Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

L11a327

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L11a326.gif

L11a326

L11a328.gif

L11a328

Contents

L11a327.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a327's page at Knotilus.

Visit L11a327's page at the original Knot Atlas.


Link Presentations

[edit Notes on L11a327's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,10,19,9 X6,13,7,14 X14,7,15,8 X8,15,1,16 X22,18,9,17 X16,22,17,21 X4,20,5,19 X20,6,21,5
Gauss code {1, -2, 3, -10, 11, -5, 6, -7}, {4, -1, 2, -3, 5, -6, 7, -9, 8, -4, 10, -11, 9, -8}
A Braid Representative
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation L11a327 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u^3 v^5+u^3 v^4-u^3 v^3+u^3 v^2+u^2 v^5-4 u^2 v^4+5 u^2 v^3-5 u^2 v^2+2 u^2 v+2 u v^4-5 u v^3+5 u v^2-4 u v+u+v^3-v^2+v-1}{u^{3/2} v^{5/2}} (db)
Jones polynomial q^{9/2}-2 q^{7/2}+5 q^{5/2}-9 q^{3/2}+11 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+6 a^3 z^5+13 a^3 z^3+10 a^3 z+2 a^3 z^{-1} -a z^9-8 a z^7+z^7 a^{-1} -25 a z^5+6 z^5 a^{-1} -36 a z^3+13 z^3 a^{-1} -22 a z+10 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-3 a^7 z^3+2 a^7 z+2 a^6 z^6-4 a^6 z^4+a^6 z^2+3 a^5 z^7-5 a^5 z^5+2 a^5 z^3-a^5 z+4 a^4 z^8-10 a^4 z^6+z^6 a^{-4} +14 a^4 z^4-4 z^4 a^{-4} -7 a^4 z^2+4 z^2 a^{-4} +3 a^3 z^9-6 a^3 z^7+2 z^7 a^{-3} +10 a^3 z^5-6 z^5 a^{-3} -10 a^3 z^3+4 z^3 a^{-3} +9 a^3 z-2 a^3 z^{-1} +a^2 z^{10}+4 a^2 z^8+3 z^8 a^{-2} -17 a^2 z^6-9 z^6 a^{-2} +30 a^2 z^4+10 z^4 a^{-2} -18 a^2 z^2-7 z^2 a^{-2} +3 a^2+ a^{-2} +6 a z^9+3 z^9 a^{-1} -21 a z^7-10 z^7 a^{-1} +43 a z^5+21 z^5 a^{-1} -47 a z^3-28 z^3 a^{-1} +24 a z+12 z a^{-1} -3 a z^{-1} - a^{-1} z^{-1} +z^{10}+3 z^8-15 z^6+26 z^4-21 z^2+3 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=-1 is the signature of L11a327. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
10           1-1
8          1 1
6         41 -3
4        51  4
2       64   -2
0      85    3
-2     67     1
-4    67      -1
-6   36       3
-8  26        -4
-10 14         3
-12 1          -1
-141           1
Integral Khovanov Homology

(db, data source)

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

L11a326.gif

L11a326

L11a328.gif

L11a328