L11a337

From Knot Atlas
Jump to: navigation, search

L11a336.gif

L11a336

L11a338.gif

L11a338

Contents

L11a337.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a337 at Knotilus!


Link Presentations

[edit Notes on L11a337's Link Presentations]

Planar diagram presentation X10,1,11,2 X22,5,9,6 X14,3,15,4 X4,15,5,16 X16,7,17,8 X20,17,21,18 X18,11,19,12 X12,19,13,20 X2,9,3,10 X8,13,1,14 X6,21,7,22
Gauss code {1, -9, 3, -4, 2, -11, 5, -10}, {9, -1, 7, -8, 10, -3, 4, -5, 6, -7, 8, -6, 11, -2}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a337 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(2)^5-4 t(1)^2 t(2)^4+5 t(1) t(2)^4-t(2)^4-2 t(1)^3 t(2)^3+9 t(1)^2 t(2)^3-10 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-10 t(1)^2 t(2)^2+9 t(1) t(2)^2-2 t(2)^2-t(1)^3 t(2)+5 t(1)^2 t(2)-4 t(1) t(2)-t(1)^2}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial -\frac{8}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{9}{q^{23/2}}-\frac{15}{q^{21/2}}+\frac{19}{q^{19/2}}-\frac{22}{q^{17/2}}+\frac{22}{q^{15/2}}-\frac{19}{q^{13/2}}+\frac{13}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z a^{13}-a^{13} z^{-1} +4 z^3 a^{11}+7 z a^{11}+3 a^{11} z^{-1} -3 z^5 a^9-7 z^3 a^9-5 z a^9-2 a^9 z^{-1} -3 z^5 a^7-7 z^3 a^7-4 z a^7-z^5 a^5-2 z^3 a^5-z a^5 (db)
Kauffman polynomial -z^6 a^{16}+2 z^4 a^{16}-z^2 a^{16}-4 z^7 a^{15}+9 z^5 a^{15}-6 z^3 a^{15}+z a^{15}-7 z^8 a^{14}+15 z^6 a^{14}-8 z^4 a^{14}+2 z^2 a^{14}-a^{14}-6 z^9 a^{13}+6 z^7 a^{13}+8 z^5 a^{13}-6 z^3 a^{13}+a^{13} z^{-1} -2 z^{10} a^{12}-12 z^8 a^{12}+36 z^6 a^{12}-31 z^4 a^{12}+15 z^2 a^{12}-3 a^{12}-12 z^9 a^{11}+21 z^7 a^{11}-16 z^5 a^{11}+19 z^3 a^{11}-14 z a^{11}+3 a^{11} z^{-1} -2 z^{10} a^{10}-12 z^8 a^{10}+31 z^6 a^{10}-28 z^4 a^{10}+12 z^2 a^{10}-3 a^{10}-6 z^9 a^9+5 z^7 a^9-4 z^5 a^9+8 z^3 a^9-8 z a^9+2 a^9 z^{-1} -7 z^8 a^8+8 z^6 a^8-3 z^4 a^8-z^2 a^8-6 z^7 a^7+10 z^5 a^7-9 z^3 a^7+4 z a^7-3 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+2 z^3 a^5-z a^5 (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          31-2
-8         5  5
-10        83  -5
-12       115   6
-14      118    -3
-16     1111     0
-18    811      3
-20   711       -4
-22  39        6
-24 16         -5
-26 3          3
-281           -1
Integral Khovanov Homology

(db, data source)

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

L11a336.gif

L11a336

L11a338.gif

L11a338