L11a74

From Knot Atlas
Jump to: navigation, search

L11a73.gif

L11a73

L11a75.gif

L11a75

Contents

L11a74.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a74 at Knotilus!


Link Presentations

[edit Notes on L11a74's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X16,8,17,7 X22,13,5,14 X20,18,21,17 X18,9,19,10 X8,19,9,20 X14,21,15,22 X10,16,11,15 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -7, 6, -9, 11, -2, 4, -8, 9, -3, 5, -6, 7, -5, 8, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a74 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^5+2 t(1) t(2)^4-6 t(2)^4-8 t(1) t(2)^3+12 t(2)^3+12 t(1) t(2)^2-8 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{16}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{15}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^9 z^{-1} -4 z a^7-3 a^7 z^{-1} +5 z^3 a^5+8 z a^5+4 a^5 z^{-1} -2 z^5 a^3-4 z^3 a^3-6 z a^3-2 a^3 z^{-1} -z^5 a+z^3 a^{-1} (db)
Kauffman polynomial a^9 z^7-4 a^9 z^5+6 a^9 z^3-4 a^9 z+a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+13 a^8 z^4-6 a^8 z^2+a^8+3 a^7 z^9-4 a^7 z^7-14 a^7 z^5+28 a^7 z^3-16 a^7 z+3 a^7 z^{-1} +a^6 z^{10}+10 a^6 z^8-42 a^6 z^6+46 a^6 z^4-18 a^6 z^2+3 a^6+8 a^5 z^9-9 a^5 z^7-30 a^5 z^5+50 a^5 z^3-25 a^5 z+4 a^5 z^{-1} +a^4 z^{10}+17 a^4 z^8-52 a^4 z^6+45 a^4 z^4-15 a^4 z^2+2 a^4+5 a^3 z^9+7 a^3 z^7-39 a^3 z^5+38 a^3 z^3-15 a^3 z+2 a^3 z^{-1} +10 a^2 z^8-13 a^2 z^6+4 a^2 z^4+z^4 a^{-2} -2 a^2 z^2+a^2+11 a z^7-15 a z^5+4 z^5 a^{-1} +8 a z^3-2 z^3 a^{-1} -2 a z+8 z^6-7 z^4+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 \
-8-7-6-5-4-3-2-10123χ
6           1-1
4          3 3
2         51 -4
0        83  5
-2       96   -3
-4      107    3
-6     89     1
-8    710      -3
-10   59       4
-12  26        -4
-14 15         4
-16 2          -2
-181           1
Integral Khovanov Homology

(db, data source)

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

L11a73.gif

L11a73

L11a75.gif

L11a75