K11n121

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K11n120.gif

K11n120

K11n122.gif

K11n122

Contents

K11n121.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n121 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,16,6,17 X7,12,8,13 X18,9,19,10 X2,11,3,12 X13,20,14,21 X15,6,16,7 X22,18,1,17 X8,19,9,20 X21,14,22,15
Gauss code 1, -6, 2, -1, -3, 8, -4, -10, 5, -2, 6, 4, -7, 11, -8, 3, 9, -5, 10, 7, -11, -9
Dowker-Thistlethwaite code 4 10 -16 -12 18 2 -20 -6 22 8 -14
A Braid Representative
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A Morse Link Presentation K11n121 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n121/ThurstonBennequinNumber
Hyperbolic Volume 12.2709
A-Polynomial See Data:K11n121/A-polynomial

[edit Notes for K11n121's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n121's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+6 t^2-10 t+11-10 t^{-1} +6 t^{-2} - t^{-3}
Conway polynomial -z^6+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 45, -4 }
Jones polynomial 1-3 q^{-1} +5 q^{-2} -6 q^{-3} +8 q^{-4} -7 q^{-5} +7 q^{-6} -5 q^{-7} +2 q^{-8} - q^{-9}
HOMFLY-PT polynomial (db, data sources) -z^2 a^8-2 a^8+2 z^4 a^6+5 z^2 a^6+2 a^6-z^6 a^4-3 z^4 a^4-z^2 a^4+a^4+z^4 a^2+2 z^2 a^2
Kauffman polynomial (db, data sources) z^3 a^{11}-z a^{11}+2 z^4 a^{10}-z^2 a^{10}+4 z^5 a^9-5 z^3 a^9+4 z a^9+z^8 a^8-z^6 a^8+2 z^4 a^8+2 z^2 a^8-2 a^8+z^9 a^7-2 z^7 a^7+6 z^5 a^7-10 z^3 a^7+6 z a^7+4 z^8 a^6-11 z^6 a^6+9 z^4 a^6-2 z^2 a^6-2 a^6+z^9 a^5+z^7 a^5-8 z^5 a^5+3 z^3 a^5+z a^5+3 z^8 a^4-9 z^6 a^4+6 z^4 a^4-3 z^2 a^4+a^4+3 z^7 a^3-10 z^5 a^3+7 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2
The A2 invariant Data:K11n121/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n121/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n14,}

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

Vassiliev invariants

V2 and V3: (5, -11)
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 -88 200 \frac{1558}{3} \frac{290}{3} -1760 -\frac{9904}{3} -\frac{1696}{3} -568 \frac{4000}{3} 3872 \frac{31160}{3} \frac{5800}{3} \frac{126895}{6} -\frac{1894}{3} \frac{89798}{9} \frac{3925}{18} \frac{8719}{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=-4 is the signature of K11n121. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-1012χ
1         11
-1        2 -2
-3       31 2
-5      43  -1
-7     42   2
-9    34    1
-11   44     0
-13  13      2
-15 14       -3
-17 1        1
-191         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11n120.gif

K11n120

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K11n122