K11a352

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K11a351.gif

K11a351

K11a353.gif

K11a353

Contents

K11a352.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a352 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X20,9,21,10 X4,12,5,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X10,19,11,20 X8,21,9,22
Gauss code 1, -7, 2, -6, 3, -1, 4, -11, 5, -10, 6, -9, 7, -4, 8, -3, 9, -2, 10, -5, 11, -8
Dowker-Thistlethwaite code 6 18 16 14 20 4 2 22 12 10 8
A Braid Representative
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A Morse Link Presentation K11a352 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a352/ThurstonBennequinNumber
Hyperbolic Volume 17.1582
A-Polynomial See Data:K11a352/A-polynomial

[edit Notes for K11a352's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,3]
Rasmussen s-Invariant -2

[edit Notes for K11a352's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-13 t^2+32 t-41+32 t^{-1} -13 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \left\{3,t^2+2 t+1\right\}
Determinant and Signature { 135, 2 }
Jones polynomial q^7-4 q^6+8 q^5-13 q^4+19 q^3-21 q^2+21 q-19+14 q^{-1} -9 q^{-2} +5 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6-a^2 z^4+z^4 a^{-2} -2 z^4 a^{-4} +z^4+z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -2 z^2+2 a^2+2 a^{-2} -3
Kauffman polynomial (db, data sources) 4 z^{10} a^{-2} +4 z^{10}+8 a z^9+18 z^9 a^{-1} +10 z^9 a^{-3} +5 a^2 z^8+6 z^8 a^{-2} +12 z^8 a^{-4} -z^8+a^3 z^7-28 a z^7-57 z^7 a^{-1} -17 z^7 a^{-3} +11 z^7 a^{-5} -16 a^2 z^6-37 z^6 a^{-2} -17 z^6 a^{-4} +8 z^6 a^{-6} -28 z^6-2 a^3 z^5+27 a z^5+51 z^5 a^{-1} +6 z^5 a^{-3} -12 z^5 a^{-5} +4 z^5 a^{-7} +11 a^2 z^4+28 z^4 a^{-2} +3 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +28 z^4-6 a z^3-12 z^3 a^{-1} -2 z^3 a^{-3} +2 z^3 a^{-5} -2 z^3 a^{-7} +2 a^2 z^2+2 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3
The A2 invariant Data:K11a352/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a352/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a6, K11a132,}

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

Vassiliev invariants

V2 and V3: (-2, 0)
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
-8 0 32 \frac{164}{3} \frac{76}{3} 0 0 64 -64 -\frac{256}{3} 0 -\frac{1312}{3} -\frac{608}{3} -\frac{6271}{15} \frac{1964}{15} -\frac{19564}{45} \frac{319}{9} -\frac{991}{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 K11a352. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
15           11
13          3 -3
11         51 4
9        83  -5
7       115   6
5      108    -2
3     1111     0
1    911      2
-1   510       -5
-3  49        5
-5 15         -4
-7 4          4
-91           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11a351.gif

K11a351

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K11a353