K11a333

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K11a332.gif

K11a332

K11a334.gif

K11a334

Contents

K11a333.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a333 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X14,3,15,4 X12,5,13,6 X16,8,17,7 X20,9,21,10 X18,11,19,12 X4,13,5,14 X2,15,3,16 X22,18,1,17 X10,19,11,20 X8,21,9,22
Gauss code 1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -3, 7, -2, 8, -4, 9, -6, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 14 12 16 20 18 4 2 22 10 8
A Braid Representative
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A Morse Link Presentation K11a333 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:K11a333/ThurstonBennequinNumber
Hyperbolic Volume 10.2751
A-Polynomial See Data:K11a333/A-polynomial

[edit Notes for K11a333's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11a333's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^2-16 t+25-16 t^{-1} +4 t^{-2}
Conway polynomial 4 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 65, 0 }
Jones polynomial -q^3+3 q^2-5 q+8-9 q^{-1} +10 q^{-2} -9 q^{-3} +8 q^{-4} -6 q^{-5} +3 q^{-6} -2 q^{-7} + q^{-8}
HOMFLY-PT polynomial (db, data sources) a^8-2 z^2 a^6-2 a^6+z^4 a^4+2 z^4 a^2+3 z^2 a^2+2 a^2+z^4-z^2 a^{-2}
Kauffman polynomial (db, data sources) a^6 z^{10}+a^4 z^{10}+2 a^7 z^9+5 a^5 z^9+3 a^3 z^9+a^8 z^8-2 a^6 z^8+2 a^4 z^8+5 a^2 z^8-12 a^7 z^7-24 a^5 z^7-5 a^3 z^7+7 a z^7-6 a^8 z^6-8 a^6 z^6-17 a^4 z^6-8 a^2 z^6+7 z^6+24 a^7 z^5+39 a^5 z^5-3 a^3 z^5-13 a z^5+5 z^5 a^{-1} +11 a^8 z^4+22 a^6 z^4+19 a^4 z^4-5 a^2 z^4+3 z^4 a^{-2} -10 z^4-18 a^7 z^3-28 a^5 z^3-a^3 z^3+5 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -6 a^8 z^2-14 a^6 z^2-6 a^4 z^2+7 a^2 z^2-z^2 a^{-2} +4 z^2+4 a^7 z+8 a^5 z+4 a^3 z+a^8+2 a^6-2 a^2
The A2 invariant Data:K11a333/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a333/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_33,}

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

Vassiliev invariants

V2 and V3: (0, 2)
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
0 16 0 -96 -32 0 \frac{928}{3} -\frac{128}{3} 208 0 128 0 0 -432 \frac{2368}{3} -\frac{2752}{3} -\frac{592}{3} -176

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=0 is the signature of K11a333. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-10123χ
7           1-1
5          2 2
3         31 -2
1        52  3
-1       54   -1
-3      54    1
-5     45     1
-7    45      -1
-9   24       2
-11  14        -3
-13 12         1
-15 1          -1
-171           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\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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K11a332

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K11a334