K11a343

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K11a342.gif

K11a342

K11a344.gif

K11a344

Contents

K11a343.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X6271 X14,4,15,3 X16,6,17,5 X22,8,1,7 X20,10,21,9 X18,12,19,11 X4,14,5,13 X2,16,3,15 X12,18,13,17 X10,20,11,19 X8,22,9,21
Gauss code 1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -9, 7, -2, 8, -3, 9, -6, 10, -5, 11, -4
Dowker-Thistlethwaite code 6 14 16 22 20 18 4 2 12 10 8
A Braid Representative
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A Morse Link Presentation K11a343 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 1
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a343/ThurstonBennequinNumber
Hyperbolic Volume 5.89363
A-Polynomial See Data:K11a343/A-polynomial

[edit Notes for K11a343'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 K11a343's four dimensional invariants]

Polynomial invariants

Alexander polynomial 8 t-15+8 t^{-1}
Conway polynomial 8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 31, 2 }
Jones polynomial -q^{12}+q^{11}-2 q^{10}+3 q^9-3 q^8+4 q^7-4 q^6+4 q^5-3 q^4+3 q^3-2 q^2+q
HOMFLY-PT polynomial (db, data sources) z^2 a^{-2} +2 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2 a^{-8} +z^2 a^{-10} + a^{-4} + a^{-8} - a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +3 z^9 a^{-11} +z^9 a^{-13} +3 z^8 a^{-8} -4 z^8 a^{-10} -7 z^8 a^{-12} +3 z^7 a^{-7} -9 z^7 a^{-9} -20 z^7 a^{-11} -8 z^7 a^{-13} +3 z^6 a^{-6} -13 z^6 a^{-8} +16 z^6 a^{-12} +3 z^5 a^{-5} -9 z^5 a^{-7} +9 z^5 a^{-9} +43 z^5 a^{-11} +22 z^5 a^{-13} +3 z^4 a^{-4} -6 z^4 a^{-6} +15 z^4 a^{-8} +10 z^4 a^{-10} -14 z^4 a^{-12} +2 z^3 a^{-3} -4 z^3 a^{-5} +4 z^3 a^{-7} -34 z^3 a^{-11} -24 z^3 a^{-13} +z^2 a^{-2} -3 z^2 a^{-4} -6 z^2 a^{-8} -5 z^2 a^{-10} +5 z^2 a^{-12} +8 z a^{-11} +8 z a^{-13} + a^{-4} + a^{-8} - a^{-12}
The A2 invariant Data:K11a343/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a343/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: (8, 24)
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
32 192 512 \frac{4336}{3} \frac{752}{3} 6144 11968 2048 1984 \frac{16384}{3} 18432 \frac{138752}{3} \frac{24064}{3} \frac{1493884}{15} -\frac{37136}{15} \frac{2065936}{45} \frac{10628}{9} \frac{97084}{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=2 is the signature of K11a343. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
25           1-1
23            0
21         21 -1
19        1   1
17       22   0
15      21    1
13     22     0
11    22      0
9   12       1
7  22        0
5  1         1
312          -1
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}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a342.gif

K11a342

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K11a344