K11a197

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K11a196.gif

K11a196

K11a198.gif

K11a198

Contents

K11a197.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a197 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X16,5,17,6 X14,8,15,7 X18,10,19,9 X2,12,3,11 X8,14,9,13 X22,15,1,16 X20,18,21,17 X10,20,11,19 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -7, 5, -10, 6, -2, 7, -4, 8, -3, 9, -5, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 12 16 14 18 2 8 22 20 10 6
A Braid Representative
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A Morse Link Presentation K11a197 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a197/ThurstonBennequinNumber
Hyperbolic Volume 17.4106
A-Polynomial See Data:K11a197/A-polynomial

[edit Notes for K11a197'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 K11a197's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-14 t^2+33 t-43+33 t^{-1} -14 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+4 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 143, 2 }
Jones polynomial q^9-5 q^8+10 q^7-16 q^6+21 q^5-23 q^4+23 q^3-19 q^2+14 q-7+3 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^6 a^{-4} +2 z^4 a^{-2} +6 z^4 a^{-4} -3 z^4 a^{-6} -z^4+3 z^2 a^{-2} +7 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} -2 z^2+2 a^{-2} +2 a^{-4} -2 a^{-6} -1
Kauffman polynomial (db, data sources) 2 z^{10} a^{-4} +2 z^{10} a^{-6} +6 z^9 a^{-3} +13 z^9 a^{-5} +7 z^9 a^{-7} +7 z^8 a^{-2} +15 z^8 a^{-4} +17 z^8 a^{-6} +9 z^8 a^{-8} +5 z^7 a^{-1} -3 z^7 a^{-3} -18 z^7 a^{-5} -5 z^7 a^{-7} +5 z^7 a^{-9} -9 z^6 a^{-2} -42 z^6 a^{-4} -51 z^6 a^{-6} -20 z^6 a^{-8} +z^6 a^{-10} +3 z^6+a z^5-5 z^5 a^{-1} -5 z^5 a^{-3} -6 z^5 a^{-5} -17 z^5 a^{-7} -10 z^5 a^{-9} +6 z^4 a^{-2} +41 z^4 a^{-4} +42 z^4 a^{-6} +11 z^4 a^{-8} -z^4 a^{-10} -5 z^4-2 a z^3+8 z^3 a^{-3} +16 z^3 a^{-5} +14 z^3 a^{-7} +4 z^3 a^{-9} +z^2 a^{-2} -15 z^2 a^{-4} -15 z^2 a^{-6} -2 z^2 a^{-8} +3 z^2+a z+z a^{-1} -2 z a^{-3} -6 z a^{-5} -3 z a^{-7} +z a^{-9} -2 a^{-2} +2 a^{-4} +2 a^{-6} -1
The A2 invariant Data:K11a197/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a197/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: (4, 6)
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
16 48 128 \frac{728}{3} \frac{88}{3} 768 1248 224 144 \frac{2048}{3} 1152 \frac{11648}{3} \frac{1408}{3} \frac{101102}{15} \frac{5432}{15} \frac{105608}{45} \frac{274}{9} \frac{4142}{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=2 is the signature of K11a197. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          4 -4
15         61 5
13        104  -6
11       116   5
9      1210    -2
7     1111     0
5    812      4
3   611       -5
1  29        7
-1 15         -4
-3 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a196.gif

K11a196

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K11a198