L11a37

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L11a36.gif

L11a36

L11a38.gif

L11a38

Contents

L11a37.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a37 at Knotilus!


Link Presentations

[edit Notes on L11a37's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,15,19,16 X16,7,17,8 X8,17,9,18 X22,11,5,12 X20,13,21,14 X14,19,15,20 X12,21,13,22 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 6, -9, 7, -8, 3, -4, 5, -3, 8, -7, 9, -6}
A Braid Representative
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A Morse Link Presentation L11a37 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 t(2)^3+4 t(1) t(2)^2-6 t(2)^2-6 t(1) t(2)+4 t(2)+3 t(1)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{5}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{5}{q^{19/2}}+\frac{4}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{13} z^{-1} +3 z a^{11}+2 a^{11} z^{-1} -2 z^3 a^9-2 z a^9-a^9 z^{-1} -2 z^3 a^7+a^7 z^{-1} -2 z^3 a^5-2 z a^5-a^5 z^{-1} -z^3 a^3-z a^3 (db)
Kauffman polynomial -z^8 a^{14}+6 z^6 a^{14}-12 z^4 a^{14}+10 z^2 a^{14}-3 a^{14}-2 z^9 a^{13}+11 z^7 a^{13}-18 z^5 a^{13}+10 z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -z^{10} a^{12}+z^8 a^{12}+16 z^6 a^{12}-39 z^4 a^{12}+28 z^2 a^{12}-7 a^{12}-5 z^9 a^{11}+25 z^7 a^{11}-38 z^5 a^{11}+23 z^3 a^{11}-8 z a^{11}+2 a^{11} z^{-1} -z^{10} a^{10}-z^8 a^{10}+19 z^6 a^{10}-31 z^4 a^{10}+18 z^2 a^{10}-4 a^{10}-3 z^9 a^9+11 z^7 a^9-14 z^5 a^9+13 z^3 a^9-5 z a^9+a^9 z^{-1} -3 z^8 a^8+6 z^6 a^8-3 z^7 a^7+3 z^5 a^7+4 z^3 a^7-4 z a^7+a^7 z^{-1} -3 z^6 a^6+2 z^4 a^6+z^2 a^6-a^6-3 z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1} -2 z^4 a^4+z^2 a^4-z^3 a^3+z a^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).   
\ r
  \  
j \
-11-10-9-8-7-6-5-4-3-2-10χ
-2           11
-4          21-1
-6         2  2
-8        32  -1
-10       42   2
-12      44    0
-14     43     1
-16    24      2
-18   34       -1
-20  12        1
-22 13         -2
-24 1          1
-261           -1
Integral Khovanov Homology

(db, data source)

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

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L11a36

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L11a38