K11a211

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K11a210.gif

K11a210

K11a212.gif

K11a212

Contents

K11a211.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a211 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X16,5,17,6 X20,7,21,8 X18,9,19,10 X14,12,15,11 X2,13,3,14 X22,15,1,16 X10,17,11,18 X8,19,9,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -6, 8, -3, 9, -5, 10, -4, 11, -8
Dowker-Thistlethwaite code 4 12 16 20 18 14 2 22 10 8 6
A Braid Representative
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A Morse Link Presentation K11a211 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a211/ThurstonBennequinNumber
Hyperbolic Volume 10.8805
A-Polynomial See Data:K11a211/A-polynomial

[edit Notes for K11a211's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a211's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+17 t-27+17 t^{-1} -3 t^{-2}
Conway polynomial -3 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 67, -2 }
Jones polynomial q-3+5 q^{-1} -7 q^{-2} +10 q^{-3} -10 q^{-4} +10 q^{-5} -8 q^{-6} +6 q^{-7} -4 q^{-8} +2 q^{-9} - q^{-10}
HOMFLY-PT polynomial (db, data sources) -a^{10}+2 z^2 a^8+a^8-z^4 a^6-a^6-z^4 a^4+2 z^2 a^4+2 a^4-z^4 a^2+z^2
Kauffman polynomial (db, data sources) z^7 a^{11}-5 z^5 a^{11}+7 z^3 a^{11}-3 z a^{11}+2 z^8 a^{10}-9 z^6 a^{10}+11 z^4 a^{10}-5 z^2 a^{10}+a^{10}+2 z^9 a^9-7 z^7 a^9+4 z^5 a^9+z^3 a^9+z^{10} a^8-12 z^6 a^8+18 z^4 a^8-9 z^2 a^8+a^8+5 z^9 a^7-20 z^7 a^7+27 z^5 a^7-17 z^3 a^7+4 z a^7+z^{10} a^6+2 z^8 a^6-16 z^6 a^6+24 z^4 a^6-12 z^2 a^6+a^6+3 z^9 a^5-8 z^7 a^5+10 z^5 a^5-5 z^3 a^5+z a^5+4 z^8 a^4-9 z^6 a^4+11 z^4 a^4-6 z^2 a^4+2 a^4+4 z^7 a^3-5 z^5 a^3+2 z^3 a^3+4 z^6 a^2-5 z^4 a^2+z^2 a^2+3 z^5 a-4 z^3 a+z^4-z^2
The A2 invariant Data:K11a211/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a211/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: (5, -12)
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
20 -96 200 \frac{1798}{3} \frac{362}{3} -1920 -3968 -640 -800 \frac{4000}{3} 4608 \frac{35960}{3} \frac{7240}{3} \frac{157567}{6} -\frac{6062}{3} \frac{125582}{9} \frac{7237}{18} \frac{12799}{6}

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=-2 is the signature of K11a211. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-1012χ
3           11
1          2 -2
-1         31 2
-3        53  -2
-5       52   3
-7      55    0
-9     55     0
-11    35      2
-13   35       -2
-15  13        2
-17 13         -2
-19 1          1
-211           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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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K11a210

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K11a212