K11a350

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K11a349.gif

K11a349

K11a351.gif

K11a351

Contents

K11a350.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a350 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X4,10,5,9 X2,11,3,12 X20,13,21,14 X22,16,1,15 X10,18,11,17 X12,19,13,20 X8,21,9,22
Gauss code 1, -6, 2, -5, 3, -1, 4, -11, 5, -9, 6, -10, 7, -4, 8, -3, 9, -2, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 18 16 14 4 2 20 22 10 12 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a350 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a350/ThurstonBennequinNumber
Hyperbolic Volume 19.3243
A-Polynomial See Data:K11a350/A-polynomial

[edit Notes for K11a350's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a350's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-7 t^3+21 t^2-39 t+49-39 t^{-1} +21 t^{-2} -7 t^{-3} + t^{-4}
Conway polynomial z^8+z^6-z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 185, 0 }
Jones polynomial q^6-5 q^5+12 q^4-19 q^3+26 q^2-30 q+30-26 q^{-1} +19 q^{-2} -11 q^{-3} +5 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +4 z^6-2 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} +5 z^4-2 z^2 a^{-2} +z^2 a^{-4} -z^2+2 a^2+2 a^{-2} -3
Kauffman polynomial (db, data sources) 5 z^{10} a^{-2} +5 z^{10}+13 a z^9+25 z^9 a^{-1} +12 z^9 a^{-3} +15 a^2 z^8+13 z^8 a^{-2} +11 z^8 a^{-4} +17 z^8+11 a^3 z^7-13 a z^7-51 z^7 a^{-1} -22 z^7 a^{-3} +5 z^7 a^{-5} +5 a^4 z^6-20 a^2 z^6-49 z^6 a^{-2} -23 z^6 a^{-4} +z^6 a^{-6} -50 z^6+a^5 z^5-13 a^3 z^5-4 a z^5+27 z^5 a^{-1} +9 z^5 a^{-3} -8 z^5 a^{-5} -4 a^4 z^4+5 a^2 z^4+35 z^4 a^{-2} +13 z^4 a^{-4} -z^4 a^{-6} +30 z^4+3 a^3 z^3+7 a z^3+z^3 a^{-1} -z^3 a^{-3} +2 z^3 a^{-5} +3 a^2 z^2-3 z^2 a^{-2} -3 z^2 a^{-4} +3 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3
The A2 invariant -q^{14}+3 q^{12}-3 q^{10}+4 q^8+2 q^6-5 q^4+5 q^2-7+3 q^{-2} - q^{-4} - q^{-6} +6 q^{-8} -4 q^{-10} +3 q^{-12} -2 q^{-16} + q^{-18}
The G2 invariant Data:K11a350/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, 0)
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 0 32 \frac{164}{3} \frac{76}{3} 0 0 32 -32 -\frac{256}{3} 0 -\frac{1312}{3} -\frac{608}{3} -\frac{6271}{15} \frac{1964}{15} -\frac{19564}{45} \frac{319}{9} -\frac{1471}{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 K11a350. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          4 -4
9         81 7
7        114  -7
5       158   7
3      1511    -4
1     1515     0
-1    1216      4
-3   714       -7
-5  412        8
-7 17         -6
-9 4          4
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a349.gif

K11a349

K11a351.gif

K11a351