K11a210

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K11a209.gif

K11a209

K11a211.gif

K11a211

Contents

K11a210.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a210 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X16,6,17,5 X20,7,21,8 X18,9,19,10 X14,12,15,11 X2,13,3,14 X10,16,11,15 X22,18,1,17 X8,19,9,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -2, 7, -6, 8, -3, 9, -5, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 12 16 20 18 14 2 10 22 8 6
A Braid Representative
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A Morse Link Presentation K11a210 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a210/ThurstonBennequinNumber
Hyperbolic Volume 11.5224
A-Polynomial See Data:K11a210/A-polynomial

[edit Notes for K11a210's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11a210's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^2-18 t+29-18 t^{-1} +4 t^{-2}
Conway polynomial 4 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 73, 0 }
Jones polynomial -q^5+3 q^4-5 q^3+8 q^2-10 q+12-11 q^{-1} +9 q^{-2} -7 q^{-3} +4 q^{-4} -2 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-2 z^2 a^4-a^4+z^4 a^2-z^2 a^2-a^2+2 z^4+2 z^2+2+z^4 a^{-2} -z^2 a^{-4}
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+2 a^3 z^9+5 a z^9+3 z^9 a^{-1} +2 a^4 z^8+4 z^8 a^{-2} +2 z^8+2 a^5 z^7-5 a^3 z^7-19 a z^7-8 z^7 a^{-1} +4 z^7 a^{-3} +a^6 z^6-4 a^4 z^6-6 a^2 z^6-10 z^6 a^{-2} +3 z^6 a^{-4} -14 z^6-7 a^5 z^5+4 a^3 z^5+32 a z^5+11 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} -4 a^6 z^4-3 a^4 z^4+11 a^2 z^4+10 z^4 a^{-2} -7 z^4 a^{-4} +27 z^4+6 a^5 z^3-3 a^3 z^3-20 a z^3-5 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +4 a^6 z^2+5 a^4 z^2-7 a^2 z^2-4 z^2 a^{-2} +2 z^2 a^{-4} -14 z^2-2 a^5 z+4 a z+2 z a^{-1} -a^6-a^4+a^2+2
The A2 invariant Data:K11a210/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a210/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (-2, 3)
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
-8 24 32 -\frac{76}{3} -\frac{44}{3} -192 -208 -128 56 -\frac{256}{3} 288 \frac{608}{3} \frac{352}{3} \frac{13049}{15} -\frac{1036}{15} \frac{27836}{45} -\frac{713}{9} \frac{1529}{15}

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 K11a210. 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          2 2
7         31 -2
5        52  3
3       53   -2
1      75    2
-1     56     1
-3    46      -2
-5   35       2
-7  14        -3
-9 13         2
-11 1          -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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