K11n33

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K11n32.gif

K11n32

K11n34.gif

K11n34

Contents

K11n33.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n33 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,12,6,13 X2837 X9,17,10,16 X11,6,12,7 X13,20,14,21 X15,11,16,10 X17,1,18,22 X19,14,20,15 X21,19,22,18
Gauss code 1, -4, 2, -1, -3, 6, 4, -2, -5, 8, -6, 3, -7, 10, -8, 5, -9, 11, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 8 -12 2 -16 -6 -20 -10 -22 -14 -18
A Braid Representative
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A Morse Link Presentation K11n33 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n33/ThurstonBennequinNumber
Hyperbolic Volume 12.6362
A-Polynomial See Data:K11n33/A-polynomial

[edit Notes for K11n33's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11n33's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+12 t-13+12 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 51, 2 }
Jones polynomial -q^6+4 q^5-6 q^4+8 q^3-9 q^2+8 q-7+5 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +3 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} -5 z^2+2 a^2+ a^{-4} -2
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8+2 a z^7+z^7 a^{-1} +z^7 a^{-5} +a^2 z^6-8 z^6 a^{-2} -4 z^6 a^{-4} -3 z^6-6 a z^5-5 z^5 a^{-1} +3 z^5 a^{-3} +2 z^5 a^{-5} -4 a^2 z^4+5 z^4 a^{-2} +10 z^4 a^{-4} +4 z^4 a^{-6} -5 z^4+4 a z^3-3 z^3 a^{-1} -9 z^3 a^{-3} -z^3 a^{-5} +z^3 a^{-7} +5 a^2 z^2-4 z^2 a^{-2} -8 z^2 a^{-4} -2 z^2 a^{-6} +7 z^2+4 z a^{-1} +5 z a^{-3} +z a^{-5} -2 a^2+ a^{-4} -2
The A2 invariant Data:K11n33/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n33/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: (-3, -1)
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
-12 -8 72 98 38 96 \frac{400}{3} \frac{64}{3} 24 -288 32 -1176 -456 -\frac{10351}{10} \frac{1346}{15} -\frac{12422}{15} \frac{655}{6} -\frac{1711}{10}

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 K11n33. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
13         1-1
11        3 3
9       31 -2
7      53  2
5     43   -1
3    45    -1
1   45     1
-1  13      -2
-3 14       3
-5 1        -1
-71         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11n32.gif

K11n32

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K11n34