L10a104

From Knot Atlas
Jump to: navigation, search

L10a103.gif

L10a103

L10a105.gif

L10a105

Contents

L10a104.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a104 at Knotilus!


Link Presentations

[edit Notes on L10a104's Link Presentations]

Planar diagram presentation X10,1,11,2 X16,11,17,12 X8,9,1,10 X20,17,9,18 X12,4,13,3 X14,8,15,7 X6,14,7,13 X18,6,19,5 X4,20,5,19 X2,16,3,15
Gauss code {1, -10, 5, -9, 8, -7, 6, -3}, {3, -1, 2, -5, 7, -6, 10, -2, 4, -8, 9, -4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a104 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 \left(-v^3\right)+2 u^3 v^2-u^3 v+3 u^2 v^3-7 u^2 v^2+7 u^2 v-2 u^2-2 u v^3+7 u v^2-7 u v+3 u-v^2+2 v-1}{u^{3/2} v^{3/2}} (db)
Jones polynomial -7 q^{9/2}+11 q^{7/2}-\frac{1}{q^{7/2}}-15 q^{5/2}+\frac{4}{q^{5/2}}+15 q^{3/2}-\frac{8}{q^{3/2}}-q^{13/2}+3 q^{11/2}-15 \sqrt{q}+\frac{12}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} +6 z^3 a^{-3} +6 z a^{-3} + a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-6 z^3 a^{-1} +a z-3 z a^{-1} (db)
Kauffman polynomial z^5 a^{-7} -2 z^3 a^{-7} +z a^{-7} +3 z^6 a^{-6} -5 z^4 a^{-6} +2 z^2 a^{-6} +5 z^7 a^{-5} -8 z^5 a^{-5} +6 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -5 z^6 a^{-4} +2 z^2 a^{-4} - a^{-4} +2 z^9 a^{-3} +9 z^7 a^{-3} +a^3 z^5-25 z^5 a^{-3} -a^3 z^3+23 z^3 a^{-3} -9 z a^{-3} + a^{-3} z^{-1} +11 z^8 a^{-2} +4 a^2 z^6-17 z^6 a^{-2} -6 a^2 z^4+6 z^4 a^{-2} +2 a^2 z^2+z^2 a^{-2} +2 z^9 a^{-1} +7 a z^7+11 z^7 a^{-1} -12 a z^5-29 z^5 a^{-1} +6 a z^3+22 z^3 a^{-1} -2 a z-6 z a^{-1} +6 z^8-5 z^6-5 z^4+3 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 \
-4-3-2-10123456χ
14          11
12         2 -2
10        51 4
8       73  -4
6      84   4
4     77    0
2    88     0
0   58      3
-2  37       -4
-4 15        4
-6 3         -3
-81          1
Integral Khovanov Homology

(db, data source)

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

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.

L10a103.gif

L10a103

L10a105.gif

L10a105