K11a114

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K11a113.gif

K11a113

K11a115.gif

K11a115

Contents

K11a114.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a114 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,6,15,5 X18,7,19,8 X2,10,3,9 X16,12,17,11 X6,14,7,13 X22,16,1,15 X20,17,21,18 X8,19,9,20 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, -7, 4, -10, 5, -2, 6, -11, 7, -3, 8, -6, 9, -4, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 10 14 18 2 16 6 22 20 8 12
A Braid Representative
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A Morse Link Presentation K11a114 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a114/ThurstonBennequinNumber
Hyperbolic Volume 17.5584
A-Polynomial See Data:K11a114/A-polynomial

[edit Notes for K11a114's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a114's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-15 t^2+35 t-45+35 t^{-1} -15 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+3 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 151, 2 }
Jones polynomial q^9-4 q^8+9 q^7-16 q^6+21 q^5-24 q^4+25 q^3-21 q^2+16 q-9+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^6 a^{-4} +z^4 a^{-2} +6 z^4 a^{-4} -3 z^4 a^{-6} -z^4+8 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} -z^2+4 a^{-4} -4 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-4} +2 z^{10} a^{-6} +7 z^9 a^{-3} +13 z^9 a^{-5} +6 z^9 a^{-7} +10 z^8 a^{-2} +19 z^8 a^{-4} +16 z^8 a^{-6} +7 z^8 a^{-8} +8 z^7 a^{-1} +2 z^7 a^{-3} -12 z^7 a^{-5} -2 z^7 a^{-7} +4 z^7 a^{-9} -13 z^6 a^{-2} -46 z^6 a^{-4} -44 z^6 a^{-6} -14 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-11 z^5 a^{-1} -20 z^5 a^{-3} -18 z^5 a^{-5} -19 z^5 a^{-7} -9 z^5 a^{-9} +7 z^4 a^{-2} +37 z^4 a^{-4} +35 z^4 a^{-6} +8 z^4 a^{-8} -2 z^4 a^{-10} -5 z^4-a z^3+6 z^3 a^{-1} +16 z^3 a^{-3} +22 z^3 a^{-5} +20 z^3 a^{-7} +7 z^3 a^{-9} -2 z^2 a^{-2} -16 z^2 a^{-4} -15 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} +2 z^2-z a^{-1} -3 z a^{-3} -7 z a^{-5} -7 z a^{-7} -2 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8}
The A2 invariant Data:K11a114/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a114/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: (2, 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
8 16 32 \frac{76}{3} -\frac{28}{3} 128 \frac{160}{3} \frac{64}{3} -48 \frac{256}{3} 128 \frac{608}{3} -\frac{224}{3} \frac{1111}{15} \frac{3316}{15} -\frac{13076}{45} -\frac{439}{9} -\frac{809}{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 K11a114. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          3 -3
15         61 5
13        103  -7
11       116   5
9      1310    -3
7     1211     1
5    913      4
3   712       -5
1  310        7
-1 16         -5
-3 3          3
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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K11a113.gif

K11a113

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K11a115