L11a485

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L11a484.gif

L11a484

L11a486.gif

L11a486

Contents

L11a485.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a485 at Knotilus!


Link Presentations

[edit Notes on L11a485's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X22,11,19,12 X18,22,5,21 X20,10,21,9 X10,17,11,18 X16,19,17,20 X2,14,3,13
Gauss code {1, -11, 5, -3}, {10, -8, 7, -6}, {4, -1, 2, -5, 8, -9, 6, -4, 11, -2, 3, -10, 9, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L11a485 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) (w-1)^3 \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{5/2}} (db)
Jones polynomial  q^{-6} -q^5-6 q^{-5} +5 q^4+13 q^{-4} -11 q^3-20 q^{-3} +19 q^2+28 q^{-2} -26 q-30 q^{-1} +32 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4 z^4-2 a^4-2 a^2 z^6-z^6 a^{-2} -4 a^2 z^4-2 z^4 a^{-2} +2 a^2 z^2+a^2 z^{-2} + a^{-2} z^{-2} +6 a^2+2 a^{-2} +z^8+4 z^6+5 z^4-2 z^2-2 z^{-2} -6 (db)
Kauffman polynomial a^6 z^6+6 a^5 z^7-8 a^5 z^5+z^5 a^{-5} +2 a^5 z+13 a^4 z^8-26 a^4 z^6+5 z^6 a^{-4} +12 a^4 z^4-4 z^4 a^{-4} +2 a^4 z^2-4 a^4+12 a^3 z^9-14 a^3 z^7+11 z^7 a^{-3} -8 a^3 z^5-13 z^5 a^{-3} +2 a^3 z^3+4 z^3 a^{-3} +6 a^3 z+4 a^2 z^{10}+24 a^2 z^8+15 z^8 a^{-2} -69 a^2 z^6-20 z^6 a^{-2} +38 a^2 z^4+8 z^4 a^{-2} +10 a^2 z^2+2 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -12 a^2-4 a^{-2} +24 a z^9+12 z^9 a^{-1} -36 a z^7-5 z^7 a^{-1} -3 a z^5-17 z^5 a^{-1} +10 a z^3+12 z^3 a^{-1} +6 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +4 z^{10}+26 z^8-67 z^6+38 z^4+10 z^2+2 z^{-2} -11 (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
11           1-1
9          4 4
7         71 -6
5        124  8
3       147   -7
1      1812    6
-1     1618     2
-3    1214      -2
-5   816       8
-7  512        -7
-9 18         7
-11 5          -5
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=0 {\mathbb Z}^{18}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{18}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11a484.gif

L11a484

L11a486.gif

L11a486