K11a357

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K11a356.gif

K11a356

K11a358.gif

K11a358

Contents

K11a357.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a357 at Knotilus!



Knot presentations

Planar diagram presentation X8291 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X6,14,7,13 X4,16,5,15 X2,18,3,17 X12,20,13,19 X10,22,11,21
Gauss code 1, -9, 2, -8, 3, -7, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6
Dowker-Thistlethwaite code 8 14 16 18 20 22 6 4 2 12 10
A Braid Representative
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A Morse Link Presentation K11a357 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{3,4\}
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a357/ThurstonBennequinNumber
Hyperbolic Volume 13.05
A-Polynomial See Data:K11a357/A-polynomial

[edit Notes for K11a357'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 K11a357's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^3-11 t^2+19 t-23+19 t^{-1} -11 t^{-2} +4 t^{-3}
Conway polynomial 4 z^6+13 z^4+11 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 91, 6 }
Jones polynomial -q^{14}+3 q^{13}-7 q^{12}+10 q^{11}-13 q^{10}+15 q^9-14 q^8+12 q^7-8 q^6+5 q^5-2 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +2 z^6 a^{-8} +z^6 a^{-10} +4 z^4 a^{-6} +8 z^4 a^{-8} +2 z^4 a^{-10} -z^4 a^{-12} +4 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} -2 z^2 a^{-12} +4 a^{-8} -2 a^{-10} - a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +5 z^9 a^{-11} +3 z^9 a^{-13} +3 z^8 a^{-8} +2 z^8 a^{-10} +4 z^8 a^{-12} +5 z^8 a^{-14} +2 z^7 a^{-7} -2 z^7 a^{-9} -10 z^7 a^{-11} -z^7 a^{-13} +5 z^7 a^{-15} +z^6 a^{-6} -10 z^6 a^{-8} -7 z^6 a^{-10} -7 z^6 a^{-12} -8 z^6 a^{-14} +3 z^6 a^{-16} -6 z^5 a^{-7} -2 z^5 a^{-9} +11 z^5 a^{-11} -3 z^5 a^{-13} -9 z^5 a^{-15} +z^5 a^{-17} -4 z^4 a^{-6} +15 z^4 a^{-8} +10 z^4 a^{-10} +z^4 a^{-12} +5 z^4 a^{-14} -5 z^4 a^{-16} +4 z^3 a^{-7} +2 z^3 a^{-9} -6 z^3 a^{-11} +3 z^3 a^{-13} +5 z^3 a^{-15} -2 z^3 a^{-17} +4 z^2 a^{-6} -14 z^2 a^{-8} -9 z^2 a^{-10} +6 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -2 z a^{-9} +2 z a^{-11} -3 z a^{-15} +z a^{-17} +4 a^{-8} +2 a^{-10} - a^{-12}
The A2 invariant Data:K11a357/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a357/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: (11, 35)
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
44 280 968 \frac{7066}{3} \frac{1118}{3} 12320 \frac{65392}{3} \frac{11584}{3} 2968 \frac{42592}{3} 39200 \frac{310904}{3} \frac{49192}{3} \frac{6155861}{30} \frac{68398}{15} \frac{3672562}{45} \frac{23659}{18} \frac{324341}{30}

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 K11a357. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          2 2
25         51 -4
23        52  3
21       85   -3
19      75    2
17     78     1
15    57      -2
13   37       4
11  25        -3
9  3         3
712          -1
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}^{2}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=8 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a356.gif

K11a356

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K11a358