L11n434

From Knot Atlas
Jump to: navigation, search

L11n433.gif

L11n433

L11n435.gif

L11n435

Contents

L11n434.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n434 at Knotilus!


Link Presentations

[edit Notes on L11n434's Link Presentations]

Planar diagram presentation X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X15,20,16,21 X5,17,6,16 X18,11,19,12 X10,17,11,18 X19,5,20,4
Gauss code {1, -4, 2, 11, -8, -6}, {4, -1, 5, -10, 9, -3}, {3, -2, -7, 8, 10, -9, -11, 7, 6, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n434 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v w-u^2 v-u^2 w+u^2+u v^2 w^3-u v^2 w^2+u v^2 w-u v^2-2 u v w^3+3 u v w^2-3 u v w+2 u v+u w^3-u w^2+u w-u-v^2 w^3+v^2 w^2+v w^3-2 v w^2}{u v w^{3/2}} (db)
Jones polynomial q^6-3 q^5+6 q^4-7 q^3- q^{-3} +10 q^2+4 q^{-2} -9 q-6 q^{-1} +9 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +2 z^4-a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +4 z^2-5 a^{-2} +2 a^{-4} +3-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} (db)
Kauffman polynomial z^6 a^{-6} -3 z^4 a^{-6} +2 z^2 a^{-6} +3 z^7 a^{-5} -9 z^5 a^{-5} +5 z^3 a^{-5} +4 z^8 a^{-4} -13 z^6 a^{-4} +13 z^4 a^{-4} -11 z^2 a^{-4} - a^{-4} z^{-2} +6 a^{-4} +2 z^9 a^{-3} -2 z^7 a^{-3} -7 z^5 a^{-3} +a^3 z^3+8 z^3 a^{-3} -a^3 z-6 z a^{-3} +2 a^{-3} z^{-1} +8 z^8 a^{-2} -30 z^6 a^{-2} +4 a^2 z^4+44 z^4 a^{-2} -4 a^2 z^2-33 z^2 a^{-2} -2 a^{-2} z^{-2} +a^2+12 a^{-2} +2 z^9 a^{-1} +2 a z^7-3 z^7 a^{-1} -4 a z^5-2 z^5 a^{-1} +8 a z^3+10 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-16 z^6+32 z^4-24 z^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 \
-3-2-10123456χ
13         11
11        2 -2
9       41 3
7      43  -1
5     63   3
3    45    1
1   55     0
-1  25      3
-3 24       -2
-5 3        3
-71         -1
Integral Khovanov Homology

(db, data source)

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

L11n433.gif

L11n433

L11n435.gif

L11n435