K11a344

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K11a343.gif

K11a343

K11a345.gif

K11a345

Contents

K11a344.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a344 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X14,4,15,3 X18,6,19,5 X16,7,17,8 X20,10,21,9 X22,12,1,11 X4,14,5,13 X8,15,9,16 X2,18,3,17 X12,20,13,19 X10,22,11,21
Gauss code 1, -9, 2, -7, 3, -1, 4, -8, 5, -11, 6, -10, 7, -2, 8, -4, 9, -3, 10, -5, 11, -6
Dowker-Thistlethwaite code 6 14 18 16 20 22 4 8 2 12 10
A Braid Representative
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A Morse Link Presentation K11a344 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a344/ThurstonBennequinNumber
Hyperbolic Volume 16.4336
A-Polynomial See Data:K11a344/A-polynomial

[edit Notes for K11a344's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a344's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+15 t^2-29 t+35-29 t^{-1} +15 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-3 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 129, 4 }
Jones polynomial -q^{11}+4 q^{10}-9 q^9+14 q^8-19 q^7+21 q^6-20 q^5+18 q^4-12 q^3+7 q^2-3 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -z^4 a^{-4} -6 z^4 a^{-6} +3 z^4 a^{-8} +2 z^2 a^{-2} +3 z^2 a^{-4} -6 z^2 a^{-6} +6 z^2 a^{-8} -z^2 a^{-10} +3 a^{-4} -3 a^{-6} +2 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-6} +2 z^{10} a^{-8} +5 z^9 a^{-5} +11 z^9 a^{-7} +6 z^9 a^{-9} +5 z^8 a^{-4} +7 z^8 a^{-6} +11 z^8 a^{-8} +9 z^8 a^{-10} +3 z^7 a^{-3} -10 z^7 a^{-5} -22 z^7 a^{-7} -z^7 a^{-9} +8 z^7 a^{-11} +z^6 a^{-2} -13 z^6 a^{-4} -27 z^6 a^{-6} -30 z^6 a^{-8} -13 z^6 a^{-10} +4 z^6 a^{-12} -8 z^5 a^{-3} +8 z^5 a^{-5} +18 z^5 a^{-7} -12 z^5 a^{-9} -13 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +13 z^4 a^{-4} +34 z^4 a^{-6} +29 z^4 a^{-8} +6 z^4 a^{-10} -5 z^4 a^{-12} +5 z^3 a^{-3} -4 z^3 a^{-5} -8 z^3 a^{-7} +9 z^3 a^{-9} +7 z^3 a^{-11} -z^3 a^{-13} +2 z^2 a^{-2} -9 z^2 a^{-4} -20 z^2 a^{-6} -13 z^2 a^{-8} -3 z^2 a^{-10} +z^2 a^{-12} +2 z a^{-7} -2 z a^{-11} +3 a^{-4} +3 a^{-6} +2 a^{-8} + a^{-10}
The A2 invariant Data:K11a344/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a344/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a275,}

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

Vassiliev invariants

V2 and V3: (4, 9)
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
16 72 128 \frac{1256}{3} \frac{256}{3} 1152 2544 416 520 \frac{2048}{3} 2592 \frac{20096}{3} \frac{4096}{3} \frac{230822}{15} -\frac{18368}{15} \frac{370448}{45} \frac{2698}{9} \frac{18182}{15}

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=4 is the signature of K11a344. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          3 3
19         61 -5
17        83  5
15       116   -5
13      108    2
11     1011     1
9    810      -2
7   410       6
5  38        -5
3 15         4
1 2          -2
-11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\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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K11a343.gif

K11a343

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K11a345