L11a435

From Knot Atlas
Jump to: navigation, search

L11a434.gif

L11a434

L11a436.gif

L11a436

Contents

L11a435.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a435 at Knotilus!


Link Presentations

[edit Notes on L11a435's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,12,19,11 X14,8,15,7 X8,14,9,13 X22,20,13,19 X20,16,21,15 X16,22,17,21 X12,18,5,17 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, {5, -4, 7, -8, 9, -3, 6, -7, 8, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a435 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(v-1) \left(-u v w^3+4 u v w^2-4 u v w+2 u v+u w^3-2 u w^2+2 u w+2 v^2 w^2-2 v^2 w+v^2+2 v w^3-4 v w^2+4 v w-v\right)}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial -q^8+3 q^7-7 q^6+13 q^5-18 q^4+21 q^3-20 q^2+19 q-13+9 q^{-1} -3 q^{-2} + q^{-3} (db)
Signature 2 (db)
HOMFLY-PT polynomial - a^{-8} +3 z^2 a^{-6} - a^{-6} z^{-2} +2 a^{-6} -3 z^4 a^{-4} -2 z^2 a^{-4} +4 a^{-4} z^{-2} +4 a^{-4} +z^6 a^{-2} +a^2 z^2-5 z^2 a^{-2} -5 a^{-2} z^{-2} +a^2-9 a^{-2} -2 z^4-z^2+2 z^{-2} +3 (db)
Kauffman polynomial z^{10} a^{-2} +z^{10} a^{-4} +4 z^9 a^{-1} +8 z^9 a^{-3} +4 z^9 a^{-5} +16 z^8 a^{-2} +16 z^8 a^{-4} +6 z^8 a^{-6} +6 z^8+3 a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +7 z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-46 z^6 a^{-2} -38 z^6 a^{-4} -5 z^6 a^{-6} +3 z^6 a^{-8} -15 z^6-6 a z^5-32 z^5 a^{-1} -48 z^5 a^{-3} -29 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+52 z^4 a^{-2} +35 z^4 a^{-4} -z^4 a^{-6} -5 z^4 a^{-8} +18 z^4+3 a z^3+39 z^3 a^{-1} +68 z^3 a^{-3} +37 z^3 a^{-5} +3 z^3 a^{-7} -2 z^3 a^{-9} +3 a^2 z^2-44 z^2 a^{-2} -22 z^2 a^{-4} +4 z^2 a^{-6} +3 z^2 a^{-8} -18 z^2-23 z a^{-1} -41 z a^{-3} -21 z a^{-5} -2 z a^{-7} +z a^{-9} -a^2+23 a^{-2} +14 a^{-4} - a^{-8} +10+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -2 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 \
-4-3-2-101234567χ
17           1-1
15          2 2
13         51 -4
11        82  6
9       116   -5
7      107    3
5     1011     1
3    910      -1
1   612       6
-1  37        -4
-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}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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.

L11a434.gif

L11a434

L11a436.gif

L11a436