K11n9

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K11n8.gif

K11n8

K11n10.gif

K11n10

Contents

K11n9.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n9 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,19,12,18 X6,14,7,13 X15,21,16,20 X17,1,18,22 X19,13,20,12 X21,17,22,16
Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 10, 7, -4, -8, 11, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 8 10 14 2 -18 6 -20 -22 -12 -16
A Braid Representative
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A Morse Link Presentation K11n9 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 3
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n9/ThurstonBennequinNumber
Hyperbolic Volume 10.3175
A-Polynomial See Data:K11n9/A-polynomial

[edit Notes for K11n9's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -6

[edit Notes for K11n9's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+3 t^3-t^2-4 t+7-4 t^{-1} - t^{-2} +3 t^{-3} - t^{-4}
Conway polynomial -z^8-5 z^6-3 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 5, 4 }
Jones polynomial -q^{11}+2 q^{10}-2 q^9+2 q^8-2 q^7+q^6-q^4+2 q^3-q^2+q
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -16 z^4 a^{-6} +7 z^4 a^{-8} +10 z^2 a^{-4} -17 z^2 a^{-6} +12 z^2 a^{-8} -2 z^2 a^{-10} +5 a^{-4} -8 a^{-6} +6 a^{-8} -2 a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +3 z^8 a^{-6} +2 z^8 a^{-8} -6 z^7 a^{-5} -6 z^7 a^{-7} +z^7 a^{-9} +z^7 a^{-11} -7 z^6 a^{-4} -21 z^6 a^{-6} -15 z^6 a^{-8} +z^6 a^{-10} +2 z^6 a^{-12} +9 z^5 a^{-5} +6 z^5 a^{-7} -7 z^5 a^{-9} -3 z^5 a^{-11} +z^5 a^{-13} +16 z^4 a^{-4} +43 z^4 a^{-6} +31 z^4 a^{-8} -3 z^4 a^{-10} -7 z^4 a^{-12} -2 z^3 a^{-5} +5 z^3 a^{-7} +11 z^3 a^{-9} +z^3 a^{-11} -3 z^3 a^{-13} -15 z^2 a^{-4} -32 z^2 a^{-6} -22 z^2 a^{-8} -z^2 a^{-10} +4 z^2 a^{-12} -2 z a^{-5} -4 z a^{-7} -4 z a^{-9} -z a^{-11} +z a^{-13} +5 a^{-4} +8 a^{-6} +6 a^{-8} +2 a^{-10}
The A2 invariant  q^{-4} + q^{-6} + q^{-8} +2 q^{-10} + q^{-12} - q^{-14} - q^{-16} - q^{-18} -2 q^{-20} + q^{-22} +2 q^{-26} + q^{-32} -2 q^{-34}
The G2 invariant Data:K11n9/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: (3, 7)
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
12 56 72 318 58 672 \frac{5264}{3} \frac{992}{3} 248 288 1568 3816 696 \frac{97551}{10} \frac{5054}{15} \frac{57542}{15} \frac{593}{6} \frac{4751}{10}

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=4 is the signature of K11n9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          1 1
19         11 0
17       121  0
15      121   0
13     122    -1
11    122     1
9   111      -1
7  111       1
5 12         1
3            0
11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5 i=7
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2} {\mathbb Z}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\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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K11n8.gif

K11n8

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K11n10