L11n97

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L11n96.gif

L11n96

L11n98.gif

L11n98

Contents

L11n97.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n97 at Knotilus!


Link Presentations

[edit Notes on L11n97's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X16,8,17,7 X18,10,19,9 X8,18,9,17 X19,22,20,5 X13,20,14,21 X21,14,22,15 X10,16,11,15 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 11, -2, -7, 8, 9, -3, 5, -4, -6, 7, -8, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n97 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^5-t(2)^5-2 t(1) t(2)^4+2 t(2)^4+3 t(1) t(2)^3-4 t(2)^3-4 t(1) t(2)^2+3 t(2)^2+2 t(1) t(2)-2 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -q^{7/2}+3 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-a^3 z^5+5 a z^5-z^5 a^{-1} -3 a^3 z^3+8 a z^3-3 z^3 a^{-1} -3 a^3 z+3 a z-2 z a^{-1} +a^5 z^{-1} -a^3 z^{-1} (db)
Kauffman polynomial a^6 z^4-a^6 z^2+4 a^5 z^5-6 a^5 z^3-a^5 z+a^5 z^{-1} +a^4 z^8-a^4 z^6+3 a^4 z^4-2 a^4 z^2-a^4+a^3 z^9-2 a^3 z^7+7 a^3 z^5+z^5 a^{-3} -6 a^3 z^3-2 z^3 a^{-3} +a^3 z^{-1} +4 a^2 z^8-10 a^2 z^6+3 z^6 a^{-2} +13 a^2 z^4-7 z^4 a^{-2} -4 a^2 z^2+2 z^2 a^{-2} +a z^9+2 a z^7+4 z^7 a^{-1} -8 a z^5-10 z^5 a^{-1} +9 a z^3+7 z^3 a^{-1} -a z-2 z a^{-1} +3 z^8-6 z^6+4 z^4-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 \
-5-4-3-2-101234χ
8         11
6        2 -2
4       31 2
2      42  -2
0     53   2
-2    55    0
-4   34     -1
-6  25      3
-8 23       -1
-10 3        3
-121         -1
Integral Khovanov Homology

(db, data source)

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

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L11n96

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L11n98