L11a234

From Knot Atlas
Jump to: navigation, search

L11a233.gif

L11a233

L11a235.gif

L11a235

Contents

L11a234.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a234 at Knotilus!


Link Presentations

[edit Notes on L11a234's Link Presentations]

Planar diagram presentation X8192 X12,4,13,3 X18,6,19,5 X22,12,7,11 X20,17,21,18 X16,21,17,22 X14,10,15,9 X10,16,11,15 X4,20,5,19 X2738 X6,14,1,13
Gauss code {1, -10, 2, -9, 3, -11}, {10, -1, 7, -8, 4, -2, 11, -7, 8, -6, 5, -3, 9, -5, 6, -4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a234 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-3 t(1) t(2)^4+2 t(2)^4-3 t(1)^2 t(2)^3+8 t(1) t(2)^3-4 t(2)^3+4 t(1)^2 t(2)^2-9 t(1) t(2)^2+4 t(2)^2-4 t(1)^2 t(2)+8 t(1) t(2)-3 t(2)+2 t(1)^2-3 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+16 q^{9/2}-19 q^{7/2}+18 q^{5/2}-17 q^{3/2}+12 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -7 z a^{-5} -3 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +8 z a^{-3} +2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-5 z^3 a^{-1} +a z-3 z a^{-1} (db)
Kauffman polynomial z^4 a^{-10} -z^2 a^{-10} +3 z^5 a^{-9} -2 z^3 a^{-9} +6 z^6 a^{-8} -6 z^4 a^{-8} +4 z^2 a^{-8} - a^{-8} +9 z^7 a^{-7} -14 z^5 a^{-7} +13 z^3 a^{-7} -5 z a^{-7} + a^{-7} z^{-1} +9 z^8 a^{-6} -12 z^6 a^{-6} +z^4 a^{-6} +7 z^2 a^{-6} -3 a^{-6} +6 z^9 a^{-5} -z^7 a^{-5} -23 z^5 a^{-5} +27 z^3 a^{-5} -13 z a^{-5} +3 a^{-5} z^{-1} +2 z^{10} a^{-4} +11 z^8 a^{-4} -38 z^6 a^{-4} +26 z^4 a^{-4} -2 z^2 a^{-4} -3 a^{-4} +11 z^9 a^{-3} -25 z^7 a^{-3} +3 z^5 a^{-3} +15 z^3 a^{-3} -10 z a^{-3} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +6 z^8 a^{-2} -34 z^6 a^{-2} +32 z^4 a^{-2} -7 z^2 a^{-2} +5 z^9 a^{-1} +a z^7-14 z^7 a^{-1} -3 a z^5+6 z^5 a^{-1} +3 a z^3+6 z^3 a^{-1} -a z-3 z a^{-1} +4 z^8-14 z^6+14 z^4-3 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
18           1-1
16          2 2
14         51 -4
12        72  5
10       95   -4
8      107    3
6     910     1
4    89      -1
2   510       5
0  37        -4
-2 15         4
-4 3          -3
-61           1
Integral Khovanov Homology

(db, data source)

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

L11a233.gif

L11a233

L11a235.gif

L11a235