K11a119

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K11a118.gif

K11a118

K11a120.gif

K11a120

Contents

K11a119.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a119 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X18,7,19,8 X12,10,13,9 X2,11,3,12 X8,14,9,13 X22,16,1,15 X20,18,21,17 X6,19,7,20 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -5, 7, -3, 8, -11, 9, -4, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 10 14 18 12 2 8 22 20 6 16
A Braid Representative
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A Morse Link Presentation K11a119 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a119/ThurstonBennequinNumber
Hyperbolic Volume 12.058
A-Polynomial See Data:K11a119/A-polynomial

[edit Notes for K11a119's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11a119's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^2-19 t+31-19 t^{-1} +4 t^{-2}
Conway polynomial 4 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 77, 0 }
Jones polynomial -q^7+3 q^6-5 q^5+8 q^4-10 q^3+12 q^2-12 q+10-8 q^{-1} +5 q^{-2} -2 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) a^4-2 z^2 a^2+z^4-2 z^2-2+2 z^4 a^{-2} +2 z^2 a^{-2} +2 a^{-2} +z^4 a^{-4} -z^2 a^{-6}
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} +2 z^8 a^{-2} +2 z^8 a^{-4} +3 z^8 a^{-6} +3 z^8+4 a z^7+2 z^7 a^{-1} -14 z^7 a^{-3} -11 z^7 a^{-5} +z^7 a^{-7} +3 a^2 z^6-5 z^6 a^{-2} -15 z^6 a^{-4} -13 z^6 a^{-6} +2 a^3 z^5-6 a z^5-12 z^5 a^{-1} +11 z^5 a^{-3} +11 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-2 a^2 z^4-6 z^4 a^{-2} +15 z^4 a^{-4} +16 z^4 a^{-6} -8 z^4-2 a^3 z^3+8 a z^3+12 z^3 a^{-1} -6 z^3 a^{-3} -4 z^3 a^{-5} +4 z^3 a^{-7} -2 a^4 z^2+8 z^2 a^{-2} -5 z^2 a^{-4} -5 z^2 a^{-6} +10 z^2-4 a z-4 z a^{-1} +z a^{-3} +z a^{-5} +a^4-2 a^{-2} -2
The A2 invariant Data:K11a119/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a119/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a98,}

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

Vassiliev invariants

V2 and V3: (-3, 1)
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
-12 8 72 66 6 -96 -\frac{592}{3} -\frac{352}{3} 8 -288 32 -792 -72 -\frac{6671}{10} -\frac{2058}{5} \frac{2618}{15} -\frac{145}{6} \frac{689}{10}

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=0 is the signature of K11a119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
15           1-1
13          2 2
11         31 -2
9        52  3
7       53   -2
5      75    2
3     55     0
1    57      -2
-1   46       2
-3  14        -3
-5 14         3
-7 1          -1
-91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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K11a120