K11n101

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K11n100.gif

K11n100

K11n102.gif

K11n102

Contents

K11n101.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n101 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X18,9,19,10 X2,11,3,12 X13,6,14,7 X22,16,1,15 X20,18,21,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -6, 2, -1, -3, 7, -4, -10, 5, -2, 6, 4, -7, 3, 8, -11, 9, -5, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 10 -14 -12 18 2 -6 22 20 8 16
A Braid Representative
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A Morse Link Presentation K11n101 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n101/ThurstonBennequinNumber
Hyperbolic Volume 11.167
A-Polynomial See Data:K11n101/A-polynomial

[edit Notes for K11n101'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 K11n101's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+10 t-15+10 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 39, -2 }
Jones polynomial q^3-3 q^2+4 q-5+7 q^{-1} -6 q^{-2} +6 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+a^2-z^4-z^2+z^2 a^{-2}
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8-3 a^3 z^7+3 z^7 a^{-1} -3 a^4 z^6-15 a^2 z^6+z^6 a^{-2} -11 z^6+3 a^5 z^5+6 a^3 z^5-8 a z^5-11 z^5 a^{-1} +2 a^6 z^4+8 a^4 z^4+19 a^2 z^4-3 z^4 a^{-2} +10 z^4+a^7 z^3-3 a^5 z^3-6 a^3 z^3+6 a z^3+8 z^3 a^{-1} -2 a^6 z^2-6 a^4 z^2-8 a^2 z^2+z^2 a^{-2} -3 z^2-a^7 z+a^5 z+3 a^3 z+a z+a^6+a^4-a^2
The A2 invariant Data:K11n101/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n101/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_15, 10_165, K11n63,}

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{316}{3} \frac{92}{3} -192 -432 -32 -152 \frac{256}{3} 288 \frac{2528}{3} \frac{736}{3} \frac{26911}{15} -\frac{7124}{15} \frac{60484}{45} \frac{593}{9} \frac{3871}{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=-2 is the signature of K11n101. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-101234χ
7         11
5        2 -2
3       21 1
1      32  -1
-1     42   2
-3    34    1
-5   33     0
-7  13      2
-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=-3 i=-1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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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K11n100.gif

K11n100

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K11n102