K11n180

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K11n179.gif

K11n179

K11n181.gif

K11n181

Contents

K11n180.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n180 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,13,4,12 X5,17,6,16 X14,8,15,7 X9,19,10,18 X11,21,12,20 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, 5, -10, 6, -11, -8
Dowker-Thistlethwaite code 6 -12 -16 14 -18 -20 -2 22 -4 -10 -8
A Braid Representative
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A Morse Link Presentation K11n180 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{3,4\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n180/ThurstonBennequinNumber
Hyperbolic Volume 12.6965
A-Polynomial See Data:K11n180/A-polynomial

[edit Notes for K11n180'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 K11n180's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-7 t^2+11 t-13+11 t^{-1} -7 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+11 z^4+10 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 55, 6 }
Jones polynomial -2 q^{12}+4 q^{11}-7 q^{10}+9 q^9-9 q^8+9 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} +4 z^4 a^{-6} +9 z^4 a^{-8} -2 z^4 a^{-10} +4 z^2 a^{-6} +12 z^2 a^{-8} -6 z^2 a^{-10} +5 a^{-8} -4 a^{-10}
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} +2 z^7 a^{-11} +2 z^7 a^{-13} +z^6 a^{-6} -11 z^6 a^{-8} -13 z^6 a^{-10} +z^6 a^{-14} -6 z^5 a^{-7} -10 z^5 a^{-9} -4 z^5 a^{-11} -4 z^4 a^{-6} +17 z^4 a^{-8} +16 z^4 a^{-10} -3 z^4 a^{-12} +2 z^4 a^{-14} +4 z^3 a^{-7} +7 z^3 a^{-9} +3 z^3 a^{-15} +4 z^2 a^{-6} -16 z^2 a^{-8} -15 z^2 a^{-10} +4 z^2 a^{-12} -z^2 a^{-14} -3 z a^{-9} +z a^{-11} +z a^{-13} -3 z a^{-15} +5 a^{-8} +4 a^{-10}
The A2 invariant Data:K11n180/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n180/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a339,}

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

Vassiliev invariants

V2 and V3: (10, 30)
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
40 240 800 \frac{5756}{3} \frac{916}{3} 9600 16864 3008 2288 \frac{32000}{3} 28800 \frac{230240}{3} \frac{36640}{3} \frac{452311}{3} 3516 \frac{539404}{9} \frac{8269}{9} \frac{23863}{3}

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 K11n180. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
25         2-2
23        2 2
21       52 -3
19      42  2
17     55   0
15    44    0
13   35     2
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}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11n179.gif

K11n179

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K11n181