K11a339

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K11a338.gif

K11a338

K11a340.gif

K11a340

Contents

K11a339.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a339 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X14,4,15,3 X16,6,17,5 X20,8,21,7 X22,10,1,9 X18,12,19,11 X2,14,3,13 X4,16,5,15 X12,18,13,17 X10,20,11,19 X8,22,9,21
Gauss code 1, -7, 2, -8, 3, -1, 4, -11, 5, -10, 6, -9, 7, -2, 8, -3, 9, -6, 10, -4, 11, -5
Dowker-Thistlethwaite code 6 14 16 20 22 18 2 4 12 10 8
A Braid Representative
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A Morse Link Presentation K11a339 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:K11a339/ThurstonBennequinNumber
Hyperbolic Volume 10.1171
A-Polynomial See Data:K11a339/A-polynomial

[edit Notes for K11a339'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 K11a339's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-7 t^2+11 t-13+11 t^{-1} -7 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+11 z^4+10 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 55, 6 }
Jones polynomial -q^{14}+2 q^{13}-4 q^{12}+6 q^{11}-8 q^{10}+9 q^9-8 q^8+7 q^7-5 q^6+3 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} +3 z^4 a^{-8} +4 z^4 a^{-10} -z^4 a^{-12} +7 z^2 a^{-6} +z^2 a^{-8} +5 z^2 a^{-10} -3 z^2 a^{-12} +2 a^{-6} - a^{-8} +2 a^{-10} -2 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +3 z^9 a^{-11} +2 z^9 a^{-13} +z^8 a^{-8} -5 z^8 a^{-10} -3 z^8 a^{-12} +3 z^8 a^{-14} +z^7 a^{-7} -3 z^7 a^{-9} -13 z^7 a^{-11} -6 z^7 a^{-13} +3 z^7 a^{-15} +z^6 a^{-6} -2 z^6 a^{-8} +15 z^6 a^{-10} +6 z^6 a^{-12} -10 z^6 a^{-14} +2 z^6 a^{-16} -3 z^5 a^{-7} +6 z^5 a^{-9} +26 z^5 a^{-11} +7 z^5 a^{-13} -9 z^5 a^{-15} +z^5 a^{-17} -5 z^4 a^{-6} -z^4 a^{-8} -19 z^4 a^{-10} -5 z^4 a^{-12} +13 z^4 a^{-14} -5 z^4 a^{-16} +z^3 a^{-7} -8 z^3 a^{-9} -19 z^3 a^{-11} +z^3 a^{-13} +8 z^3 a^{-15} -3 z^3 a^{-17} +7 z^2 a^{-6} +2 z^2 a^{-8} +8 z^2 a^{-10} +7 z^2 a^{-12} -5 z^2 a^{-14} +z^2 a^{-16} +z a^{-7} +z a^{-9} +4 z a^{-11} -3 z a^{-15} +z a^{-17} -2 a^{-6} - a^{-8} -2 a^{-10} -2 a^{-12}
The A2 invariant Data:K11a339/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a339/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n180,}

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

Vassiliev invariants

V2 and V3: (10, 32)
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
40 256 800 \frac{6188}{3} \frac{916}{3} 10240 \frac{55552}{3} \frac{9568}{3} 2464 \frac{32000}{3} 32768 \frac{247520}{3} \frac{36640}{3} \frac{507727}{3} 4044 \frac{590524}{9} \frac{12949}{9} \frac{25423}{3}

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 K11a339. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          1 1
25         31 -2
23        31  2
21       53   -2
19      43    1
17     45     1
15    34      -1
13   24       2
11  13        -2
9  2         2
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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2 {\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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K11a338.gif

K11a338

K11a340.gif

K11a340