K11n93

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K11n92.gif

K11n92

K11n94.gif

K11n94

Contents

K11n93.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n93 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,6,15,5 X7,13,8,12 X2,10,3,9 X11,21,12,20 X18,14,19,13 X22,16,1,15 X6,18,7,17 X19,9,20,8 X16,22,17,21
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 K11n93 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n93/ThurstonBennequinNumber
Hyperbolic Volume 12.5839
A-Polynomial See Data:K11n93/A-polynomial

[edit Notes for K11n93's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11n93's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-7 t^2+9 t-9+9 t^{-1} -7 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+11 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 47, 6 }
Jones polynomial -q^{12}+3 q^{11}-6 q^{10}+7 q^9-8 q^8+8 q^7-6 q^6+5 q^5-2 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +2 z^6 a^{-8} +4 z^4 a^{-6} +9 z^4 a^{-8} -2 z^4 a^{-10} +4 z^2 a^{-6} +11 z^2 a^{-8} -7 z^2 a^{-10} + a^{-6} +4 a^{-8} -5 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) z^9 a^{-9} +z^9 a^{-11} +3 z^8 a^{-8} +5 z^8 a^{-10} +2 z^8 a^{-12} +2 z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^7 a^{-13} +z^6 a^{-6} -11 z^6 a^{-8} -17 z^6 a^{-10} -5 z^6 a^{-12} -6 z^5 a^{-7} -15 z^5 a^{-9} -9 z^5 a^{-11} -4 z^4 a^{-6} +14 z^4 a^{-8} +23 z^4 a^{-10} +8 z^4 a^{-12} +3 z^4 a^{-14} +3 z^3 a^{-7} +17 z^3 a^{-9} +15 z^3 a^{-11} +2 z^3 a^{-13} +z^3 a^{-15} +4 z^2 a^{-6} -12 z^2 a^{-8} -17 z^2 a^{-10} -3 z^2 a^{-12} -2 z^2 a^{-14} -7 z a^{-9} -7 z a^{-11} -z a^{-13} -z a^{-15} - a^{-6} +4 a^{-8} +5 a^{-10} + a^{-12}
The A2 invariant Data:K11n93/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n93/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a242,}

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

Vassiliev invariants

V2 and V3: (8, 21)
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
32 168 512 \frac{3472}{3} \frac{488}{3} 5376 8880 1536 1032 \frac{16384}{3} 14112 \frac{111104}{3} \frac{15616}{3} \frac{1049884}{15} \frac{53624}{15} \frac{1092376}{45} \frac{3476}{9} \frac{45244}{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=6 is the signature of K11n93. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
25         1-1
23        2 2
21       41 -3
19      32  1
17     54   -1
15    33    0
13   35     2
11  23      -1
9  3       3
712        -1
51         1
Integral Khovanov Homology

(db, data source)

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