L11n370

From Knot Atlas
Jump to: navigation, search

L11n369.gif

L11n369

L11n371.gif

L11n371

Contents

L11n370.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n370 at Knotilus!


Link Presentations

[edit Notes on L11n370's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,8,15,7 X15,20,16,21 X11,19,12,18 X17,13,18,12 X19,22,20,17 X21,16,22,5 X8,14,9,13 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {-6, 5, -7, 4, -8, 7}, {10, -1, 3, -9, 11, -2, -5, 6, 9, -3, -4, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n370 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) (w-1)^3}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial - q^{-6} +5 q^{-5} -7 q^{-4} +q^3+11 q^{-3} -3 q^2-10 q^{-2} +6 q+11 q^{-1} -9 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^6 z^{-2} -a^4 z^4-a^4 z^2-2 a^4 z^{-2} -2 a^4+a^2 z^6+3 a^2 z^4+4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +3 a^2+ a^{-2} -2 z^4-4 z^2-2 (db)
Kauffman polynomial a^7 z^3+5 a^6 z^4-a^6 z^2+a^6 z^{-2} -2 a^6+2 a^5 z^7+2 a^5 z^3+2 a^5 z-2 a^5 z^{-1} +3 a^4 z^8-4 a^4 z^6+6 a^4 z^4-3 a^4 z^2+2 a^4 z^{-2} -2 a^4+a^3 z^9+6 a^3 z^7-16 a^3 z^5+12 a^3 z^3-2 a^3 z-2 a^3 z^{-1} +6 a^2 z^8-9 a^2 z^6+z^6 a^{-2} -3 a^2 z^4-3 z^4 a^{-2} +3 a^2 z^2+3 z^2 a^{-2} +a^2 z^{-2} -a^2- a^{-2} +a z^9+7 a z^7+3 z^7 a^{-1} -25 a z^5-9 z^5 a^{-1} +19 a z^3+8 z^3 a^{-1} -6 a z-2 z a^{-1} +3 z^8-4 z^6-7 z^4+8 z^2-1 (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 \
-5-4-3-2-101234χ
7         11
5        2 -2
3       41 3
1      52  -3
-1     64   2
-3    67    1
-5   54     1
-7  26      4
-9 35       -2
-11 4        4
-131         -1
Integral Khovanov Homology

(db, data source)

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

L11n369.gif

L11n369

L11n371.gif

L11n371