K11a285

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K11a284.gif

K11a284

K11a286.gif

K11a286

Contents

K11a285.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a285 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,5,17,6 X14,8,15,7 X20,9,21,10 X18,12,19,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X4,19,5,20 X8,21,9,22
Gauss code 1, -7, 2, -10, 3, -1, 4, -11, 5, -2, 6, -9, 7, -4, 8, -3, 9, -6, 10, -5, 11, -8
Dowker-Thistlethwaite code 6 10 16 14 20 18 2 22 12 4 8
A Braid Representative
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A Morse Link Presentation K11a285 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a285/ThurstonBennequinNumber
Hyperbolic Volume 18.3328
A-Polynomial See Data:K11a285/A-polynomial

[edit Notes for K11a285's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a285's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+14 t^2-38 t+53-38 t^{-1} +14 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 161, 0 }
Jones polynomial -q^5+4 q^4-9 q^3+16 q^2-22 q+26-26 q^{-1} +23 q^{-2} -17 q^{-3} +11 q^{-4} -5 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6-z^6+a^4 z^4-a^2 z^4+2 z^4 a^{-2} -a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +z^2+1
Kauffman polynomial (db, data sources) 4 a^2 z^{10}+4 z^{10}+10 a^3 z^9+20 a z^9+10 z^9 a^{-1} +10 a^4 z^8+10 a^2 z^8+11 z^8 a^{-2} +11 z^8+5 a^5 z^7-19 a^3 z^7-47 a z^7-15 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-22 a^4 z^6-40 a^2 z^6-18 z^6 a^{-2} +4 z^6 a^{-4} -39 z^6-9 a^5 z^5+9 a^3 z^5+41 a z^5+11 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -a^6 z^4+11 a^4 z^4+32 a^2 z^4+13 z^4 a^{-2} -5 z^4 a^{-4} +38 z^4+2 a^5 z^3-6 a^3 z^3-17 a z^3-4 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a^4 z^2-8 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} -12 z^2+a^5 z+3 a^3 z+3 a z+z a^{-1} +1
The A2 invariant Data:K11a285/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a285/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a138,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a138,}

Vassiliev invariants

V2 and V3: (0, 0)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
0 0 0 -16 -16 0 32 -64 96 0 0 0 0 -40 \frac{784}{3} -\frac{800}{3} -\frac{104}{3} -88

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=0 is the signature of K11a285. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         61 -5
5        103  7
3       126   -6
1      1410    4
-1     1313     0
-3    1013      -3
-5   713       6
-7  410        -6
-9 17         6
-11 4          -4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11a284.gif

K11a284

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K11a286