K11n181

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K11n180.gif

K11n180

K11n182.gif

K11n182

Contents

K11n181.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n181 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,13,4,12 X5,17,6,16 X14,8,15,7 X9,21,10,20 X11,19,12,18 X13,3,14,2 X22,16,1,15 X17,5,18,4 X19,11,20,10 X21,9,22,8
Gauss code 1, 7, -2, 9, -3, -1, 4, 11, -5, 10, -6, 2, -7, -4, 8, 3, -9, 6, -10, 5, -11, -8
Dowker-Thistlethwaite code 6 -12 -16 14 -20 -18 -2 22 -4 -10 -8
A Braid Representative
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A Morse Link Presentation K11n181 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n181/ThurstonBennequinNumber
Hyperbolic Volume 11.345
A-Polynomial See Data:K11n181/A-polynomial

[edit Notes for K11n181's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n181's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t^2-11 t+13-11 t^{-1} +5 t^{-2}
Conway polynomial 5 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 45, 4 }
Jones polynomial -2 q^{11}+3 q^{10}-5 q^9+7 q^8-7 q^7+8 q^6-6 q^5+4 q^4-2 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +2 z^4 a^{-6} +2 z^4 a^{-8} +2 z^2 a^{-4} +4 z^2 a^{-6} +5 z^2 a^{-8} -2 z^2 a^{-10} + a^{-6} +3 a^{-8} -3 a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-9} +z^9 a^{-11} +3 z^8 a^{-8} +4 z^8 a^{-10} +z^8 a^{-12} +3 z^7 a^{-7} -3 z^7 a^{-11} +3 z^6 a^{-6} -10 z^6 a^{-8} -15 z^6 a^{-10} -2 z^6 a^{-12} +2 z^5 a^{-5} -6 z^5 a^{-7} -4 z^5 a^{-9} +7 z^5 a^{-11} +3 z^5 a^{-13} +z^4 a^{-4} -6 z^4 a^{-6} +18 z^4 a^{-8} +25 z^4 a^{-10} -3 z^3 a^{-5} +6 z^3 a^{-7} +7 z^3 a^{-9} -12 z^3 a^{-11} -10 z^3 a^{-13} -2 z^2 a^{-4} +5 z^2 a^{-6} -11 z^2 a^{-8} -19 z^2 a^{-10} -z^2 a^{-12} -2 z a^{-9} +5 z a^{-11} +7 z a^{-13} - a^{-6} +3 a^{-8} +3 a^{-10}
The A2 invariant Data:K11n181/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n181/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 1610 278 7488 \frac{40768}{3} \frac{7264}{3} 2064 7776 21632 57960 10008 \frac{1159773}{10} -\frac{1438}{15} \frac{756026}{15} \frac{5123}{6} \frac{70013}{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 K11n181. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
23         2-2
21        1 1
19       42 -2
17      31  2
15     44   0
13    43    1
11   24     2
9  24      -2
7  2       2
512        -1
31         1
Integral Khovanov Homology

(db, data source)

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

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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K11n180

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K11n182