K11n94

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K11n93.gif

K11n93

K11n95.gif

K11n95

Contents

K11n94.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,6,15,5 X12,7,13,8 X2,10,3,9 X20,11,21,12 X18,14,19,13 X15,22,16,1 X6,18,7,17 X8,19,9,20 X21,16,22,17
Gauss code 1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -4, 7, -3, -8, 11, 9, -7, 10, -6, -11, 8
Dowker-Thistlethwaite code 4 10 14 12 2 20 18 -22 6 8 -16
A Braid Representative
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A Morse Link Presentation K11n94 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:K11n94/ThurstonBennequinNumber
Hyperbolic Volume 14.0065
A-Polynomial See Data:K11n94/A-polynomial

[edit Notes for K11n94's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n94's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+6 t^2-13 t+17-13 t^{-1} +6 t^{-2} - t^{-3}
Conway polynomial -z^6+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 57, 0 }
Jones polynomial q^6-4 q^5+6 q^4-8 q^3+10 q^2-9 q+9-6 q^{-1} +3 q^{-2} - q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -3 z^4 a^{-2} +z^4 a^{-4} +2 z^4-a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} +4 z^2-a^2+ a^{-2} - a^{-4} +2
Kauffman polynomial (db, data sources) 2 z^9 a^{-1} +2 z^9 a^{-3} +8 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8+a z^7-3 z^7 a^{-1} +4 z^7 a^{-5} -27 z^6 a^{-2} -16 z^6 a^{-4} +z^6 a^{-6} -10 z^6-2 z^5 a^{-1} -14 z^5 a^{-3} -12 z^5 a^{-5} +3 a^2 z^4+29 z^4 a^{-2} +12 z^4 a^{-4} -2 z^4 a^{-6} +18 z^4+a^3 z^3+3 a z^3+7 z^3 a^{-1} +11 z^3 a^{-3} +6 z^3 a^{-5} -3 a^2 z^2-10 z^2 a^{-2} -3 z^2 a^{-4} -10 z^2-a^3 z-2 a z-2 z a^{-1} +z a^{-5} +a^2- a^{-2} - a^{-4} +2
The A2 invariant Data:K11n94/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n94/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, 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
8 8 32 \frac{172}{3} \frac{44}{3} 64 \frac{368}{3} \frac{128}{3} 8 \frac{256}{3} 32 \frac{1376}{3} \frac{352}{3} \frac{8791}{15} -\frac{4}{15} \frac{12484}{45} \frac{137}{9} \frac{871}{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=0 is the signature of K11n94. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-10123456χ
13         11
11        3 -3
9       31 2
7      53  -2
5     53   2
3    45    1
1   55     0
-1  25      3
-3 14       -3
-5 2        2
-71         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11n93.gif

K11n93

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K11n95