K11n2

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K11n1.gif

K11n1

K11n3.gif

K11n3

Contents

K11n2.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n2 at Knotilus!


Knot K11n2.
A graph, knot K11n2.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,15,8,14 X2,9,3,10 X16,12,17,11 X20,14,21,13 X15,7,16,6 X22,18,1,17 X12,20,13,19 X18,22,19,21
Gauss code 1, -5, 2, -1, 3, 8, -4, -2, 5, -3, 6, -10, 7, 4, -8, -6, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 10 -14 2 16 20 -6 22 12 18
A Braid Representative
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A Morse Link Presentation K11n2 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:K11n2/ThurstonBennequinNumber
Hyperbolic Volume 12.5932
A-Polynomial See Data:K11n2/A-polynomial

[edit Notes for K11n2'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 K11n2's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+8 t^2-12 t+13-12 t^{-1} +8 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-4 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 57, 4 }
Jones polynomial -2 q^9+5 q^8-7 q^7+9 q^6-10 q^5+9 q^4-7 q^3+5 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -2 z^2 a^{-4} -2 z^2 a^{-6} +3 z^2 a^{-8} +2 a^{-2} - a^{-4} - a^{-6} +2 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +5 z^8 a^{-6} +3 z^8 a^{-8} +2 z^7 a^{-3} +2 z^7 a^{-5} +3 z^7 a^{-7} +3 z^7 a^{-9} +z^6 a^{-2} -3 z^6 a^{-4} -11 z^6 a^{-6} -6 z^6 a^{-8} +z^6 a^{-10} -6 z^5 a^{-3} -9 z^5 a^{-5} -7 z^5 a^{-7} -4 z^5 a^{-9} -4 z^4 a^{-2} -4 z^4 a^{-4} +8 z^4 a^{-6} +12 z^4 a^{-8} +4 z^4 a^{-10} +4 z^3 a^{-3} +4 z^3 a^{-5} +3 z^3 a^{-7} +6 z^3 a^{-9} +3 z^3 a^{-11} +5 z^2 a^{-2} +5 z^2 a^{-4} -5 z^2 a^{-6} -10 z^2 a^{-8} -5 z^2 a^{-10} -2 z a^{-9} -2 z a^{-11} -2 a^{-2} - a^{-4} + a^{-6} +2 a^{-8} + a^{-10}
The A2 invariant Data:K11n2/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n2/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_14, K11a161,}

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

Vassiliev invariants

V2 and V3: (2, 5)
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 40 32 \frac{652}{3} \frac{140}{3} 320 \frac{3280}{3} \frac{640}{3} 200 \frac{256}{3} 800 \frac{5216}{3} \frac{1120}{3} \frac{81511}{15} -\frac{1684}{15} \frac{118564}{45} \frac{905}{9} \frac{5191}{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=4 is the signature of K11n2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
19         2-2
17        3 3
15       42 -2
13      53  2
11     54   -1
9    45    -1
7   35     2
5  24      -2
3 14       3
1 1        -1
-11         1
Integral Khovanov Homology

(db, data source)

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

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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K11n1.gif

K11n1

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K11n3