K11a137

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K11a136.gif

K11a136

K11a138.gif

K11a138

Contents

K11a137.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a137 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,6,17,5 X14,8,15,7 X18,9,19,10 X2,11,3,12 X20,13,21,14 X6,16,7,15 X22,18,1,17 X12,19,13,20 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -11, 5, -2, 6, -10, 7, -4, 8, -3, 9, -5, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 10 16 14 18 2 20 6 22 12 8
A Braid Representative
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A Morse Link Presentation K11a137 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a137/ThurstonBennequinNumber
Hyperbolic Volume 15.5095
A-Polynomial See Data:K11a137/A-polynomial

[edit Notes for K11a137'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 K11a137's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-12 t^2+26 t-31+26 t^{-1} -12 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 111, -2 }
Jones polynomial -q^4+4 q^3-7 q^2+12 q-15+17 q^{-1} -18 q^{-2} +15 q^{-3} -11 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) z^2 a^6+a^6-2 z^4 a^4-3 z^2 a^4+z^6 a^2+z^4 a^2-2 z^2 a^2-2 a^2+z^6+2 z^4+z^2+1-z^4 a^{-2} -z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+11 a z^9+5 z^9 a^{-1} +8 a^4 z^8+9 a^2 z^8+4 z^8 a^{-2} +5 z^8+8 a^5 z^7-7 a^3 z^7-32 a z^7-16 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-11 a^4 z^6-38 a^2 z^6-15 z^6 a^{-2} -36 z^6+3 a^7 z^5-11 a^5 z^5-2 a^3 z^5+27 a z^5+12 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-7 a^6 z^4+4 a^4 z^4+40 a^2 z^4+16 z^4 a^{-2} +44 z^4-2 a^7 z^3+8 a^5 z^3-14 a z^3-2 z^3 a^{-1} +2 z^3 a^{-3} -a^8 z^2+5 a^6 z^2-19 a^2 z^2-4 z^2 a^{-2} -17 z^2-2 a^5 z+2 a^3 z+6 a z+2 z a^{-1} -a^6+2 a^2- a^{-2} +1
The A2 invariant Data:K11a137/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a137/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a202,}

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

Vassiliev invariants

V2 and V3: (-4, 4)
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
-16 32 128 \frac{424}{3} \frac{200}{3} -512 -\frac{2464}{3} -\frac{640}{3} -160 -\frac{2048}{3} 512 -\frac{6784}{3} -\frac{3200}{3} -\frac{9902}{15} \frac{6088}{15} -\frac{58088}{45} \frac{2606}{9} -\frac{5582}{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 K11a137. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
9           1-1
7          3 3
5         41 -3
3        83  5
1       74   -3
-1      108    2
-3     98     -1
-5    69      -3
-7   59       4
-9  26        -4
-11 15         4
-13 2          -2
-151           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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K11a136.gif

K11a136

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K11a138