L11a445

From Knot Atlas
Jump to: navigation, search

L11a444.gif

L11a444

L11a446.gif

L11a446

Contents

L11a445.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a445 at Knotilus!


Link Presentations

[edit Notes on L11a445's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X18,10,19,9 X16,8,17,7 X20,11,21,12 X22,20,13,19 X10,21,11,22 X8,14,9,13 X12,18,5,17 X2536 X4,16,1,15
Gauss code {1, -10, 2, -11}, {10, -1, 4, -8, 3, -7, 5, -9}, {8, -2, 11, -4, 9, -3, 6, -5, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L11a445 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^2 t(2)^3-t(3)^2 t(2)^3-t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-5 t(1) t(3)^2 t(2)^2+5 t(3)^2 t(2)^2-t(1) t(2)^2+4 t(1) t(3) t(2)^2-6 t(3) t(2)^2+2 t(2)^2-2 t(1) t(3)^3 t(2)+t(3)^3 t(2)+6 t(1) t(3)^2 t(2)-4 t(3)^2 t(2)+t(1) t(2)-5 t(1) t(3) t(2)+5 t(3) t(2)-t(2)+t(1) t(3)^3-2 t(1) t(3)^2+t(3)^2+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial -q^8+4 q^7-9 q^6+14 q^5-17 q^4+21 q^3+ q^{-3} -19 q^2-3 q^{-2} +17 q+7 q^{-1} -11 (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^4 a^{-6} -z^2 a^{-6} - a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} +3 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} +z^6 a^{-2} +z^4 a^{-2} +a^2 z^2-z^2 a^{-2} -2 a^{-2} z^{-2} +a^2-3 a^{-2} -2 z^4-3 z^2+ z^{-2} (db)
Kauffman polynomial z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -5 z^4 a^{-8} +z^2 a^{-8} +8 z^7 a^{-7} -13 z^5 a^{-7} +6 z^3 a^{-7} -2 z a^{-7} +9 z^8 a^{-6} -15 z^6 a^{-6} +11 z^4 a^{-6} -7 z^2 a^{-6} +3 a^{-6} +5 z^9 a^{-5} +3 z^7 a^{-5} -20 z^5 a^{-5} +20 z^3 a^{-5} -7 z a^{-5} +z^{10} a^{-4} +16 z^8 a^{-4} -42 z^6 a^{-4} +47 z^4 a^{-4} -31 z^2 a^{-4} - a^{-4} z^{-2} +11 a^{-4} +8 z^9 a^{-3} -6 z^7 a^{-3} -15 z^5 a^{-3} +24 z^3 a^{-3} -12 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +11 z^8 a^{-2} +a^2 z^6-31 z^6 a^{-2} -3 a^2 z^4+35 z^4 a^{-2} +3 a^2 z^2-26 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2+11 a^{-2} +3 z^9 a^{-1} +3 a z^7+2 z^7 a^{-1} -8 a z^5-17 z^5 a^{-1} +6 a z^3+17 z^3 a^{-1} -a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-7 z^6+z^4- z^{-2} +3 (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 \
-4-3-2-101234567χ
17           1-1
15          3 3
13         61 -5
11        83  5
9       107   -3
7      117    4
5     810     2
3    911      -2
1   511       6
-1  26        -4
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

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

L11a444.gif

L11a444

L11a446.gif

L11a446