K11a258

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K11a257.gif

K11a257

K11a259.gif

K11a259

Contents

K11a258.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a258 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8493 X14,5,15,6 X2837 X16,10,17,9 X20,11,21,12 X18,13,19,14 X4,15,5,16 X22,18,1,17 X12,19,13,20 X10,21,11,22
Gauss code 1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -7, 10, -6, 11, -9
Dowker-Thistlethwaite code 6 8 14 2 16 20 18 4 22 12 10
A Braid Representative
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A Morse Link Presentation K11a258 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a258/ThurstonBennequinNumber
Hyperbolic Volume 11.9877
A-Polynomial See Data:K11a258/A-polynomial

[edit Notes for K11a258'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 K11a258's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+17 t-19+17 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 75, 2 }
Jones polynomial -q^6+3 q^5-6 q^4+9 q^3-10 q^2+12 q-11+9 q^{-1} -7 q^{-2} +4 q^{-3} -2 q^{-4} + q^{-5}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6-2 a^2 z^4+3 z^4 a^{-2} -z^4 a^{-4} +3 z^4+a^4 z^2-6 a^2 z^2+3 z^2 a^{-2} -2 z^2 a^{-4} +3 z^2+2 a^4-4 a^2+2 a^{-2} - a^{-4} +2
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+2 a^3 z^9+6 a z^9+4 z^9 a^{-1} +a^4 z^8+7 z^8 a^{-2} +6 z^8-11 a^3 z^7-26 a z^7-6 z^7 a^{-1} +9 z^7 a^{-3} -6 a^4 z^6-18 a^2 z^6-13 z^6 a^{-2} +9 z^6 a^{-4} -34 z^6+20 a^3 z^5+35 a z^5-7 z^5 a^{-1} -16 z^5 a^{-3} +6 z^5 a^{-5} +12 a^4 z^4+39 a^2 z^4-z^4 a^{-2} -14 z^4 a^{-4} +3 z^4 a^{-6} +43 z^4-14 a^3 z^3-19 a z^3+4 z^3 a^{-1} +4 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} -9 a^4 z^2-25 a^2 z^2+4 z^2 a^{-2} +7 z^2 a^{-4} -19 z^2+4 a^3 z+6 a z+2 z a^{-1} +z a^{-3} +z a^{-5} +2 a^4+4 a^2-2 a^{-2} - a^{-4} +2
The A2 invariant Data:K11a258/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a258/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n150,}

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

Vassiliev invariants

V2 and V3: (-1, 3)
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 24 8 -\frac{62}{3} -\frac{58}{3} -96 -144 -128 56 -\frac{32}{3} 288 \frac{248}{3} \frac{232}{3} \frac{18929}{30} -\frac{846}{5} \frac{25978}{45} -\frac{1361}{18} \frac{2609}{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 K11a258. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
13           1-1
11          2 2
9         41 -3
7        52  3
5       54   -1
3      75    2
1     56     1
-1    46      -2
-3   35       2
-5  14        -3
-7 13         2
-9 1          -1
-111           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a257.gif

K11a257

K11a259.gif

K11a259