K11a325

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K11a324.gif

K11a324

K11a326.gif

K11a326

Contents

K11a325.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a325 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,3,13,4 X16,5,17,6 X22,8,1,7 X20,10,21,9 X18,11,19,12 X2,13,3,14 X10,15,11,16 X4,17,5,18 X14,19,15,20 X8,22,9,21
Gauss code 1, -7, 2, -9, 3, -1, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -6, 10, -5, 11, -4
Dowker-Thistlethwaite code 6 12 16 22 20 18 2 10 4 14 8
A Braid Representative
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A Morse Link Presentation K11a325 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a325/ThurstonBennequinNumber
Hyperbolic Volume 14.4126
A-Polynomial See Data:K11a325/A-polynomial

[edit Notes for K11a325's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a325's four dimensional invariants]

Polynomial invariants

Alexander polynomial -6 t^2+24 t-35+24 t^{-1} -6 t^{-2}
Conway polynomial 1-6 z^4
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 95, -2 }
Jones polynomial q^3-3 q^2+6 q-9+13 q^{-1} -15 q^{-2} +15 q^{-3} -13 q^{-4} +10 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^8+3 z^2 a^6+2 a^6-2 z^4 a^4-z^2 a^4-3 z^4 a^2-4 z^2 a^2-2 a^2-z^4+z^2+2+z^2 a^{-2}
Kauffman polynomial (db, data sources) 2 a^4 z^{10}+2 a^2 z^{10}+4 a^5 z^9+9 a^3 z^9+5 a z^9+4 a^6 z^8+a^2 z^8+5 z^8+4 a^7 z^7-8 a^5 z^7-32 a^3 z^7-17 a z^7+3 z^7 a^{-1} +3 a^8 z^6-3 a^6 z^6-9 a^4 z^6-21 a^2 z^6+z^6 a^{-2} -17 z^6+a^9 z^5-5 a^7 z^5+10 a^5 z^5+45 a^3 z^5+20 a z^5-9 z^5 a^{-1} -6 a^8 z^4-6 a^6 z^4+17 a^4 z^4+37 a^2 z^4-3 z^4 a^{-2} +17 z^4-2 a^9 z^3-2 a^7 z^3-6 a^5 z^3-21 a^3 z^3-11 a z^3+4 z^3 a^{-1} +3 a^8 z^2+6 a^6 z^2-6 a^4 z^2-19 a^2 z^2+z^2 a^{-2} -9 z^2+a^9 z+a^7 z+a^5 z+3 a^3 z+2 a z-a^8-2 a^6+2 a^2+2
The A2 invariant Data:K11a325/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a325/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a7,}

Vassiliev invariants

V2 and V3: (0, -2)
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
0 -16 0 112 48 0 -\frac{1024}{3} -\frac{64}{3} -176 0 128 0 0 920 -\frac{1424}{3} 864 168 136

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 K11a325. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
7           11
5          2 -2
3         41 3
1        52  -3
-1       84   4
-3      86    -2
-5     77     0
-7    68      2
-9   47       -3
-11  26        4
-13 14         -3
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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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K11a324

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K11a326