K11a120

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K11a119.gif

K11a119

K11a121.gif

K11a121

Contents

K11a120.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a120 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,5,15,6 X18,7,19,8 X12,10,13,9 X2,11,3,12 X22,13,1,14 X6,15,7,16 X20,17,21,18 X8,19,9,20 X16,21,17,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -5, 7, -3, 8, -11, 9, -4, 10, -9, 11, -7
Dowker-Thistlethwaite code 4 10 14 18 12 2 22 6 20 8 16
A Braid Representative
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A Morse Link Presentation K11a120 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a120/ThurstonBennequinNumber
Hyperbolic Volume 14.3126
A-Polynomial See Data:K11a120/A-polynomial

[edit Notes for K11a120's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 4

[edit Notes for K11a120's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+12 t^2-25 t+31-25 t^{-1} +12 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 109, -4 }
Jones polynomial 1-3 q^{-1} +7 q^{-2} -11 q^{-3} +15 q^{-4} -17 q^{-5} +18 q^{-6} -15 q^{-7} +11 q^{-8} -7 q^{-9} +3 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-a^{10}+2 z^4 a^8+3 z^2 a^8-z^6 a^6-z^4 a^6+2 z^2 a^6+2 a^6-z^6 a^4-2 z^4 a^4-z^2 a^4-a^4+z^4 a^2+2 z^2 a^2+a^2
Kauffman polynomial (db, data sources) z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+5 z^7 a^{11}-8 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+5 z^8 a^{10}-5 z^6 a^{10}+2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+3 z^9 a^9+4 z^7 a^9-13 z^5 a^9+10 z^3 a^9-2 z a^9+z^{10} a^8+8 z^8 a^8-14 z^6 a^8+6 z^4 a^8+z^2 a^8+6 z^9 a^7-4 z^7 a^7-7 z^5 a^7+5 z^3 a^7+z^{10} a^6+7 z^8 a^6-14 z^6 a^6+z^4 a^6+6 z^2 a^6-2 a^6+3 z^9 a^5-11 z^5 a^5+8 z^3 a^5-2 z a^5+4 z^8 a^4-7 z^6 a^4-z^4 a^4+4 z^2 a^4-a^4+3 z^7 a^3-8 z^5 a^3+6 z^3 a^3-z a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2
The A2 invariant Data:K11a120/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a120/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a295,}

Vassiliev invariants

V2 and V3: (5, -13)
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
20 -104 200 \frac{1846}{3} \frac{290}{3} -2080 -\frac{12272}{3} -\frac{2048}{3} -616 \frac{4000}{3} 5408 \frac{36920}{3} \frac{5800}{3} \frac{165583}{6} \frac{26}{3} \frac{103910}{9} \frac{5941}{18} \frac{9199}{6}

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 K11a120. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          2 -2
-3         51 4
-5        73  -4
-7       84   4
-9      97    -2
-11     98     1
-13    69      3
-15   59       -4
-17  26        4
-19 15         -4
-21 2          2
-231           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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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K11a119.gif

K11a119

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K11a121