L11a423

From Knot Atlas
Jump to: navigation, search

L11a422.gif

L11a422

L11a424.gif

L11a424

Contents

L11a423.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a423 at Knotilus!


Link Presentations

[edit Notes on L11a423's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,10,21,9 X16,8,17,7 X18,14,19,13 X14,20,15,19 X22,16,11,15 X10,18,5,17 X8,22,9,21 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, {11, -2, 5, -6, 7, -4, 8, -5, 6, -3, 9, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a423 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(3)^4+t(1) t(2) t(3)^4-2 t(2) t(3)^4+t(3)^4+t(1) t(2)^2 t(3)^3-3 t(2)^2 t(3)^3+3 t(1) t(3)^3-4 t(1) t(2) t(3)^3+6 t(2) t(3)^3-2 t(3)^3-2 t(1) t(2)^2 t(3)^2+4 t(2)^2 t(3)^2-4 t(1) t(3)^2+7 t(1) t(2) t(3)^2-7 t(2) t(3)^2+2 t(3)^2+2 t(1) t(2)^2 t(3)-3 t(2)^2 t(3)+3 t(1) t(3)-6 t(1) t(2) t(3)+4 t(2) t(3)-t(3)-t(1) t(2)^2+t(2)^2+2 t(1) t(2)-t(2)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial -q^8+4 q^7-10 q^6+16 q^5-21 q^4+25 q^3+ q^{-3} -23 q^2-3 q^{-2} +21 q+9 q^{-1} -14 (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^4 a^{-6} -z^2 a^{-6} - a^{-6} z^{-2} -2 a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} +6 z^2 a^{-4} +4 a^{-4} z^{-2} +9 a^{-4} +z^6 a^{-2} -z^4 a^{-2} +a^2 z^2-8 z^2 a^{-2} -5 a^{-2} z^{-2} +a^2-11 a^{-2} -2 z^4-z^2+2 z^{-2} +3 (db)
Kauffman polynomial z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -4 z^4 a^{-8} +9 z^7 a^{-7} -14 z^5 a^{-7} +9 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +12 z^8 a^{-6} -22 z^6 a^{-6} +19 z^4 a^{-6} -9 z^2 a^{-6} - a^{-6} z^{-2} +3 a^{-6} +8 z^9 a^{-5} -2 z^7 a^{-5} -20 z^5 a^{-5} +29 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +2 z^{10} a^{-4} +20 z^8 a^{-4} -54 z^6 a^{-4} +52 z^4 a^{-4} -29 z^2 a^{-4} -4 a^{-4} z^{-2} +15 a^{-4} +13 z^9 a^{-3} -14 z^7 a^{-3} -21 z^5 a^{-3} +45 z^3 a^{-3} -33 z a^{-3} +9 a^{-3} z^{-1} +2 z^{10} a^{-2} +14 z^8 a^{-2} +a^2 z^6-43 z^6 a^{-2} -3 a^2 z^4+48 z^4 a^{-2} +3 a^2 z^2-38 z^2 a^{-2} -5 a^{-2} z^{-2} -a^2+20 a^{-2} +5 z^9 a^{-1} +3 a z^7-6 a z^5-22 z^5 a^{-1} +3 a z^3+29 z^3 a^{-1} -18 z a^{-1} +5 a^{-1} z^{-1} +6 z^8-14 z^6+16 z^4-15 z^2-2 z^{-2} +8 (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         71 -6
11        93  6
9       127   -5
7      139    4
5     1214     2
3    911      -2
1   613       7
-1  38        -5
-3  6         6
-513          -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} {\mathbb Z}
r=-3 {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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.

L11a422.gif

L11a422

L11a424.gif

L11a424