K11a247

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K11a246.gif

K11a246

K11a248.gif

K11a248

Contents

K11a247.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X22,6,1,5 X20,8,21,7 X18,10,19,9 X16,12,17,11 X2,14,3,13 X12,16,13,15 X10,18,11,17 X8,20,9,19 X6,22,7,21
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -8, 7, -2, 8, -6, 9, -5, 10, -4, 11, -3
Dowker-Thistlethwaite code 4 14 22 20 18 16 2 12 10 8 6
A Braid Representative
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A Morse Link Presentation K11a247 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a247/ThurstonBennequinNumber
Hyperbolic Volume 3.55382
A-Polynomial See Data:K11a247/A-polynomial

[edit Notes for K11a247's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant -2

[edit Notes for K11a247's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t-9+5 t^{-1}
Conway polynomial 5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 19, 2 }
Jones polynomial -q^{12}+q^{11}-q^{10}+2 q^9-2 q^8+2 q^7-2 q^6+2 q^5-2 q^4+2 q^3-q^2+q
HOMFLY-PT polynomial (db, data sources) z^2 a^{-2} +z^2 a^{-4} +z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} + a^{-2} + a^{-10} - 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} -8 z^8 a^{-12} +z^7 a^{-7} -5 z^7 a^{-9} -14 z^7 a^{-11} -8 z^7 a^{-13} +z^6 a^{-6} -4 z^6 a^{-8} +17 z^6 a^{-10} +22 z^6 a^{-12} +z^5 a^{-5} -3 z^5 a^{-7} +6 z^5 a^{-9} +31 z^5 a^{-11} +21 z^5 a^{-13} +z^4 a^{-4} -2 z^4 a^{-6} +3 z^4 a^{-8} -19 z^4 a^{-10} -25 z^4 a^{-12} +z^3 a^{-3} -z^3 a^{-5} +z^3 a^{-7} -z^3 a^{-9} -24 z^3 a^{-11} -20 z^3 a^{-13} +z^2 a^{-2} +10 z^2 a^{-10} +11 z^2 a^{-12} +5 z a^{-11} +5 z a^{-13} - a^{-2} - a^{-10} - a^{-12}
The A2 invariant Data:K11a247/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a247/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: (5, 15)
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
20 120 200 \frac{2230}{3} \frac{290}{3} 2400 5200 800 760 \frac{4000}{3} 7200 \frac{44600}{3} \frac{5800}{3} \frac{223375}{6} -\frac{1142}{3} \frac{138182}{9} \frac{12565}{18} \frac{11791}{6}

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 K11a247. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
25           1-1
23            0
21         11 0
19        1   1
17       11   0
15      11    0
13     11     0
11    11      0
9   11       0
7  11        0
5  1         1
311          0
11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\mathbb Z}_2 {\mathbb Z}
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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K11a246

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K11a248