K11a199

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K11a198.gif

K11a198

K11a200.gif

K11a200

Contents

K11a199.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a199 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X16,6,17,5 X14,8,15,7 X18,9,19,10 X20,11,21,12 X2,13,3,14 X6,16,7,15 X22,18,1,17 X10,19,11,20 X8,21,9,22
Gauss code 1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 12 16 14 18 20 2 6 22 10 8
A Braid Representative
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A Morse Link Presentation K11a199 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a199/ThurstonBennequinNumber
Hyperbolic Volume 13.9644
A-Polynomial See Data:K11a199/A-polynomial

[edit Notes for K11a199's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 2

[edit Notes for K11a199's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-11 t^2+23 t-27+23 t^{-1} -11 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 99, -2 }
Jones polynomial -q^4+3 q^3-5 q^2+10 q-13+15 q^{-1} -16 q^{-2} +14 q^{-3} -11 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) z^2 a^6+a^6-2 z^4 a^4-3 z^2 a^4+z^6 a^2+z^4 a^2-3 z^2 a^2-3 a^2+z^6+3 z^4+4 z^2+3-z^4 a^{-2} -2 z^2 a^{-2}
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +7 a^4 z^8+10 a^2 z^8+3 z^8 a^{-2} +6 z^8+8 a^5 z^7+a^3 z^7-17 a z^7-9 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-8 a^4 z^6-32 a^2 z^6-13 z^6 a^{-2} -31 z^6+3 a^7 z^5-12 a^5 z^5-14 a^3 z^5+11 a z^5+6 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-7 a^6 z^4+31 a^2 z^4+18 z^4 a^{-2} +41 z^4-2 a^7 z^3+10 a^5 z^3+9 a^3 z^3-8 a z^3-z^3 a^{-1} +4 z^3 a^{-3} -a^8 z^2+5 a^6 z^2+3 a^4 z^2-17 a^2 z^2-8 z^2 a^{-2} -22 z^2-3 a^5 z-a^3 z+4 a z+2 z a^{-1} -a^6+3 a^2+3
The A2 invariant Data:K11a199/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a199/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a102, K11a181,}

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

Vassiliev invariants

V2 and V3: (-3, 4)
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 32 72 50 30 -384 -\frac{1792}{3} -\frac{544}{3} -96 -288 512 -600 -360 \frac{8129}{10} \frac{826}{15} \frac{2258}{15} \frac{671}{6} -\frac{351}{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=-2 is the signature of K11a199. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
9           1-1
7          2 2
5         31 -2
3        72  5
1       63   -3
-1      97    2
-3     87     -1
-5    68      -2
-7   58       3
-9  26        -4
-11 15         4
-13 2          -2
-151           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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K11a198.gif

K11a198

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K11a200