L11n298

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L11n297.gif

L11n297

L11n299.gif

L11n299

Contents

L11n298.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n298 at Knotilus!


Link Presentations

[edit Notes on L11n298's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,14,10,15 X15,20,16,21 X19,5,20,10 X2536 X11,1,12,4
Gauss code {1, -10, -2, 11}, {10, -1, -5, 6, -7, 9}, {-11, 2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n298 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v^2 w^3+3 u v^2 w^2-3 u v^2 w+u v^2+2 u v w^3-5 u v w^2+3 u v w-u v-u w^3+u w^2-v^2 w^2+v^2 w+v w^4-3 v w^3+5 v w^2-2 v w-w^4+3 w^3-3 w^2+w}{\sqrt{u} v w^2} (db)
Jones polynomial q^6-4 q^5+8 q^4-11 q^3-2 q^{-3} +14 q^2+7 q^{-2} -14 q-9 q^{-1} +14 (db)
Signature 0 (db)
HOMFLY-PT polynomial z^4 a^{-4} +z^2 a^{-4} + a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} -2 a^2 z^2-5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^{-2} +3 z^4+5 z^2-2 z^{-2} +1 (db)
Kauffman polynomial 3 z^9 a^{-1} +3 z^9 a^{-3} +14 z^8 a^{-2} +6 z^8 a^{-4} +8 z^8+6 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-40 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -23 z^6-11 a z^5-28 z^5 a^{-1} -27 z^5 a^{-3} -10 z^5 a^{-5} +8 a^2 z^4+39 z^4 a^{-2} +8 z^4 a^{-4} -2 z^4 a^{-6} +37 z^4+3 a^3 z^3+14 a z^3+30 z^3 a^{-1} +25 z^3 a^{-3} +6 z^3 a^{-5} -9 a^2 z^2-20 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -25 z^2-a^3 z-3 a z-7 z a^{-1} -7 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 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 \
-3-2-10123456χ
13         11
11        3 -3
9       51 4
7      63  -3
5     85   3
3    77    0
1   77     0
-1  49      5
-3 35       -2
-5 5        5
-72         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-3 {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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