L11n234

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L11n233.gif

L11n233

L11n235.gif

L11n235

Contents

L11n234.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n234 at Knotilus!


Link Presentations

[edit Notes on L11n234's Link Presentations]

Planar diagram presentation X10,1,11,2 X5,20,6,21 X3,15,4,14 X15,5,16,4 X19,22,20,9 X7,17,8,16 X11,18,12,19 X17,12,18,13 X2,9,3,10 X13,1,14,8 X21,6,22,7
Gauss code {1, -9, -3, 4, -2, 11, -6, 10}, {9, -1, -7, 8, -10, 3, -4, 6, -8, 7, -5, 2, -11, 5}
A Braid Representative
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BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
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A Morse Link Presentation L11n234 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(1)^2 t(2)^2-t(1) t(2)^2+5 t(1) t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{5}{q^{9/2}}-q^{7/2}+\frac{8}{q^{7/2}}+3 q^{5/2}-\frac{12}{q^{5/2}}-7 q^{3/2}+\frac{12}{q^{3/2}}+\frac{2}{q^{11/2}}+10 \sqrt{q}-\frac{12}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^5 z-a^3 z^5-2 a^3 z^3-2 a^3 z+a z^7+5 a z^5-z^5 a^{-1} +10 a z^3-3 z^3 a^{-1} +7 a z-4 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-2 a^4 z^8-6 a^2 z^8-4 z^8-a^5 z^7-5 a^3 z^7-9 a z^7-5 z^7 a^{-1} +6 a^2 z^6-3 z^6 a^{-2} +3 z^6-3 a^5 z^5+9 a^3 z^5+22 a z^5+9 z^5 a^{-1} -z^5 a^{-3} -3 a^6 z^4+a^2 z^4+5 z^4 a^{-2} +3 z^4+6 a^5 z^3-5 a^3 z^3-19 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} +4 a^6 z^2+a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -z^2-3 a^5 z+8 a z+4 z a^{-1} -z a^{-3} +1-a z^{-1} - a^{-1} z^{-1} (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       51 4
2      52  -3
0     75   2
-2    77    0
-4   55     0
-6  37      4
-8 25       -3
-10 3        3
-122         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L11n233.gif

L11n233

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L11n235