L10a9

From Knot Atlas
Jump to: navigation, search

L10a8.gif

L10a8

L10a10.gif

L10a10

Contents

L10a9.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a9 at Knotilus!


Link Presentations

[edit Notes on L10a9's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X18,14,19,13 X16,9,17,10 X8,17,9,18 X20,16,5,15 X14,20,15,19 X2536 X4,12,1,11
Gauss code {1, -9, 2, -10}, {9, -1, 3, -6, 5, -2, 10, -3, 4, -8, 7, -5, 6, -4, 8, -7}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10a9 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial 9 q^{9/2}-10 q^{7/2}+10 q^{5/2}-\frac{1}{q^{5/2}}-10 q^{3/2}+\frac{2}{q^{3/2}}-q^{15/2}+3 q^{13/2}-6 q^{11/2}+7 \sqrt{q}-\frac{5}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-5} -3 z^3 a^{-5} -3 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +10 z^3 a^{-3} +10 z a^{-3} +3 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-10 z a^{-1} +2 a z^{-1} -4 a^{-1} z^{-1} (db)
Kauffman polynomial z^3 a^{-9} +3 z^4 a^{-8} +6 z^5 a^{-7} -4 z^3 a^{-7} +z a^{-7} +9 z^6 a^{-6} -14 z^4 a^{-6} +7 z^2 a^{-6} - a^{-6} +9 z^7 a^{-5} -18 z^5 a^{-5} +10 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -4 z^6 a^{-4} -15 z^4 a^{-4} +14 z^2 a^{-4} -3 a^{-4} +z^9 a^{-3} +11 z^7 a^{-3} -44 z^5 a^{-3} +45 z^3 a^{-3} -19 z a^{-3} +3 a^{-3} z^{-1} +7 z^8 a^{-2} -21 z^6 a^{-2} +11 z^4 a^{-2} +6 z^2 a^{-2} -3 a^{-2} +z^9 a^{-1} +a z^7+3 z^7 a^{-1} -5 a z^5-25 z^5 a^{-1} +9 a z^3+39 z^3 a^{-1} -7 a z-21 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +2 z^8-8 z^6+9 z^4-z^2-2 (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 \
-4-3-2-10123456χ
16          11
14         2 -2
12        41 3
10       52  -3
8      54   1
6     55    0
4    55     0
2   47      3
0  13       -2
-2 14        3
-4 1         -1
-61          1
Integral Khovanov Homology

(db, data source)

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

Read me first: Modifying Knot Pages

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

See/edit the Link_Splice_Base (expert).

Back to the top.

L10a8.gif

L10a8

L10a10.gif

L10a10