L11a157

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L11a156.gif

L11a156

L11a158.gif

L11a158

Contents

L11a157.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a157's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X2738 X18,13,19,14 X6,12,1,11 X20,18,21,17 X16,22,17,21 X4,16,5,15 X14,6,15,5 X12,19,13,20
Gauss code {1, -4, 2, -9, 10, -6}, {4, -1, 3, -2, 6, -11, 5, -10, 9, -8, 7, -5, 11, -7, 8, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
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A Morse Link Presentation L11a157 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^4-t(2)^4+2 t(1)^2 t(2)^3-8 t(1) t(2)^3+5 t(2)^3-6 t(1)^2 t(2)^2+15 t(1) t(2)^2-6 t(2)^2+5 t(1)^2 t(2)-8 t(1) t(2)+2 t(2)-t(1)^2+t(1)}{t(1) t(2)^2} (db)
Jones polynomial q^{15/2}-4 q^{13/2}+8 q^{11/2}-13 q^{9/2}+17 q^{7/2}-20 q^{5/2}+19 q^{3/2}-17 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z a^{-7} -3 z^3 a^{-5} -3 z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} +4 z^3 a^{-3} +a^3 z+6 z a^{-3} +2 a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-2 z^3 a^{-1} -4 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -4 z^9 a^{-1} -8 z^9 a^{-3} -4 z^9 a^{-5} -14 z^8 a^{-2} -14 z^8 a^{-4} -6 z^8 a^{-6} -6 z^8-5 a z^7-3 z^7 a^{-1} +4 z^7 a^{-3} -2 z^7 a^{-5} -4 z^7 a^{-7} -3 a^2 z^6+36 z^6 a^{-2} +39 z^6 a^{-4} +13 z^6 a^{-6} -z^6 a^{-8} +8 z^6-a^3 z^5+7 a z^5+18 z^5 a^{-1} +25 z^5 a^{-3} +25 z^5 a^{-5} +10 z^5 a^{-7} +5 a^2 z^4-34 z^4 a^{-2} -31 z^4 a^{-4} -6 z^4 a^{-6} +2 z^4 a^{-8} -6 z^4+2 a^3 z^3-4 a z^3-24 z^3 a^{-1} -35 z^3 a^{-3} -24 z^3 a^{-5} -7 z^3 a^{-7} -2 a^2 z^2+11 z^2 a^{-2} +8 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} +3 z^2-a^3 z+3 a z+13 z a^{-1} +15 z a^{-3} +8 z a^{-5} +2 z a^{-7} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} 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 \
-4-3-2-101234567χ
16           1-1
14          3 3
12         51 -4
10        83  5
8       95   -4
6      118    3
4     910     1
2    810      -2
0   510       5
-2  27        -5
-4 15         4
-6 2          -2
-81           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L11a156

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L11a158