^3 a^{-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 \
-3-2-1012345678χ
21           11
19          31-2
17         4  4
15        63  -3
13       84   4
11      87    -1
9     87     1
7    58      3
5   58       -3
3  37        4
1 13         -2
-1 3          3
-31           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\mathbb Z} {\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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L11a460

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L11a462