L11a165

From Knot Atlas
Jump to: navigation, search

L11a164.gif

L11a164

L11a166.gif

L11a166

Contents

L11a165.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a165 at Knotilus!


Link Presentations

[edit Notes on L11a165's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X20,18,21,17 X18,12,19,11 X12,20,13,19 X22,16,7,15 X16,22,17,21 X6718 X4,13,5,14
Gauss code {1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -7, 11, -4, 8, -9, 5, -6, 7, -5, 9, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a165 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^4-4 u^2 v^3+3 u^2 v^2+3 u v^3-5 u v^2+3 u v+3 v^2-4 v+2}{u v^2} (db)
Jones polynomial \frac{6}{q^{9/2}}+q^{7/2}-\frac{8}{q^{7/2}}-2 q^{5/2}+\frac{8}{q^{5/2}}+4 q^{3/2}-\frac{9}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{4}{q^{11/2}}-6 \sqrt{q}+\frac{7}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^5 z^5+4 a^5 z^3+4 a^5 z+a^5 z^{-1} -a^3 z^7-5 a^3 z^5-8 a^3 z^3-5 a^3 z-a z^7-5 a z^5+z^5 a^{-1} -8 a z^3+4 z^3 a^{-1} -6 a z+4 z a^{-1} -2 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-3 a^3 z^9-5 a z^9-2 z^9 a^{-1} -4 a^4 z^8-2 a^2 z^8-z^8 a^{-2} +z^8-5 a^5 z^7+7 a^3 z^7+23 a z^7+11 z^7 a^{-1} -4 a^6 z^6+8 a^4 z^6+20 a^2 z^6+6 z^6 a^{-2} +14 z^6-3 a^7 z^5+12 a^5 z^5+2 a^3 z^5-32 a z^5-19 z^5 a^{-1} -2 a^8 z^4+5 a^6 z^4+2 a^4 z^4-26 a^2 z^4-12 z^4 a^{-2} -33 z^4-a^9 z^3+2 a^7 z^3-11 a^5 z^3-7 a^3 z^3+19 a z^3+12 z^3 a^{-1} +a^8 z^2-2 a^6 z^2-5 a^4 z^2+12 a^2 z^2+9 z^2 a^{-2} +23 z^2+a^9 z-a^7 z+6 a^5 z+4 a^3 z-8 a z-4 z a^{-1} +a^4-3 a^2-2 a^{-2} -5-a^5 z^{-1} +2 a z^{-1} + a^{-1} z^{-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 \
-6-5-4-3-2-1012345χ
8           1-1
6          1 1
4         31 -2
2        31  2
0       43   -1
-2      53    2
-4     45     1
-6    44      0
-8   24       2
-10  24        -2
-12  2         2
-1412          -1
-161           1
Integral Khovanov Homology

(db, data source)

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

L11a164.gif

L11a164

L11a166.gif

L11a166