K11a349

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K11a348.gif

K11a348

K11a350.gif

K11a350

Contents

K11a349.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X6271 X18,4,19,3 X16,5,17,6 X12,8,13,7 X4,10,5,9 X2,11,3,12 X22,14,1,13 X20,16,21,15 X10,18,11,17 X8,19,9,20 X14,22,15,21
Gauss code 1, -6, 2, -5, 3, -1, 4, -10, 5, -9, 6, -4, 7, -11, 8, -3, 9, -2, 10, -8, 11, -7
Dowker-Thistlethwaite code 6 18 16 12 4 2 22 20 10 8 14
A Braid Representative
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A Morse Link Presentation K11a349 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:K11a349/ThurstonBennequinNumber
Hyperbolic Volume 17.9314
A-Polynomial See Data:K11a349/A-polynomial

[edit Notes for K11a349'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 K11a349's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-14 t^2+37 t-49+37 t^{-1} -14 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-2 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 155, 2 }
Jones polynomial q^9-4 q^8+9 q^7-15 q^6+21 q^5-25 q^4+25 q^3-22 q^2+17 q-10+5 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} -2 z^4 a^{-6} -z^4+z^2 a^{-2} -2 z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} + a^{-2} -2 a^{-4} + a^{-6} +1
Kauffman polynomial (db, data sources) 4 z^{10} a^{-4} +4 z^{10} a^{-6} +10 z^9 a^{-3} +19 z^9 a^{-5} +9 z^9 a^{-7} +12 z^8 a^{-2} +9 z^8 a^{-4} +5 z^8 a^{-6} +8 z^8 a^{-8} +10 z^7 a^{-1} -12 z^7 a^{-3} -49 z^7 a^{-5} -23 z^7 a^{-7} +4 z^7 a^{-9} -15 z^6 a^{-2} -30 z^6 a^{-4} -32 z^6 a^{-6} -21 z^6 a^{-8} +z^6 a^{-10} +5 z^6+a z^5-13 z^5 a^{-1} +4 z^5 a^{-3} +49 z^5 a^{-5} +22 z^5 a^{-7} -9 z^5 a^{-9} +21 z^4 a^{-4} +35 z^4 a^{-6} +17 z^4 a^{-8} -2 z^4 a^{-10} -5 z^4+2 z^3 a^{-1} -4 z^3 a^{-3} -19 z^3 a^{-5} -10 z^3 a^{-7} +3 z^3 a^{-9} +2 z^2 a^{-2} -8 z^2 a^{-6} -6 z^2 a^{-8} +z a^{-3} +3 z a^{-5} +2 z a^{-7} - a^{-2} -2 a^{-4} - a^{-6} +1
The A2 invariant Data:K11a349/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a349/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: (-1, -2)
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 -16 8 -\frac{14}{3} \frac{62}{3} 64 \frac{416}{3} \frac{320}{3} 16 -\frac{32}{3} 128 \frac{56}{3} -\frac{248}{3} \frac{14849}{30} \frac{4702}{15} \frac{178}{45} -\frac{161}{18} -\frac{991}{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 K11a349. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          3 -3
15         61 5
13        93  -6
11       126   6
9      139    -4
7     1212     0
5    1013      3
3   712       -5
1  411        7
-1 16         -5
-3 4          4
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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K11a348.gif

K11a348

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K11a350