K11a244

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K11a243.gif

K11a243

K11a245.gif

K11a245

Contents

K11a244.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a244 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X18,6,19,5 X22,8,1,7 X16,10,17,9 X20,12,21,11 X2,14,3,13 X8,16,9,15 X12,18,13,17 X6,20,7,19 X10,22,11,21
Gauss code 1, -7, 2, -1, 3, -10, 4, -8, 5, -11, 6, -9, 7, -2, 8, -5, 9, -3, 10, -6, 11, -4
Dowker-Thistlethwaite code 4 14 18 22 16 20 2 8 12 6 10
A Braid Representative
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A Morse Link Presentation K11a244 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a244/ThurstonBennequinNumber
Hyperbolic Volume 17.3477
A-Polynomial See Data:K11a244/A-polynomial

[edit Notes for K11a244'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 K11a244's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t^3-17 t^2+32 t-39+32 t^{-1} -17 t^{-2} +5 t^{-3}
Conway polynomial 5 z^6+13 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 147, 6 }
Jones polynomial -q^{14}+4 q^{13}-10 q^{12}+16 q^{11}-21 q^{10}+24 q^9-24 q^8+20 q^7-14 q^6+9 q^5-3 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +3 z^4 a^{-6} +11 z^4 a^{-8} -z^4 a^{-12} +3 z^2 a^{-6} +12 z^2 a^{-8} -5 z^2 a^{-10} -z^2 a^{-12} + a^{-6} +3 a^{-8} -3 a^{-10}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-10} +2 z^{10} a^{-12} +5 z^9 a^{-9} +12 z^9 a^{-11} +7 z^9 a^{-13} +6 z^8 a^{-8} +12 z^8 a^{-10} +17 z^8 a^{-12} +11 z^8 a^{-14} +3 z^7 a^{-7} -z^7 a^{-9} -10 z^7 a^{-11} +3 z^7 a^{-13} +9 z^7 a^{-15} +z^6 a^{-6} -14 z^6 a^{-8} -31 z^6 a^{-10} -36 z^6 a^{-12} -16 z^6 a^{-14} +4 z^6 a^{-16} -6 z^5 a^{-7} -17 z^5 a^{-9} -20 z^5 a^{-11} -24 z^5 a^{-13} -14 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +15 z^4 a^{-8} +22 z^4 a^{-10} +17 z^4 a^{-12} +9 z^4 a^{-14} -4 z^4 a^{-16} +3 z^3 a^{-7} +19 z^3 a^{-9} +25 z^3 a^{-11} +20 z^3 a^{-13} +10 z^3 a^{-15} -z^3 a^{-17} +3 z^2 a^{-6} -11 z^2 a^{-8} -10 z^2 a^{-10} -3 z^2 a^{-14} +z^2 a^{-16} -6 z a^{-9} -7 z a^{-11} -5 z a^{-13} -4 z a^{-15} - a^{-6} +3 a^{-8} +3 a^{-10}
The A2 invariant Data:K11a244/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a244/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: (9, 26)
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
36 208 648 1546 214 7488 \frac{38464}{3} \frac{6592}{3} 1520 7776 21632 55656 7704 \frac{1090333}{10} \frac{73282}{15} \frac{578746}{15} \frac{4483}{6} \frac{47933}{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=6 is the signature of K11a244. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          3 3
25         71 -6
23        93  6
21       127   -5
19      129    3
17     1212     0
15    812      -4
13   612       6
11  38        -5
9  6         6
713          -2
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}^{3}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=6 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=7 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=8 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=9 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a243.gif

K11a243

K11a245.gif

K11a245