L11a168

From Knot Atlas
Jump to: navigation, search

L11a167.gif

L11a167

L11a169.gif

L11a169

Contents

L11a168.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a168 at Knotilus!


Link Presentations

[edit Notes on L11a168's Link Presentations]

Planar diagram presentation X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X20,13,21,14 X12,19,13,20 X22,16,7,15 X4,22,5,21
Gauss code {1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -9, 8, -5, 10, -7, 4, -2, 9, -8, 11, -10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation L11a168 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^6-3 u^2 v^5+4 u^2 v^4-4 u^2 v^3+3 u^2 v^2-u^2 v-u v^6+4 u v^5-7 u v^4+7 u v^3-7 u v^2+4 u v-u-v^5+3 v^4-4 v^3+4 v^2-3 v+1}{u v^3} (db)
Jones polynomial -q^{5/2}+4 q^{3/2}-8 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{19}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^5 z^7-4 a^5 z^5-4 a^5 z^3+a^3 z^9+6 a^3 z^7+12 a^3 z^5+8 a^3 z^3+a^3 z^{-1} -a z^7-4 a z^5-4 a z^3-a z-a z^{-1} (db)
Kauffman polynomial a^{10} z^4+4 a^9 z^5-2 a^9 z^3+8 a^8 z^6-7 a^8 z^4+2 a^8 z^2+11 a^7 z^7-13 a^7 z^5+4 a^7 z^3+11 a^6 z^8-14 a^6 z^6+4 a^6 z^2+8 a^5 z^9-9 a^5 z^7-4 a^5 z^5+2 a^5 z^3+3 a^4 z^{10}+7 a^4 z^8-30 a^4 z^6+18 a^4 z^4+14 a^3 z^9-42 a^3 z^7+38 a^3 z^5-14 a^3 z^3+a^3 z-a^3 z^{-1} +3 a^2 z^{10}-22 a^2 z^6+23 a^2 z^4-4 a^2 z^2+a^2+6 a z^9-21 a z^7+z^7 a^{-1} +22 a z^5-3 z^5 a^{-1} -8 a z^3+2 z^3 a^{-1} +a z-a z^{-1} +4 z^8-14 z^6+13 z^4-2 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 \
-7-6-5-4-3-2-101234χ
6           11
4          3 -3
2         51 4
0        73  -4
-2       115   6
-4      98    -1
-6     1110     1
-8    810      2
-10   510       -5
-12  38        5
-14 15         -4
-16 3          3
-181           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.

L11a167.gif

L11a167

L11a169.gif

L11a169