K11a358

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K11a357.gif

K11a357

K11a359.gif

K11a359

Contents

K11a358.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a358 at Knotilus!



Knot presentations

Planar diagram presentation X8291 X14,4,15,3 X16,6,17,5 X18,8,19,7 X22,10,1,9 X20,12,21,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X12,20,13,19 X10,22,11,21
Gauss code 1, -7, 2, -8, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, -6, 11, -5
Dowker-Thistlethwaite code 8 14 16 18 22 20 2 4 6 12 10
A Braid Representative
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A Morse Link Presentation K11a358 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a358/ThurstonBennequinNumber
Hyperbolic Volume 6.1028
A-Polynomial See Data:K11a358/A-polynomial

[edit Notes for K11a358's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a358's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-5 t^2+5 t-5+5 t^{-1} -5 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+13 z^4+12 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 31, 6 }
Jones polynomial -q^{14}+q^{13}-2 q^{12}+3 q^{11}-4 q^{10}+5 q^9-4 q^8+4 q^7-3 q^6+2 q^5-q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +z^6 a^{-8} +z^6 a^{-10} +5 z^4 a^{-6} +4 z^4 a^{-8} +5 z^4 a^{-10} -z^4 a^{-12} +6 z^2 a^{-6} +3 z^2 a^{-8} +7 z^2 a^{-10} -4 z^2 a^{-12} + a^{-6} +3 a^{-10} -3 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +2 z^9 a^{-11} +z^9 a^{-13} +z^8 a^{-8} -7 z^8 a^{-10} -7 z^8 a^{-12} +z^8 a^{-14} +z^7 a^{-7} -5 z^7 a^{-9} -12 z^7 a^{-11} -5 z^7 a^{-13} +z^7 a^{-15} +z^6 a^{-6} -4 z^6 a^{-8} +20 z^6 a^{-10} +20 z^6 a^{-12} -4 z^6 a^{-14} +z^6 a^{-16} -4 z^5 a^{-7} +9 z^5 a^{-9} +26 z^5 a^{-11} +9 z^5 a^{-13} -3 z^5 a^{-15} +z^5 a^{-17} -5 z^4 a^{-6} +4 z^4 a^{-8} -26 z^4 a^{-10} -26 z^4 a^{-12} +6 z^4 a^{-14} -3 z^4 a^{-16} +3 z^3 a^{-7} -7 z^3 a^{-9} -20 z^3 a^{-11} -4 z^3 a^{-13} +2 z^3 a^{-15} -4 z^3 a^{-17} +6 z^2 a^{-6} -2 z^2 a^{-8} +14 z^2 a^{-10} +18 z^2 a^{-12} -3 z^2 a^{-14} +z^2 a^{-16} +5 z a^{-11} +z a^{-13} -z a^{-15} +3 z a^{-17} - a^{-6} -3 a^{-10} -3 a^{-12}
The A2 invariant Data:K11a358/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a358/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: (12, 41)
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
48 328 1152 2904 464 15744 \frac{85168}{3} \frac{15136}{3} 3944 18432 53792 139392 22272 \frac{1408222}{5} \frac{76816}{15} \frac{569456}{5} 1942 \frac{75822}{5}

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=6 is the signature of K11a358. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27            0
25         21 -1
23        1   1
21       32   -1
19      21    1
17     23     1
15    22      0
13   12       1
11  12        -1
9  1         1
711          0
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=10 {\mathbb Z}
r=11 {\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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K11a357

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K11a359