L11a158

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L11a157.gif

L11a157

L11a159.gif

L11a159

Contents

L11a158.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a158 at Knotilus!


Link Presentations

[edit Notes on L11a158's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X2738 X16,12,17,11 X14,5,15,6 X4,15,5,16 X20,18,21,17 X18,13,19,14 X12,19,13,20 X6,22,1,21
Gauss code {1, -4, 2, -7, 6, -11}, {4, -1, 3, -2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8, 11, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a158 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-4 u^2 v^3+6 u^2 v^2-4 u^2 v+u^2-2 u v^4+7 u v^3-9 u v^2+7 u v-2 u+v^4-4 v^3+6 v^2-4 v+1}{u v^2} (db)
Jones polynomial -4 q^{9/2}+\frac{3}{q^{9/2}}+8 q^{7/2}-\frac{7}{q^{7/2}}-13 q^{5/2}+\frac{11}{q^{5/2}}+17 q^{3/2}-\frac{16}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-19 \sqrt{q}+\frac{18}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+9 a z^3-7 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-6 a^3 z+8 a z-5 z a^{-1} +z a^{-3} +a^5 z^{-1} -2 a^3 z^{-1} +2 a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} +a^5 z^7-4 a^5 z^5+4 z^5 a^{-5} +6 a^5 z^3-2 z^3 a^{-5} -4 a^5 z+a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+8 z^6 a^{-4} +13 a^4 z^4-7 z^4 a^{-4} -5 a^4 z^2+2 z^2 a^{-4} +3 a^3 z^9-4 a^3 z^7+11 z^7 a^{-3} -13 a^3 z^5-14 z^5 a^{-3} +26 a^3 z^3+7 z^3 a^{-3} -14 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +a^2 z^{10}+10 a^2 z^8+10 z^8 a^{-2} -40 a^2 z^6-11 z^6 a^{-2} +40 a^2 z^4-z^4 a^{-2} -12 a^2 z^2+2 z^2 a^{-2} +a^2+8 a z^9+5 z^9 a^{-1} -8 a z^7+8 z^7 a^{-1} -30 a z^5-39 z^5 a^{-1} +44 a z^3+33 z^3 a^{-1} -18 a z-10 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +z^{10}+17 z^8-48 z^6+34 z^4-7 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 \
-6-5-4-3-2-1012345χ
12           1-1
10          3 3
8         51 -4
6        83  5
4       95   -4
2      108    2
0     910     1
-2    79      -2
-4   510       5
-6  26        -4
-8 15         4
-10 2          -2
-121           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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L11a157

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L11a159