K11a191

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K11a190.gif

K11a190

K11a192.gif

K11a192

Contents

K11a191.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a191 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X14,6,15,5 X18,8,19,7 X20,10,21,9 X2,12,3,11 X10,14,11,13 X22,16,1,15 X6,18,7,17 X8,20,9,19 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -9, 4, -10, 5, -7, 6, -2, 7, -3, 8, -11, 9, -4, 10, -5, 11, -8
Dowker-Thistlethwaite code 4 12 14 18 20 2 10 22 6 8 16
A Braid Representative
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A Morse Link Presentation K11a191 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a191/ThurstonBennequinNumber
Hyperbolic Volume 12.3995
A-Polynomial See Data:K11a191/A-polynomial

[edit Notes for K11a191'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 K11a191's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^3-11 t^2+17 t-19+17 t^{-1} -11 t^{-2} +4 t^{-3}
Conway polynomial 4 z^6+13 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 83, 6 }
Jones polynomial -q^{14}+3 q^{13}-6 q^{12}+9 q^{11}-12 q^{10}+13 q^9-13 q^8+11 q^7-7 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} +z^6 a^{-10} +4 z^4 a^{-6} +8 z^4 a^{-8} +2 z^4 a^{-10} -z^4 a^{-12} +4 z^2 a^{-6} +9 z^2 a^{-8} -2 z^2 a^{-10} -2 z^2 a^{-12} + a^{-6} +3 a^{-8} -3 a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +5 z^9 a^{-11} +3 z^9 a^{-13} +3 z^8 a^{-8} +2 z^8 a^{-10} +3 z^8 a^{-12} +4 z^8 a^{-14} +2 z^7 a^{-7} -2 z^7 a^{-9} -13 z^7 a^{-11} -5 z^7 a^{-13} +4 z^7 a^{-15} +z^6 a^{-6} -10 z^6 a^{-8} -12 z^6 a^{-10} -10 z^6 a^{-12} -6 z^6 a^{-14} +3 z^6 a^{-16} -6 z^5 a^{-7} -7 z^5 a^{-9} +13 z^5 a^{-11} +7 z^5 a^{-13} -6 z^5 a^{-15} +z^5 a^{-17} -4 z^4 a^{-6} +12 z^4 a^{-8} +19 z^4 a^{-10} +12 z^4 a^{-12} +3 z^4 a^{-14} -6 z^4 a^{-16} +3 z^3 a^{-7} +11 z^3 a^{-9} +z^3 a^{-11} -5 z^3 a^{-13} -2 z^3 a^{-17} +4 z^2 a^{-6} -10 z^2 a^{-8} -12 z^2 a^{-10} -2 z^2 a^{-12} -2 z^2 a^{-14} +2 z^2 a^{-16} -5 z a^{-9} -3 z a^{-11} +z a^{-13} +z a^{-17} - a^{-6} +3 a^{-8} +3 a^{-10}
The A2 invariant Data:K11a191/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a191/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: (9, 26)
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
36 208 648 1546 214 7488 \frac{38368}{3} \frac{6592}{3} 1488 7776 21632 55656 7704 \frac{1086173}{10} \frac{81682}{15} \frac{565786}{15} \frac{3811}{6} \frac{46493}{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=6 is the signature of K11a191. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          2 2
25         41 -3
23        52  3
21       74   -3
19      65    1
17     77     0
15    46      -2
13   37       4
11  24        -2
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}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=11 {\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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K11a190

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K11a192