L11n295

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L11n294.gif

L11n294

L11n296.gif

L11n296

Contents

L11n295.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n295 at Knotilus!


Link Presentations

[edit Notes on L11n295's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X21,10,22,5 X11,19,12,18 X9,17,10,16 X17,11,18,22 X8,19,9,20 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 4, -9, -7, 5}, {-6, -2, 11, -4, 3, 7, -8, 6, 9, -3, -5, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n295 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^2 w^2-u v^2 w-3 u v w^2+4 u v w-2 u v+u w^2-2 u w+u-v^2 w^2+2 v^2 w-v^2+2 v w^2-4 v w+3 v+w-1}{\sqrt{u} v w} (db)
Jones polynomial -q^8+3 q^7-6 q^6+8 q^5-10 q^4+11 q^3-8 q^2+8 q-3+2 q^{-1} (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} z^{-2} -2 a^{-6} +z^6 a^{-4} +4 z^4 a^{-4} +8 z^2 a^{-4} +4 a^{-4} z^{-2} +8 a^{-4} -3 z^4 a^{-2} -9 z^2 a^{-2} -5 a^{-2} z^{-2} -10 a^{-2} +2 z^2+2 z^{-2} +4 (db)
Kauffman polynomial z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -6 z^4 a^{-8} +z^2 a^{-8} +5 z^7 a^{-7} -13 z^5 a^{-7} +11 z^3 a^{-7} -5 z a^{-7} + a^{-7} z^{-1} +4 z^8 a^{-6} -9 z^6 a^{-6} +9 z^4 a^{-6} -6 z^2 a^{-6} - a^{-6} z^{-2} +3 a^{-6} +z^9 a^{-5} +6 z^7 a^{-5} -23 z^5 a^{-5} +32 z^3 a^{-5} -18 z a^{-5} +5 a^{-5} z^{-1} +6 z^8 a^{-4} -18 z^6 a^{-4} +30 z^4 a^{-4} -24 z^2 a^{-4} -4 a^{-4} z^{-2} +12 a^{-4} +z^9 a^{-3} +2 z^7 a^{-3} -10 z^5 a^{-3} +24 z^3 a^{-3} -24 z a^{-3} +9 a^{-3} z^{-1} +2 z^8 a^{-2} -6 z^6 a^{-2} +18 z^4 a^{-2} -25 z^2 a^{-2} -5 a^{-2} z^{-2} +15 a^{-2} +z^7 a^{-1} -z^5 a^{-1} +5 z^3 a^{-1} -11 z a^{-1} +5 a^{-1} z^{-1} +3 z^4-8 z^2-2 z^{-2} +7 (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 \
-2-101234567χ
17         1-1
15        2 2
13       41 -3
11      42  2
9     64   -2
7    54    1
5   47     3
3  44      0
1 16       5
-112        -1
-32         2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-2 {\mathbb Z}^{2} {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11n294.gif

L11n294

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L11n296