L10a61

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L10a60.gif

L10a60

L10a62.gif

L10a62

Contents

L10a61.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a61's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X20,10,7,9 X2738 X18,14,19,13 X6,12,1,11 X16,20,17,19 X4,16,5,15 X14,6,15,5 X12,18,13,17
Gauss code {1, -4, 2, -8, 9, -6}, {4, -1, 3, -2, 6, -10, 5, -9, 8, -7, 10, -5, 7, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a61 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^4-t(2)^4+2 t(1)^2 t(2)^3-6 t(1) t(2)^3+3 t(2)^3-4 t(1)^2 t(2)^2+9 t(1) t(2)^2-4 t(2)^2+3 t(1)^2 t(2)-6 t(1) t(2)+2 t(2)-t(1)^2+t(1)}{t(1) t(2)^2} (db)
Jones polynomial -q^{19/2}+4 q^{17/2}-7 q^{15/2}+11 q^{13/2}-14 q^{11/2}+14 q^{9/2}-14 q^{7/2}+10 q^{5/2}-7 q^{3/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-3} -2 z^5 a^{-5} +z^3 a^{-1} -5 z^3 a^{-5} +3 z^3 a^{-7} +z a^{-1} +3 z a^{-3} -6 z a^{-5} +4 z a^{-7} -z a^{-9} +2 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} (db)
Kauffman polynomial z^5 a^{-11} -z^3 a^{-11} +4 z^6 a^{-10} -7 z^4 a^{-10} +3 z^2 a^{-10} +6 z^7 a^{-9} -10 z^5 a^{-9} +5 z^3 a^{-9} -2 z a^{-9} +4 z^8 a^{-8} +3 z^6 a^{-8} -18 z^4 a^{-8} +12 z^2 a^{-8} - a^{-8} +z^9 a^{-7} +14 z^7 a^{-7} -34 z^5 a^{-7} +28 z^3 a^{-7} -10 z a^{-7} + a^{-7} z^{-1} +8 z^8 a^{-6} -5 z^6 a^{-6} -13 z^4 a^{-6} +14 z^2 a^{-6} -3 a^{-6} +z^9 a^{-5} +13 z^7 a^{-5} -33 z^5 a^{-5} +32 z^3 a^{-5} -15 z a^{-5} +3 a^{-5} z^{-1} +4 z^8 a^{-4} -z^6 a^{-4} -7 z^4 a^{-4} +7 z^2 a^{-4} -3 a^{-4} +5 z^7 a^{-3} -9 z^5 a^{-3} +8 z^3 a^{-3} -6 z a^{-3} +2 a^{-3} z^{-1} +3 z^6 a^{-2} -5 z^4 a^{-2} +2 z^2 a^{-2} +z^5 a^{-1} -2 z^3 a^{-1} +z 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 \
-2-1012345678χ
20          11
18         3 -3
16        41 3
14       73  -4
12      74   3
10     77    0
8    77     0
6   48      4
4  36       -3
2 15        4
0 2         -2
-21          1
Integral Khovanov Homology

(db, data source)

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