K11a154

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K11a153.gif

K11a153

K11a155.gif

K11a155

Contents

K11a154.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a154 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X18,6,19,5 X12,8,13,7 X2,10,3,9 X8,12,9,11 X22,13,1,14 X20,15,21,16 X6,18,7,17 X16,19,17,20 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -4, 7, -11, 8, -10, 9, -3, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 10 18 12 2 8 22 20 6 16 14
A Braid Representative
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A Morse Link Presentation K11a154 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:K11a154/ThurstonBennequinNumber
Hyperbolic Volume 11.5462
A-Polynomial See Data:K11a154/A-polynomial

[edit Notes for K11a154'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 K11a154's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t^2+17 t-25+17 t^{-1} -4 t^{-2}
Conway polynomial -4 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 67, 2 }
Jones polynomial -q^8+2 q^7-4 q^6+7 q^5-9 q^4+11 q^3-10 q^2+9 q-7+4 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -2 z^4 a^{-2} -z^4 a^{-4} -z^4+a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} +2 z^2 a^{-6} -z^2+a^2+ a^{-4} + a^{-6} - a^{-8} -1
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +4 z^9 a^{-3} +2 z^9 a^{-5} -z^8 a^{-2} -z^8 a^{-4} +2 z^8 a^{-6} +2 z^8+2 a z^7-5 z^7 a^{-1} -14 z^7 a^{-3} -5 z^7 a^{-5} +2 z^7 a^{-7} +a^2 z^6-z^6 a^{-2} -2 z^6 a^{-6} +2 z^6 a^{-8} -4 z^6-7 a z^5+3 z^5 a^{-1} +22 z^5 a^{-3} +9 z^5 a^{-5} -2 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+2 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} -5 z^4 a^{-8} -3 z^4+6 a z^3-11 z^3 a^{-3} -5 z^3 a^{-5} -3 z^3 a^{-7} -3 z^3 a^{-9} +4 a^2 z^2-z^2 a^{-2} -3 z^2 a^{-4} +2 z^2 a^{-6} +3 z^2 a^{-8} +5 z^2-2 a z-2 z a^{-1} +z a^{-3} +z a^{-5} +2 z a^{-7} +2 z a^{-9} -a^2+ a^{-4} - a^{-6} - a^{-8} -1
The A2 invariant Data:K11a154/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a154/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_30,}

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

Vassiliev invariants

V2 and V3: (1, 5)
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
4 40 8 \frac{446}{3} \frac{106}{3} 160 \frac{1744}{3} \frac{160}{3} 168 \frac{32}{3} 800 \frac{1784}{3} \frac{424}{3} \frac{87631}{30} -\frac{1954}{5} \frac{74582}{45} \frac{2129}{18} \frac{6991}{30}

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 K11a154. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          1 1
13         31 -2
11        41  3
9       53   -2
7      64    2
5     45     1
3    56      -1
1   35       2
-1  14        -3
-3 13         2
-5 1          -1
-71           1
Integral Khovanov Homology

(db, data source)

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