L11a418

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L11a417.gif

L11a417

L11a419.gif

L11a419

Contents

L11a418.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a418 at Knotilus!


Link Presentations

[edit Notes on L11a418's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X18,10,19,9 X16,8,17,7 X22,14,11,13 X20,16,21,15 X10,18,5,17 X8,20,9,19 X14,22,15,21 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 4, -8, 3, -7}, {11, -2, 5, -9, 6, -4, 7, -3, 8, -6, 9, -5}
A Braid Representative
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A Morse Link Presentation L11a418 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-2 t(1) t(2)^2+2 t(1) t(3) t(2)^2-3 t(3) t(2)^2+2 t(2)^2+2 t(1) t(3)^2 t(2)-3 t(3)^2 t(2)+3 t(1) t(2)-5 t(1) t(3) t(2)+5 t(3) t(2)-2 t(2)-2 t(1) t(3)^2+2 t(3)^2+3 t(1) t(3)-2 t(3)}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial -q^7+3 q^6-5 q^5+8 q^4-10 q^3+12 q^2-11 q+11-7 q^{-1} +5 q^{-2} -2 q^{-3} + q^{-4} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4-2 z^2 a^2+a^2 z^{-2} +z^4-2 z^2-2 z^{-2} -3+2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} +z^4 a^{-4} -z^2 a^{-6} (db)
Kauffman polynomial z^7 a^{-7} -4 z^5 a^{-7} +4 z^3 a^{-7} +3 z^8 a^{-6} -13 z^6 a^{-6} +16 z^4 a^{-6} -5 z^2 a^{-6} +3 z^9 a^{-5} -11 z^7 a^{-5} +11 z^5 a^{-5} -4 z^3 a^{-5} +z^{10} a^{-4} +2 z^8 a^{-4} -15 z^6 a^{-4} +a^4 z^4+14 z^4 a^{-4} -2 a^4 z^2-3 z^2 a^{-4} +a^4+5 z^9 a^{-3} -14 z^7 a^{-3} +2 a^3 z^5+10 z^5 a^{-3} -2 a^3 z^3-4 z^3 a^{-3} +z^{10} a^{-2} +2 z^8 a^{-2} +3 a^2 z^6-6 z^6 a^{-2} -3 a^2 z^4-4 z^4 a^{-2} +3 a^2 z^2+9 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -3 a^2-5 a^{-2} +2 z^9 a^{-1} +3 a z^7+z^7 a^{-1} -7 z^5 a^{-1} -5 a z^3+z^3 a^{-1} +6 a z+6 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-z^6-6 z^4+12 z^2+2 z^{-2} -8 (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
15           1-1
13          2 2
11         31 -2
9        52  3
7       53   -2
5      75    2
3     67     1
1    55      0
-1   37       4
-3  24        -2
-5  3         3
-712          -1
-91           1
Integral Khovanov Homology

(db, data source)

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

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L11a417.gif

L11a417

L11a419.gif

L11a419